Topology Seminar
Past Talks

Achim Krause (Institute for Advanced Study)
$\begingroup $In recent and ongoing work with Antieau and Nikolaus, we compute algebraic Ktheory of $Z/p^n$ and related rings. In this talk I want to give an overview of our approach based on prismatic cohomology.
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Andrew Senger (Harvard University)
$\begingroup $In this talk, I will explain how to compute the mod $(p, v_1^{p^n})$ algebraic Ktheory of $\mathbb{Z}/p^{n+2}$ for $n \geq 0$. A key role is played by a new crystallinity result for syntomic cohomology: the mod $(p, v_1^{p^n})$ syntomic cohomology of $R$ only depends on $R/p^{n+2}$. Our proof depends crucially on Drinfeld and Bhatt–Lurie's stacky approach to syntomic cohomology. This is joint work with Jeremy Hahn and Ishan Levy.
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Stephen McKean (Harvard University)
$\begingroup $There is a quadratic formvalued version of the compactly supported Euler characteristic coming from motivic homotopy. A feature of this Euler characteristic is that it descends to a ring homomorphism out of the Grothendieck ring of varieties. In characteristic 0, this Euler characteristic was constructed by Rondigs and later ArcilaMay–Bethea–Opie–Wickelgren–Zakharevich, who used Bittner's blow up presentation of K_0(Var). In characteristic not 2, Azouri gave a characterization in terms of the six functor formalism. I will discuss a hybrid approach using a sort of universal property of K_0(Var). I will then discuss power structures on K_0(Var) and the Grothendieck–Witt ring of quadratic forms, and conclude with a conjecture relating these two power structures. This is joint work in progress with Dori Bejleri.
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Gabriel AngeliniKnoll (Sorbonne Paris Nord)
$\begingroup $Work of HahnRaksitWilson extended the BhattMorrowScholze filtration on topological cyclic homology and topological periodic cyclic homology to sufficiently nice ring spectra. This allows one to define syntomic cohomology and prismatic cohomology at this level of generality. One example of such a nice ring spectrum is real topological Ktheory. In joint work with Christian Ausoni and John Rognes, we compute the syntomic cohomology of real topological Ktheory modulo $(2,\eta,v_1)$. This computation produces a new example of pure redshift and arithmetic duality. As an application, we compute the algebraic Ktheory of real topological Ktheory modulo $(2,\eta,v_1)$ and show that it satisfies a higher chromatic complexity version of the LichtenbaumQuillen conjecture.
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Aaron Landesman (Harvard/MIT)
$\begingroup $We describe a new homological stability result for a generalized version of Hurwitz spaces. This builds on previous work of EllenbergVenkateshWesterland, showing that homology groups of certain Hurwitz spaces stabilize. We generalize this in two directions. First, we work with covers of arbitrary punctured Riemann surfaces instead of just the disc. Second, we generalize the result to 'coefficient systems,' which are essentially a sequence of compatible local systems on configurations spaces. After detailing the above homological stability result, we will then explain how both these generalizations are employed to prove versions of numerous conjectures from number theory relating to the distributions of ranks of elliptic curves and Selmer groups of elliptic curves.
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Elden Elmanto (University of Toronto/IAS)
$\begingroup $I'll construct the motivic filtration on K theory mod tau. Over C, the resulting E2 page contains unramified cohomology as studied by ColliotThélène and others. By reverse engineering the spectral sequence we prove some new theorems in birational geometry: the integral Hodge conjecture for several new Fano 4folds and derived invariance of brauer groups of surfaces. I hope to advertise the possible use of Ctaumethods in birational geometry.
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Eva Belmont (Case Western Reserve University)
$\begingroup $Abstract: The modified Adams spectral sequence, which computes the homotopy groups of a Borelequivariant spectrum, is a combination of the homotopy fixed points spectral sequence and the nonequivariant Adams spectral sequence. One can also use it to read off (completed) $\mathbb{R}$motivic homotopy groups, via a synthetic spectra construction. We will explain how to compute it using examples coming from $ko_{C_2}$ and $kq$. This is joint work with Gabriel AngeliniKnoll, Mark Behrens, and Hana Kong.
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Alexander Efimov (Max Planck Institute)
$\begingroup $I will explain some of my recent results on the category of localizing motives – the target of the universal localizing invariant commuting with filtered colimits. The main surprising result about this category is that it is rigid as a symmetric monoidal category (in the sense of Gaitsgory and Rozenblyum).
As an application of the proof of rigidity, we will deduce that the functors TR (topological restriction) and TC (topological cyclic homology) are corepresentable in this category, if we restrict to connective E_1rings.
If time permits, I will explain how rigidity of Mot^loc allows to construct refined versions of (topological) Hochschild homology and its variants, which contain much more information about the E_1algebra than the usual variants of (T)HH.
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Akhil Mathew (University of Chicago)
$\begingroup $This talk will be a largely expository introduction to the theory of 'prismatic Fgauges,' as developed by BhattLurie and Drinfeld, intended for algebraic topologists. In particular, I will discuss the close relationship between Fgauges and pdivisible groups, due to Anschutz and Le Bras ('prismatic Dieudonne theory') and explain some extensions and complements based on the ArtinLurie representability theorem.
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Kathryn Lesh (Union College)
$\begingroup $I will talk about a 'normalizer decomposition' for the classifying space of a plocal compact group. The decomposition generalizes those of Dwyer for finite groups and of Libman for plocal finite groups and (separately) for compact Lie groups. I'll show how the decomposition gives a homotopy pushout square for the exotic pcompact groups of Aguade and Zabrodsky by building on the example of SU(p). This is joint work with Belmont, Castellana, Grbic, and Strumila.
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Craig Westerland (University of Minnesota)
$\begingroup $The seminar will meet at 3:00 PM in Harvard Science Center SC507 at Arithmetic Statistics seminar.
In 2005, Conrey, Farmer, Keating, Rubinstein, and Snaith posed a conjecture on the asymptotics of moments of quadratic Lfunctions. While this conjecture originates as a question about number fields, it has a more geometric version when posed over function fields in positive characteristic. I’ll talk about how one can reinterpret the central object in this conjecture in terms of the action of the Galois group of a finite field on the cohomology of braid groups with certain coefficients coming from the braid group’s interpretation as the hyperelliptic mapping class group. We will see the “arithmetic factor” in this conjecture appear in the part of this cohomology that is accessible through tools of homological stability. This is joint work with Jonas Bergström, Adrian Diaconu, and Dan Petersen.
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Jared Weinstein (Boston University)
$\begingroup $We compute the rational homotopy groups of the K(n)local sphere for all heights n and all primes p, verifying a prediction that goes back to Morava in the early 1970s. The key ingredients are (1) the DevinatzHopkins spectral sequence, (2) the isomorphism between the Lubin–Tate tower and the Drinfeld tower at the level of perfectoid spaces, (3) integral $p$adic Hodge theory, and (4) an integral refinement of a theorem of Tate on the Galois cohomology of nonarchimedean fields. This is joint work with Tobias Barthel, Tomer Schlank, and Nathaniel Stapleton.
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Tomer Schlank (Hebrew University)
$\begingroup $We shall consider the functor $L_{T(n)}K$ of chromatically localized algebraic Ktheory. We shall discuss its interaction with pifinite colimits. This will lead to a possible alternative characterization of this functor as well as results about it's interaction with cyclotomic hyperdescent. This is a key input to the proof of the telescope conjecture. This talk is based on joint works with Shay BenMoshe, Shachar Carmeli, and Lior Yanovski, as well as with Robert Burklund, Jeremy Hahn, and Ishan Levy.
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Piotr Pstrągowski (Harvard University)
$\begingroup $The even filtration, introduced by HahnRaksitWilson, is a canonical filtration attached to a commutative ring spectrum which measures its failure to be even. Despite its simple definition, the even filtration recovers many arithmetically important constructions, such as the AdamsNovikov filtration of the sphere or the BhattMorrowScholze filtration on topological Hochschild homology, showing that they are all invariants of the commutative ring spectrum alone. I will describe a linear variant of the even filtration which is naturally defined on associative rings and can be effectively calculated through resolutions of modules, as well as joint work with Raksit on the resulting extension of prismatic cohomology to the context of $E_2$rings.
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Thomas Brazelton (Harvard University)
$\begingroup $Classical enumerative geometry asks geometric questions of the form 'how many?' and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are welldefined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of 'equivariant enumerative geometry', which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the orbits of solutions to an equivariant enumerative problem are conserved. We leverage this to compute the S4 orbits of the 27 lines on any smooth symmetric cubic surface.
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Dan BerwickEvans (University of Illinois UrbanaChampaign)
$\begingroup $Since the mid 1980s, there have been hints of a deep connection between 2dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) in which cocycles are 2dimensional supersymmetric field theories. Basic properties of these field theories lead to expected integrality and modularity properties, but the abundant torsion in TMF has always been mysterious. In this talk, I will describe deformation invariants of 2dimensional field theories that realize some of the torsion in TMF.
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Hana Jia Kong (Harvard University)
$\begingroup $The real motivic stable homotopy category has a close connection to the $C_2$equivariant stable homotopy category. From a computational perspective, the real motivic computation can be viewed as a simpler version which "removes the negative cone" in the $C_2$equivariant stable homotopy groups. On the other hand, by work of Burklund–Hahn–Senger, one can build the completed Artin–Tate real motivic category from the completed $C_2$equivariant category using the deformation construction associated to the $C_2$effective filtration.
In work with Gabriel AngeliniKnoll, Mark Behrens, and Eva Belmont, we try to build an analog of this deformation story for a general finite group $G$. We give a new interpretation of the $C_2$effective filtration in the Borel equivariant category which generalizes for $G$. Using this new interpretation, the deformation construction gives a deformation of the Borel equivariant stable homotopy category for general finite groups.
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Ben Antieau (Northwestern University)
$\begingroup $Generalizing and building on the work of Kriz, Ekedahl, Goerss, Lurie, Mandell, Mathew, Mondal, Quillen, Sullivan, Toën and Yuan, I will describe an integral cochain model for nilpotent spacees of finite type. A binomial ring is a lambdaring in which all Adams operations act as the identity. A derived binomial ring is a derived Λring equipped with simultaneous trivializations of the commuting Adams operations. For example, if X is a space, then ZX, the integral cochains on X, is naturally a derived binomial ring. The induced contravariant functor from spaces to derived binomial rings is fully faithful when restricted to nilpotent spaces of finite type. This is related, closely, to recent work of Horel and of Kubrak—Shuklin—Zakharov.
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Robert Burklund (University of Copenhagen)
$\begingroup $There is a natural dichotomy between telescopic (T(n)local) and chromatic (K(n)local) homotopy theory. Telescopic homotopy theory is more closely tied to the stable homotopy groups of spheres and through them to geometric questions, but is generally computationally intractable. Chromatic homotopy theory is more closely tied to arithmetic geometry and powerful computational tools exist in this setting. Ravenel’s telescope conjecture asserted that these two sides coincide. I will present a family of counterexamples to this conjecture based on using trace methods to analyze the algebraic Ktheory of a family of K(n)local ring spectra beginning with the K(1)local sphere. As a consequence of this we obtain a new lower bound on the average rank of the stable homotopy groups of spheres. Time permitting, I will then describe the galois group of the T(n)local sphere and how this informs our understanding of telescopic homotopy theory. This talk is based on projects joint with Carmeli, Clausen, Hahn, Levy, Schlank and Yanovski.
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Adela Zhang (MIT)
$\begingroup \def\F{\mathbb{F}} $The bar spectral sequence for algebras over a spectral operad relates Koszul duality phenomena in several contexts. In this talk, we apply this classical tool to the Koszul dual pair given by the (nonunital) $E_\infty$ operad and the spectral Lie operad over $\F_p$. The bar s.s. for $E_\infty$ algebras yields the structure of operations on mod p TAQ cohomology and spectral partition Lie algebras, building on the work of BrantnerMathew. In the colimit, the unary operations are Koszul dual to the DyerLashof algebra. On the other hand, the bar construction against certain spectral Lie algebras models labeled configuration spaces by a theorem of Knudsen. The associated bar s.s. yields new results on their mod p homology at low weights, as well as interesting patterns of universal differentials. Time permitting, I will explain an attempt with Andy Senger on detecting these differentials via deformation of comonads. Beverages relating to the title may be served after the talk.
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Allen Yuan (Columbia)
$\begingroup $Just as an ordinary commutative ring has a group of units, one can associate to any commutative ring spectrum $R$ a topological abelian group of "strict units". These strict units are a basic invariant of $R$, for instance controlling the types of roots one can adjoint to $R$. Despite this, there have not been many calculations of strict unit groups. I will survey some results in the area and report on work in progress, joint with Shachar Carmeli and Thomas Nikolaus, on calculating strict units of group rings.
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Nikolai Konovalov (University of Notre Dame)
$\begingroup $The spectral Lie operad is the Koszul dual operad to the cocommutative cooperad in the category of spectra. A spectral Lie algebra is an algebra over the spectral Lie operad. M. Behrens and J. Kjaer constructed socalled DyerLashofLie power operations acting on the modp homology groups of a spectral Lie algebra. However, they computed relations between these operations only for $p=2$. In my talk, I will explain how to compute desired relations for each prime by using functor calculus in the category of simplicial restricted Lie algebras. The latter category might be thought as an algebraic approximation of the category of spaces, and so, algebraic calculations may be also helpful in understanding of the topological Goodwillie spectral sequence.
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Peter Haine (Institute for Advanced Study)
$\begingroup $This talk is a report on joint work in progress with Piotr Pstrągowski. Pstrągowski defined a left adjoint $SH(ℂ)^{cell}$ → $Syn^{ev}$ from cellular ℂmotivic spectra to even (MUbased) synthetic spectra. This functor refines the Betti realization of a cellular motivic spectrum, and for any prime p, restricts to an equivalence on pcomplete objects. We’ll explain how to further refine the Betti realization functor $SH(ℂ)$ → $Sp$ to a left adjoint $SH(ℂ)$ → $Syn$ to all synthetic spectra. To do this, we’ll give a description of motivic spectra as sheaves on a subcategory of compact pure motives. This description also lets us show that for a complexorientable connective ring spectrum $A$, the $A$linear Betti realization $SH(ℂ)$ → $Mod_A$ refines to a left adjoint functor landing in filtered $A$modules. We’re also able to give a new construction of the Gillet–Soulé weight filtration on the compactly supported integral Betti cohomology of a complex variety.
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Ningchuan Zhang (University of Pennsylvania)
$\begingroup $Algebraic Kgroups of a number field are closely related to its Dedekind zeta function. This is illustrated by Borel’s computation of their ranks and the QuillenLichtenbaum Conjecture proved by VoevodskyRost.
In this talk, I will report my work in progress with Elden Elmanto to generalize this connection to Dirichlet Lfunctions. The key idea is to consider equivariant algebraic Ktheory of number fields with coefficients in Galois representations. Rationally, we obtain a Borel’s rank theorem for Artin Lfunctions. For torsion subgroups of equivariant algebraic Kgroups, we prove a QuillenLichtenbaum Conjecture for Dirichlet Lfunctions when the image of the Dirichlet character is cyclic of prime power. We will also discuss computations of $RO(C_2)$graded K(1)local algebraic Kgroups of the Gaussian integers if time permitted.
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Inbar Klang (Columbia)
$\begingroup $The seminar will meet at 3:00 PM in 2131.
In this talk, I will give an introduction to factorization homology and equivariant factorization homology. I will then discuss joint work with Asaf Horev and Foling Zou, with an appendix by Jeremy Hahn and Dylan Wilson, in which we prove a "nonabelian Poincaré duality" theorem for equivariant factorization homology, and study the equivariant factorization homology of equivariant Thom spectra. In particular, this provides an avenue for computing certain equivariant analogues of topological Hochschild homology.
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Shay Ben Moshe (Hebrew University)
$\begingroup $The redshift conjecture of AusoniRognes says that there is a strong interaction between algebraic Ktheory and the chromatic filtration on spectra. Namely, that if a ring spectrum R is of chromatic height n, then K(R) is of chromatic height n+1. HopkinsLurie, followed by CarmeliSchlankYanovski, showed that the category of spectra of height n is higher semiadditive, that is, colimits and limits indexed by pifinite spaces are canonically equivalent. In this talk we will describe higher semiadditive Ktheory, a variant of algebraic Ktheory that takes higher semiadditivity into account. We will explain how semiadditive methods allow us to show that it satisfies a form of the redshift conjecture. We will also explain some of its connection to chromatically localized Ktheory. Relevant background on algebraic Ktheory, chromatic homotopy and semiadditivity will be explained. This work is joint with Tomer Schlank.
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Andrew Senger (Harvard University)
$\begingroup $Let $E_n$ denote a height $n$ LubinTate theory, and let $G$ denote a finite subgroup of its Morava stabilizer group. In this talk, I will describe a new approach to the computation of the homotopy fixed points spectral sequence of $E_n^{hG}$, based on equivariant power operations.
In particular, I will show how one may compute the homotopy of $E_n^{hC_2}$ completely from scratch—without the use of Real bordism $MU_R$ or any other external input. I will conclude with some conjectures about the oddprimary case.
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Gijs Heuts (Utrecht University)
$\begingroup $I'll discuss some questions arising from Bousfield's work on localizations of unstable homotopy theory, particularly on the relation between $v_n$periodic homotopy groups and $T(n)$homology. I will indicate how the structure theory of spectral Hopf algebras plays a role in understanding the answers, although much remains to be done. Part of this is based on ongoing joint work with Brantner, Hahn, and Yuan.
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William Balderrama (University of Virginia)
$\begingroup $I will discuss some equivariant analogues of classical results about "chromatic" Bousfield localizations. Highlights will include identifying the groups G for which Gequivariant Ktheory localization satisfies an analogue of the smash product theorem, and a chromatic convergence theorem for finite $(C_p)^n$equivariant spectra. Time permitting, I will also discuss how analogues of James periodicity can be interpreted in terms of equivariant periodic selfmaps.
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Elden Elmanto (Harvard University)
$\begingroup $I want to explain what I've learned about motivic cohomology by being around two rivers the past four years: the Seine and the Charles. Topics include some resolution of Voevodsky's conjectures on slices (joint with Bachmann and Bachmann and Morrow), the construction of motivic cohomology beyond the smooth case (with Morrow), various exotic Grothendieck topologies (with Hoyois, Iwasa and Kelly), some applications to algebraic cycles (in progress each with Burklund/Raksit/Addington/Schlank).
The audience will be free to direct the talk as appropriate.
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Lukas Brantner (University of Oxford and Université ParisSaclay (CNRS))
$\begingroup $A smooth projective variety $Z$ is said to be CalabiYau if its canonical bundle is trivial. I will discuss recent joint work with Taelman, in which we use derived algebraic geometry to study how CalabiYau varieties in characteristic $p$ deform. More precisely, we show that if $Z$ has degenerating Hodgede Rham spectral sequence and torsionfree crystalline cohomology, then its mixed characteristic deformations are unobstructed; this is an analogue of the classical BTT theorem in characteristic zero. If $Z$ is ordinary, we show that it moreover admits a canonical lift to characteristic zero; this extends classical SerreTate theory. Our work generalises results of AchingerZdanowicz, BogomolovTianTodorov, DeligneNygaard, EkedahlShepherdBarron, Schröer, SerreTate, and Ward.
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Alexander Kupers (University of Toronto Scarborough)
$\begingroup $The little ddiscs operad is one of the foundational objects of algebraic topology. Through embedding calculus, its automorphism space features in an important connection between geometric topology and homotopy theory. I will discuss what we know and don't know about these automorphism spaces, and explain some intriguing results about them obtained through the aforementioned connection. This is joint work with Manuel Krannich and Geoffroy Horel.
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Stephen McKean (Harvard University)
$\begingroup $Many beautiful theorems in differential topology arise from the interplay of homotopy theory and smooth structures on manifolds. Similarly, motivic homotopy theory and algebraic structures on varieties combine to yield differentialtopological tools in algebraic geometry. I will survey various results in motivic homotopy on oriented intersections, fixed point theorems, framed cobordism, Morse theory, and the PoincaréHopf theorem. Time permitting, I will discuss how these tools relate to enumerative geometry over nonclosed fields.
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Eoin Mackall (University of Maryland, College Park)
$\begingroup $A naive $\mathbb{A}^1$homotopy between morphisms $f,g$ from a variety $X$ to a variety $Y$ is a cycle on $(X \times \mathbb{A}^1)\times Y$ whose support is finite and surjective over $X\times \mathbb{A}^1$ and whose fibers over 0 and 1 are the graphs of $f$ and $g$ respectively. Using this notion of naive $\mathbb{A}^1$homotopy, one can define naive $\mathbb{A}^1$homotopy equivalences of varieties. In this talk, we'll discuss how an analog of a theorem of Whitehead can be used to show that there are no nontrivial $\mathbb{A}^1$homotopy equivalences between smooth projective varieties.
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Manuel Rivera (Purdue University)
$\begingroup $Motivated by constructing algebraic models for homotopy types, I will discuss three different homotopy theories on the category of simplicial cocommutative coalgebras corresponding to the following notions of weak equivalence:
1. maps of simplicial coalgebras which become quasiisomorphisms of differential graded (dg) coalgebras after applying the normalized chains functor
2. maps of simplicial coalgebras which become quasiisomorphisms of dg algebras after applying the normalized chains functor followed by the dg cobar construction, and
3. maps of simplicial coalgebras which become quasiisomorphisms of dg algebras after applying a localized version of the dg cobar construction.
Notion (1) was used by Goerss to provide a fullyfaithful model for spaces up to Fhomology equivalence, for a F an algebraically closed field. I will explain how (2), which is drawn from dg Koszul duality theory, corresponds to a linearized version of the notion of categorical equivalence between simplicial sets as used in the theory of quasicategories. I will also explain how (3) leads to a fullyfaithful model for the homotopy theory of simplicial sets considered up to maps that induce isomorphisms on fundamental groups and on the Fhomology of the universal covers, for F an algebraically closed field. One of the key points is a sort of homological formulation of the fundamental group. This is based on joint work with G. Raptis and also on work with F. Wierstra and M. Zeinalian.
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Nick Kuhn (University of Virginia)
$\begingroup $The study of the action of a finite pgroup $G$ on a finite $G$CW complex $X$ is one of the oldest topics in algebraic topology. In the late 1930's, P. A. Smith proved that if $X$ is mod p acyclic, then so is $X^G$, its subspace of fixed points. A related theorem of Ed Floyd from the early 1950's says that the dimension of the mod p homology of $X$ will bound the dimension of the mod p homology of $X^G$.
The study of the Balmer spectrum of the homotopy category of $G$spectra has lead to the problem of identifying "chromatic" variants of Smith's theorem, with mod p homology replaced by the Morava Ktheories (at the prime p). One such chromatic Smith theorem is proved by Barthel et.al.: if $G$ is a cyclic pgroup and X is $K(n)$ acyclic, then $X^G$ is $K(n1)$ acyclic (and this answers questions like this for all abelian pgroups).
In work with Chris Lloyd, we have been able to show that a chromatic analogue of Floyd's theorem is true whenever a chromatic Smith theorem holds. For example, if $G$ is a cyclic pgroup, then the dimension over $K(n)_\ast$ of $K(n)_\ast(X)$ will bound the dimension over $K(n1)_\ast$ of $K(n1)_\ast(X^G)$.
The proof that chromatic Smith theorems imply the stronger chromatic Floyd theorems uses the representation theory of the symmetric groups.
These chromatic Floyd theorems open the door for many applications. We have been able to resolve open questions involving the Balmer spectrum for the extraspecial 2groups. In a different direction, at the prime 2, we can show quick collapsing of the AHSS computing the Morava Ktheory of some real Grassmanians: this is a nonequivariant result.
In my talk, I'll try to give an overview of some of this.
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Arpon Raksit (MIT)
$\begingroup $This talk will be about joint work with Jeremy Hahn and Dylan Wilson in which we define a filtration on an arbitrary commutative ring spectrum that we call the "even filtration". I'll introduce the definition, the one method we've come up with for analyzing it, and its relation to other filtrations of interest, in particular motivic filtrations on topological Hochschild homology.
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Akhil Mathew (University of Chicago)
$\begingroup $The seminar will meet at 3:00 PM in 2131.
Syntomic complexes are a form of padic motivic cohomology that filter padic étale Ktheory (or topological cyclic homology), and which are defined in terms of prismatic cohomology. I will explain a description of the syntomic complexes of ptorsionfree regular rings, based on a mixed characteristic analog of the Cartier isomorphism, closely related to the Segal conjecture for THH. (Joint with Bhargav Bhatt.)
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Tomer Schlank (Hebrew University)
$\begingroup $Hilbert's Nullstellensatz is a fundamental result in commutative algebra which gives a defining property of algebraically closed fields. This property identifies algebraically closed fields as the "points" in classical algebraic geometry.
In this talk, I will discuss joint work with Robert Burklund and Allen Yuan in which we identify certain LubinTate spectra as those that satisfy a chromatic version of Hilbert's Nullstellensatz. This will allow the definition of a "constructible" spectrum for $E_\infty$ rings. I will then sample some applications of our results to chromatic redshift and orientation theory for $E_\infty$ rings.
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Noah Riggenbach (Northwestern University)
$\begingroup $In this talk I will discuss my recent computation of the NTC groups of perfectoid rings which have a system of pth power roots of unity and thus the $K$groups of the pcompleted affine line $R\langle x\rangle$ over these rings relative to the ideal $(x)$. This includes all perfect fields of positive characteristic, for which these groups vanish in nonnegative degrees. This class of rings also contains many mixed characteristic rings, and perhaps surprisingly while the even nonnegative groups will still vanish, the odd groups will not.
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Yajit Jain (Brown University)
$\begingroup $After using smoothing theory to introduce a notion of exotic smooth structures on manifold bundles, we will discuss an equivalent class of objects: smooth bundles of hcobordisms with a topological trivialization. Using work of Dwyer, Weiss, and Williams, we will associate to such families an invariant called the smooth structure class, which is closely related to the higher FranzReidemeister torsion of Igusa and Klein. We will illustrate two proofs of a duality theorem for the smooth structure class. This theorem generalizes Milnor's duality theorem for Whitehead torsion. A consequence of this result is the rigidity conjecture of Goette and Igusa, which states that, after rationalizing, stable exotic smoothings of manifold bundles with closed even dimensional fibers do not exist.
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Robert Burklund (MIT)
$\begingroup $One of the distinguishing features of higher algebra is the difficulty of constructing quotients. In this talk I will explain a new technique for constructing algebra structures on quotients. This technique allows us to prove that $\mathbb{S}/8$ is an $\mathbb{E}_1$algebra, $\mathbb{S}/32$ is an $\mathbb{E}_2$algebra, $\mathbb{S}/p^{n+1}$ is an $\mathbb{E}_n$algebra at odd primes and, more generally, for every $h$ and $n$ there exist generalized Moore spectra of type $h$ which admit an $\mathbb{E}_n$algebra structure.
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Araminta Amabel (MIT)
$\begingroup $We will discuss naturally occurring genera (i.e. cobordism invariants) inspired by the deformation theory for supersymmetric quantum mechanics. First, we construct a canonical deformation quantization for symplectic supermanifolds. Secondly, we prove a superversion of NestTsygan’s algebraic index theorem, generalizing work of Engeli. This work is inspired by the appearance of the same genera in three related stories: index theory, trace methods in deformation theory, and partition functions in quantum field theory. Using the trace methodology, we compute the genus appearing in the story for supersymmetric quantum mechanics. This involves investigating supertraces on WeylClifford algebras and deformations of symplectic supermanifolds.
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Yuri Sulyma (Brown University)
$\begingroup $One perspective on homotopy theory is that it is an enhanced version of arithmetic which remembers combinatorics and symmetry. I will demonstrate this philosophy concretely in the case of the floor and ceiling functions from arithmetic, by explaining several situations where these appear: $K$theory of truncated polynomial algebras; Legendre's formula and its $q$analogue; hyperrepresentationgraded $\mathrm{TR}$; and equivariant homotopy theory. To understand how these examples are related, I will show how to construct a Tambara functor out of a prism, and discuss a conjectural theory of $G$crystalline/$G$de Rham cohomology generalizing $q$crystalline cohomology and the $q$de Rham complex.
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Piotr Pstragowski (Harvard University)
$\begingroup $Associated to each homology theory we have an Adams spectral sequence computing stable homotopy classes of maps. Under flatness assumptions, the $E_2$term can be identified with cohomology of a certain Hopf algebroid, giving the spectral sequence its computational power. Unfortunately, this identification fails in many important examples, such as integral homology or connective Morava $K$theory, making these spectral sequences mysterious and hard to calculate with. In this talk, I will describe a novel method of identifying these $E_2$terms in terms of cohomology in representations of certain quivers. This is based on joint work with Burklund.
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Andrew Senger (Harvard University)
$\begingroup $JohsonWilson spectra are some of the most fundamental examples of complex oriented ring spectra In this talk, I will sketch a proof of how they may be given the structure of $E_\infty$ rings at all heights and primes.
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Claudia Scheimbauer (Technical University of Munich)
$\begingroup $Since Claudia is speaking from Europe, the seminar will meet at 11:00 AM in 2131.
In this talk I will explain higher Morita categories of $E_n$algebras and bimodules and discuss dualizability therein. Important examples for applications to topological field theories are a 3category of fusion categories and a 4category of modular tensor categories. Then we will discuss why these do not suffice for ReshetikinTuraev theories and I will give an outlook on workinprogress with Freed and Teleman on how to remedy this.
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Dmitri Pavlov (Texas Tech University)
$\begingroup $I will explain my recent work with Daniel Grady on locality of functorial field theories (arXiv:2011.01208) and the geometric cobordism hypothesis (arXiv:2111.01095). The latter generalizes the Baez–Dolan cobordism hypothesis to nontopological field theories, in which bordisms can be equipped with geometric structures, such as smooth maps to a fixed target manifold, Riemannian metrics, conformal structures, principal bundles with connection, or geometric string structures.
Applications include a generalization of the Galatius–Madsen–Tillmann–Weiss theorem, a solution to a conjecture of Stolz and Teichner on representability of concordance classes of functorial field theories, a construction of power operations on the level of field theories (extending the recent work of Barthel–BerwickEvans–Stapleton), and a recent solution by Grady of a conjecture by Freed and Hopkins on deformation classes of reflection positive invertible field theories. If time permits, I will talk about the planned future work on nonperturbative quantization of functorial field theories and generalized Atiyah–Singerstyle index theorems.
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Dylan Wilson (Harvard University)
$\begingroup $The seminar will meet at 4:30 PM in 2131 (in person only).
Trying to do calculus in characteristic p returns strange results like the Cartier isomorphism: the deRham *cohomology groups* of a smooth F_pvariety looks the same as the deRham complex back again (up to a twist). This turns out to be a prototype for a result in the Hochschild homology of smooth algebras in characteristic p. In this talk I'll explain this background and give new examples of a Cartierlike isomorphism in 'higher chromatic characteristics' (e.g. for connective real Ktheory and connective topological modular forms).
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Hana Kong (Institute for Advanced Study)
$\begingroup $The seminar will meet at 4:30 PM in 2131 (in person only).
Bachmann–Hopkins defines the motivic 'imageofj' spectrum over base fields with characteristic not 2. In this talk, I will talk about the effective slice computation of this spectrum over the real numbers. Analogous to the classical story, the result captures a regular pattern that appears in the Rmotivic stable stems. This is joint work with Eva Belmont and Dan Isaksen.
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Elden Elmanto (Harvard University/CNRS)
$\begingroup $Beilinson, Macpherson and Schechtman asked us to imagine a world where topological Ktheory was first defined before singular cohomology. How would one invent the latter? This question has been influential to various approaches to motivic cohomology of smooth varieties with its relationship to Ktheory, serving a 'design principle.' I will explain an extension of this idea to define a version of motivic cohomology of singular schemes. The engine behind it is the BhattMorrowScholze prismatic sheaves.
This is all joint work with Tom Bachmann and Matthew Morrow.
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Arpon Raksit (MIT)
$\begingroup $The seminar will meet at 4:30 PM in 2131 (in person only).
Let X be an elliptic curve over a perfect field k of positive characteristic. Serre–Tate studied the deformation theory of such X, and one of their discoveries was that when X is ordinary, it admits a canonical lifting to the ring of Witt vectors W(k) (with some special features). In this talk, I'll discuss a connection between this phenomenon and properties of the moduli of elliptic curves in spectral algebraic geometry introduced by Lurie.
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Joana Cirici (Universitat de Barcelona)
$\begingroup $The notion of formality first arose in rational homotopy theory, but it makes sense in any context in which chain complexes interact with multiplicative structures and has multiple applications beyond its original purpose. The idea that purity implies formality goes back to Deligne, Griffiths, Morgan and Sullivan, who used the Hodge decomposition to show that compact Kähler manifolds are formal over the rational numbers. Following the ideas behind Deligne's philosophy of weights, I will explain how to use Galois actions on étale cohomology to study formality with torsion coefficients for algebraic structures associated to certain schemes defined over a finite field. As an application, I will review results for configuration spaces on the complex space and for the operad of little disks. This is joint work with Geoffroy Horel.
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Denis Nardin (Universität Regensburg)
$\begingroup $Hermitian Ktheory is an invariant that can be defined for every scheme X. Traditionally the focus has been on schemes X where 2 is invertible. Recently an understanding has emerged of how to deal with the many different variants that are available when 2 is not invertible. In this talk I will survey the various known and unknown but expected properties of the hermitian Ktheory presheaf, and their computational consequences. The results in this talk are joint work with a many people, among which are B. Calmés, E. Elmanto, Y. Harpaz, J. Shah and L. Yang.
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Asaf Horev (Stockholm University)
$\begingroup $Abstract: Factorization homology is a natural invariant of manifolds and Enalgebras. This talk is about an equivariant version of factorization homology, where the manifolds algebras and resulting invariants all admit a finite group action. We will use this theory to relate the following perspectives on topological Hochschild homology: (1) Topological Hochschild homology is an invariant of E1algebras. A similar invariant can be defined in any nice symmetric monoidal ∞category, for example as factorization homology over the circle. (2) Topological Hochschild homology admits cyclotomic structure, given by cyclotomic Frobenius maps in the sense of NikolausScholze. This additional structure is endemic to the category of spectra. In the talk I will review factorization homology and its equivariant extension, and describe a geometric construction of the cyclotomic Frobenius maps of topological Hochschild homology, using genuine equivariant factorization homology to integrate the Tate diagonal over the circle.
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Lior Yanovski (Max Planck Institute)
$\begingroup $The seminar will meet at 10:00 AM in 2131.
The classical discrete Fourier transform can be thought of as an isomorphism of rings between the complex group algebra of a finite abelian group A and the algebra of functions on its Pontyagin dual. Hopkins and Lurie have proved an analogous result in the chromatic world, where the field of complex numbers is replaced by the LubinTate spectrum E_n, the finite abelian group A is replaced by a suitably finite ppower torsion Zmodule spectrum, and the Pontryagin dual is modified by an nfold suspension. From this, they deduce a number of structural properties of the infinitycategory of K(n)local spectra, such as affineness and EilenbergMoore type formulas for pifinite spaces. In this talk, I will present a joint work with Barthel, Carmeli, and Sclank, in which we develop the notion of a 'higher Discrete Fourier transform' for general higher semiadditive infinitycategories. This allows us, among other things, to extend the above results of Hopkins and Lurie to the T(n)local setting. Furthermore, we study the interaction of Fourier transforms with categorification suggesting a close relationship to chromatic redshift phenomena. Finally, by replacing Pontryagin duality with BrownComenetz duality, we can contemplate the notion of Fourier transform for more general pifinite spectra than Zmodules, leading to questions intimately related to the behavior of the 'discrepancy spectrum'.
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Shachar Carmeli (Weizmann Institute of Science)
$\begingroup $The seminar will meet at 10:00 AM in 2131.
Higher semiadditivity is a property of an infinitycategory that allows, in particular, for the summation of families of morphisms between objects parametrized by pifinite spaces.
Hopkins and Lurie showed that the K(n)localizations of the infinity category of spectra are higher semiadditive. Consequently, by a work of Harpaz, the mapping objects in these infinitycategories admit the rich structure of higher commutative monoids. While many abstract properties of these higher commutative monoids are known, not many explicit computations of them have been carried out so far.
In my talk, I will present a work in progress, joint with Allen Yuan, which aims to completely determine this higher commutative monoid structure of the K(1)local sphere. Specifically, I will show how to use higher semiadditive versions of algebraic Ktheory and GrothendieckWitt theory to compute the summation maps along groupoids for the K(1)local sphere. At the prime 2, this allows us to realize some nontrivial classes in its homotopy groups as semiadditive cardinalities of pifinite spaces, and to compute explicitly certain power operations that arise from the higher semiadditivity.
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Markus Land (University of Copenhagen)
$\begingroup $The seminar will meet at 10:00 AM in 2131.
I will first give a brief overview of how one can understand classical Grothendieck—Witt theories of rings in terms of Ktheoretic and Ltheoretic pieces. Using this, I will explain how to determine various Grothendieck—Witt theories, in particular of Dedekind rings. As further application of these results, I will then give a calculation of the stable cohomology of orthogonal and symplectic groups over the integers focussing on the mod 2 cohomology.
This is all based on joint work with Calmès, Dotto, Harpaz, Hebestreit, Moi, Nardin, Nikolaus, and Steimle, and Hebestreit and Nikolaus.
$\endgroup$ 
Guozhen Wang (Shanghai Center for Mathematical Sciences)
$\begingroup $The seminar will meet at 10:00 AM in 2131.
We introduce a new method for computing topological cyclic homology of locally complete intersections over padic intergers, by using relative hochschild homology and resolving the base ring spectrum with an Adams reslolution. Using the Nygaard filtration on the E1term, we can construct algebraic Tate and algebraic homotopy fixed points spectral sequences, which are algebraic and catpture lots of informations in the Tate and homotopy fixed points spectral sequences computing TP and TC^{1}. Using this method, we can give a uniform way of computing topological cyclic homology of local fields of mixed characteristic.
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Elden Elmanto (Harvard University)
$\begingroup $We extend the DundasGoodwillieMcCarthy theorem concerning the fiber of the cyclotomic trace map from K theory to topological cyclic homology, to the context of stable categories. Our main tool is Bondarko's theory of weight structures. Applications include a new proof of cdhdescent for homotopy Ktheory of stacks (HoyoisKrishna) and new cases of Blanc's lattice conjecture in noncommutative Hodge theory (ala KatzarkovKontsevichPantev). Time permitting, I will speak about vistas, including the (equivariant) Ktheory of the equivariant sphere and padic Hodge theory for stacks.
This is all joint work with Vova Sosnilo and partly based on https://arxiv.org/abs/2010.09155
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Foling Zou (University of Michigan)
$\begingroup $The factorization homology are invariants of ndimensional manifolds with some fixed tangential structures that take coefficients in suitable $E_n$algebras. I will give a definition for the equivariant factorization homology of a framed manifold for a finite group G via a monadic bar construction following MillerKupers. I will also talk about the unital variant of symmetric sequences that is underneath this construction. Then I will talk about the equivariant nonabelian Poincare duality theorem in this case and the equivariant factorization homology on equivariant spheres for certain Thom spectra. This is joint with Asaf Horev, Inbar Klang, Peter May and Ruoqi Zhang.
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Peter Haine (MIT)
$\begingroup $A natural question arises when working with intersection cohomology and other stratified invariants of singular manifolds: what is the correct stable homotopy theory for these invariants to live in? But before answering that question one first has to identify the correct unstable homotopy theory of stratified spaces. The exitpath category construction of MacPherson, Treumann, and Lurie provides functor from suitably nice stratified topological spaces to “abstract stratified homotopy types” — ∞categories with a conservative functor to a poset. Work of Ayala–Francis–Rozenblyum even shows that their conically smooth stratified topological spaces embed into the ∞category of abstract stratified homotopy types. In this talk, we explain some of our work which goes further and produces an equivalence between the homotopy theory of all stratified topological spaces and these abstract stratified homotopy types.
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Andy Senger (MIT)
$\begingroup $We show that the oddprimary BrownPeterson spectrum does not admit the structure of an E_{2(p^2+2)} ring spectrum and that there can be no map MU–>BP of E_{2p+3} ring spectra at any prime. This extends results of Lawson at the prime 2.
$\endgroup$ 
Zhulin Li (MIT)
$\begingroup $In this talk, I will introduce unstable modules with only the top k Steenrod operations at the prime 2. I'll show that they have projective dimension at most k. Then I'll establish forgetful functors, suspension functors, loop functors and Frobenius functors between such modules. The forgetful functors induce an inverse system of Ext groups, and the inverse system stabilizes when the covariant module is bounded above. In addition, I will talk about a generalization of the Lambda algebra which computes the Ext group from such modules to suspensions of the base field.
$\endgroup$ 
Nir Gadish (MIT)
$\begingroup $Möbius inversion is classically a procedure in number theory that inverts summation of functions over the divisors of an integer. A similar construction is possible for every locally finite poset, and is governed by a so called Möbius function encoding the combinatorics. In 1936 Hall observed that the values of the Möbius function are Euler characteristics of intervals in the poset, suggesting a homotopy theoretic context for the inversion. In this talk we will discuss a functorial 'spacelevel' realization of Möbius inversion for diagrams taking values in a pointed cocomplete infinitycategory. The role of the Möbius function will be played by hömotopy types whose reduced Euler characteristics are the classical values, and inversion will hold up to extensions (think inclusionexclusion but with the alternating signs replaced by even/odd spheres).
This provides a uniform perspective to many constructions in topology and algebra. Notable examples that I hope to mention include handle decompositions, Koszul resolutions, and filtrations of configuration spaces.
$\endgroup$ 
Dylan Wilson (Harvard University)
$\begingroup $In recent work with Jeremy Hahn, we established a higher chromatic version of the LichtenbaumQuillen conjecture for truncated BrownPeterson spectra. This talk will explore some questions raised by the proof, and indicate some current and future lines of investigation. Some of what we will discuss is also joint with Akhil Mathew.
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Kirsten Wickelgren (Duke)
$\begingroup $One expects the intersection of a d and nd dimensional subscheme or submanifold of an ndimensional one to be 0 dimensional. When this is not the case, such intersections are often called excess intersections, and arise when considering questions such as 'How many conics are tangent to 5 conics in the plane?' We consider cohomology classes in oriented Chow and Hermitian Ktheory associated to excess intersections, and use some recent duality results of Eisenbud and Ulrich to give an excess intersection formula. We compute some examples giving arithmetic refinements of counts classically valid only over algebraically closed fields. This is joint work with Tom Bachmann.
$\endgroup$ 
Najib Idrissi (IMJPRG)
$\begingroup $Framed configuration spaces of a surface form a right module over the framed little disks operad. This rich algebraic structure has important consequences, for example for the computations of manifold calculus or factorization homology. Determining the homotopy type of this operadic right module remains however a difficult task. In this talk, I will explain how to compute the rational homotopy type for oriented compact surfaces. The end result is a finitedimensional purely combinatorial model. The proof involves several ingredients: Kontsevich’s formality, Tamarkin’s formality, and the cyclic formality of the framed little disks operad. (Joint work with Ricardo Campos and Thomas Willwacher.)
$\endgroup$ 
Achim Krause (Münster)
$\begingroup $The Picard group of genuine Gspectra consists of those compact objects all of whose geometric fixed points are spheres. Work by tom Dieck  Petrie classifies these objects in terms of the dimensions at all conjugacy classes of subgroups. The question of which such 'dimension functions' can be realized is quite interesting. We present a new perspective on this question, based on a version of isotropy separation which plays well with compact objects.
$\endgroup$ 
Tomer Schlank (Hebrew University)
$\begingroup $The chromatic approach to stable homotopy theory is 'divide and conquer'. That is, questions about spectra are studies through various localizations that isolate pure height phenomena and then are put back together. For each height n, there are two main candidates for pure height localization. The first is the generally more accessible K(n)localization and the second is the closely related T(n)localization. It is an open problem whether the two families of localizations coincide. One of the main reasons that the K(n)local category is more amenable to computations is the existence of well understood Galois extensions of the K(n)local sphere. In the talk, I will present a generalization, based on ambidexterity, of the classical theory of cyclotomic extensions, suitable for producing nontrivial Galois extensions in the T(n)local and K(n)local context. This construction gives a new family of Galois extensions of the T(n)local sphere and allows to lift the well known maximal abelian extension of the K(n)local sphere to the T(n)local world. I will then describe some applications, including the study of the T(n)local Picard group, a chromatic version of the Kummer theory, and interaction with algebraic Ktheory. This is a joint project with Shachar Carmeli and Lior Yanovski.
$\endgroup$

Jeremy Hahn (MIT)
$\begingroup $Ausoni and Rognes calculated that K(ku) has chromatic height 2, at least at primes larger than 3. Their redshift philosophy more generally suggests that the algebraic Ktheory of a height n ring spectrum should have height n+1. I will explain work, joint with Dylan Wilson, in which we equip BP(n) with an E_3BPalgebra structure for all primes p and heights n. The algebraic Ktheory of this E_3 ring has chromatic height n+1, giving an example of redshift at arbitrary height. To show the ideas I may present quick proofs, at the prime 2, of the facts that K(ku) is height 2 and K(tmf) is height 3.
$\endgroup$ 
Martin Speirs (Harvard University)
$\begingroup $In the 1970s Quillen proved that algebraic Ktheory is homotopy invariant for a regular noetherian base. For a nonregular base ring this is not true. Bass defined the NKgroups in order to study the failure of homotopy invariance in Ktheory. In general these groups are not well understood, though they have many interesting properties. Ten years ago, Cortiñas, Haesemeyer, Walker and Weibel used cdhdescent methods to understand the NKgroups when the input is rational. In this talk I will explain parts of their work and discuss ongoing work with Elden Elmanto where we aim to extend their methods to the mixed characteristic setting.
$\endgroup$ 
Agnes Beaudry (University of Colorado Boulder)
$\begingroup $At height $h=2^{n1}m$, the Morava stabilizer group contains a cyclic group $G$ of order $2^n$. In this talk, I will present equivariant spectra that refine the classical height $h$ Morava $K$theories. These are obtained from $G$equivariant models of LubinTate spectra which were constructed in recent joint work with HillShiZeng. I will present some preliminary results and conjectures about their slice filtration and equivariant homotopy groups and discuss how exotic transchromatic extensions lead to interesting differentials. This is joint work with HillShiZeng.
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Amnon Neeman (Australian National University)
$\begingroup $In a 2006 article Schlichting conjectured that the negative Ktheory of any abelian category must vanish. And in a 2019 article Antieau, Gepner and Heller generalized, conjecturing that the negative Ktheory of any infinitycategory with a bounded tstructure must vanish.
We will review the history, explain why both conjectures are plausible, and then sketch a counterexample disproving both.
$\endgroup$ 
Piotr Pstragowski (Harvard University)
$\begingroup $In 1996, Jens Franke conjectured that any stable infinitycategory possessing an Adams spectral sequence whose sparsity is greater than the homological dimension, admits a purely algebraic description of its homotopy category as a certain derived category. In this talk, I'll describe joint work with Irakli Patchkoria in which we prove Franke's conjecture, subsuming and improving on virtually all known algebraicity results for stable infinitycategories.
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Marcy Robertson (University of Melbourne)
$\begingroup $Welded tangles are knotted surfaces in R^4. BarNatan and Dancso described a class of welded tangles which have “foamed vertices” where one allows surfaces to merge and split. The resulting welded tangled foams carry an algebraic structure, similar to the planar algebras of Jones, called a circuit algebra. In joint work with Dancso and Halacheva we provide a onetoone correspondence between circuit algebras and a form of rigid tensor category called "wheeled props." This is a higher dimensional version of the wellknown algebraic classification of planar algebras as certain pivotal categories.
This classification allows us to connect these "welded tangled foams," to the KashiwaraVergne conjecture in Lie theory. In work in progress, we show that the group of homotopy automorphisms of the (rational completion of) the wheeled prop of welded foams is isomorphic to the group of symmetries KV, which acts on the solutions to the KashiwaraVergne conjecture. Moreover, we explain how this approach illuminates the close relationship between the group KV and the prounipotent Grothendieck–Teichmüller group.
$\endgroup$ 
Ben Knudsen (Northeastern University)
$\begingroup $We ask when embedding calculus can distinguish pairs of smooth manifolds that are homeomorphic but not diffeomorphic. We prove that, in dimension 4, the answer is "almost never". In contrast, we exhibit an infinite list of highdimensional exotic spheres detected by embedding calculus. The former result implies that the algebraic topology of knot spaces is insensitive to smooth structure in dimension 4, answering a question of Viro. The latter result gives a partial answer to a question of Francis and hints at the possibility of a new classification of exotic spheres in terms of a stratified obstruction theory applied to compactified configuration spaces. This talk represents joint work with Alexander Kupers.
$\endgroup$ 
Jonathan Campbell (Center for Communications Research La Jolla)
$\begingroup $In this talk I'll explain how one might attack Hilbert's Generalized Third Problem via homotopy theory, and describe recent progress in this direction. Two ndimensional polytopes, $P$, $Q$ are said to be scissors congruent if one can cut $P$ along a finite number of hyperplanes, and reassemble the pieces into $Q$. The scissors congruence problem, aka Hilbert's Generalized Third Problem, asks: when can we do this? What obstructs this? In two dimensions, two polygons are scissors congruent if and only if they have the same area. In three dimensions, there is volume and another invariant, the Dehn Invariant. In higher dimensions, very little is known — but the problem is known to have deep connections to motives, values of zeta functions, the weight filtration in algebraic Ktheory, and regulator maps. I'll give a leisurely introduction to this very classical problem, and explain some new results obtained via homotopy theoretic techniques. This is all joint with Inna Zakharevich.
$\endgroup$ 
Adeel Khan (IHES)
$\begingroup $Let X be a smooth complex algebraic variety. In contrast with the situation for the singular homology groups H_*(X), the construction of intersection products on the Chow groups of X is subtle due to the comparative difficulty in dealing with nontransverse intersections. I will explain one way to deal with this problem by considering cycle classes that come from derived algebraic geometry. In combination with the algebraic analogue of Steenrod's problem on resolution of singularities of homology classes (which holds by Hironaka), this yields a new construction of cup products in Chow groups. Time permitting, I may also discuss how derived cycle classes arise in motivic homotopy theory.
$\endgroup$ 
David Gepner (University of Melbourne and University of Illinois at Chicago)
$\begingroup $We aim to show that the elliptic cohomology of the (classifying stack of the) unitary group can be calculated as the ring of functions on the Hilbert scheme of points of the associated derived elliptic curve. To this end, we will begin with a discussion of (integral) equivariant elliptic cohomology, due to Jacob Lurie, using the formalism of orbispaces, as developed by myself and Andre Henriques. This is joint work with Lennart Meier.
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Oscar RandalWilliams (University of Cambridge)
$\begingroup \def\R{\mathbb{R}} $In dimensions other than 4, the difference between groups of diffeomorphisms and of homeomorphisms of an $n$manifold $M$ is governed by an $h$principle, meaning that it reduces to understanding these groups for $M=\R^n$. The group of diffeomorphisms is simple, by linearising it is equivalent to $O(n)$, but the group $Top(n)$ of homeomorphisms of $\R^n$ has little structure and is difficult to grasp. It is profitable to instead consider the $n$disc $M=D^n$, because the group of homeomorphisms of a disc (fixing the boundary) is contractible by Alexander's trick: this removes homeomorphisms from the picture entirely, and makes the problem one purely within differential topology. I will explain some of the history of this problem, as well as recent work with A. Kupers in this direction.
$\endgroup$

Rune Haugseng (Norwegian University of Science and Technology)
$\begingroup $The expected targets for many (n+1)dimensional extended TQFTs are symmetric monoidal (n+1)categories of Rlinear ncategories, i.e. Venriched ncategories where V is the category of Rmodules for some ring R. I will propose a new description of the fully dualizable objects in this target, which according to the Cobordism Hypothesis correspond to framed TQFTs, as the dualizable objects in a category of Rlinear ncategories with absolute colimits at all levels. Here the absolute colimits are those enriched (weighted) colimits that are preserved by all functors. If we take Rmodules to mean discrete modules over an ordinary ring R, there is a simple classification of these, as proposed by Gaiotto and JohnsonFreyd. (For Rmodules in the more modern sense, i.e. unbounded chain complexes of Rmodules or Rmodule spectra, such a classification is still rather mysterious, however.) This is very much work in progress, joint with David Gepner, and builds on ongoing work on (co)limits in enriched infinitycategories with Grigory Kondyrev.
$\endgroup$ 
Marc Hoyois (Universität Regensburg)
$\begingroup $It is a classical result of Weibel that homotopy invariant algebraic Ktheory satisfies excision, in the sense that for any ring A and ideal I of A, the fiber of KH(A)–>KH(A/I) depends only on I as a nonunital ring. In joint work with Elden Elmanto, Ryomei Iwasa, and Shane Kelly, we show that this is true more generally for any cohomology theory represented by a motivic spectrum.
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Thomas Nikolaus (Münster)
$\begingroup $We introduce the Grothendieck–Witt groups of the integers and the Grothendieck–Witt spectrum of the integers. Then we explain how to compute these groups and the homotopy type of the spectrum using recent work on Ktheory and Ltheory. If time permits we also explain how to resolve the homotopy limit problem for rings of integers in number fields and prove Karoubi's periodicity conjecure for arbitrart rings.
$\endgroup$ 
Benjamin Antieau (University of Illinois at Chicago and Northwestern University)
$\begingroup $I will give an introduction to the idea of higher Brauer groups, focusing on the "next" higher Brauer group, consisting of Morita equivalence classes of certain Azumaya categories. The emphasis of the talk will be on analogies, examples, calculations, and open problems.
$\endgroup$ 
Dustin Clausen (Max Planck Institute)
$\begingroup $Lazard showed that the continuous group cohomology of a large class ofpadic Lie groups, with padic coefficients, satisfies Poincare duality. Analogously to the usual Poincare duality of real manifolds, there are orientability issues, but Lazard showed that the relevant orientation local system is completely determined by the adjoint representation of the group in an explicit manner, allowing for an easy analysis. This can be compared to how the orientation local system on a real manifold is determined by the tangent bundle, a very useful "linearization" of the problem. Now, there is an analogous Poincare duality with spectrum coefficients both in the setting of padic Lie groups and in the setting of real manifolds. In the latter case the relevant orientation local system is still determined by the tangent bundle; in fact it is the suspension spectrum of the associated sphere bundle, a statement known as Atiyah duality. In the former case, there is a natural guess for how the orientation local system should still be determined by the adjoint representation. This has been highlighted by recent work of BeaudryGoerssHopkinsStojanoska in their study of duality for tmf, and they dubbed this guess the "linearization hypothesis". Neither Lazard's techniques nor the usual arguments for Atiyah duality can be used to attack the linearization hypothesis. In this talk I will explain a proof of the linearization hypothesis, whose main ingredients are a deformation of any padic Lie group to its Lie algebra, and a rather exotic "cospecialization map" which lets you use this deformation to jump from the Lie algebra to the Lie group as if the deformation were parametrized by a unit interval, even though it is only parametrized by a totally disconnected space.
$\endgroup$ 
Maria Yakerson (Universität Regensburg)
$\begingroup $Algebraic and hermitian Ktheories of smooth schemes are generalized cohomology theories, represented in the motivic stable homotopy category. In this talk, we explain how to obtain new geometric models for the corresponding motivic spectra, based on the specific kinds of transfer maps that these cohomology theories acquire. As a surprising sideeffect, we compute the motivic homotopy type of the Hilbert scheme of infinite affine space. This is joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin and Burt Totaro.
$\endgroup$ 
Markus Hausmann (Universität Bonn)
$\begingroup $I will discuss an equivariant version of Quillen's theorem that the complex bordism ring carries the universal formal group law, both over a fixed abelian group and in a global equivariant setting.
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Tyler Lawson (University of Minnesota)
$\begingroup $I'll discuss calculational methods for determining moduli of objects and maps between E_infty ring spectra, and the relation to topological AndreQuillen cohomology.
$\endgroup$ 
Zhouli Xu (MIT)
$\begingroup $In this talk, I will discuss recent progress on the computation of classical stable homotopy groups of spheres, and highlight some new results regarding certain Adams differentials and their connections to the Kervaire invariant classes. These computations use the ChowNovikov tstructure on the cellular motivic stable homotopy theory over C in an essential way. I will also discuss a recent result that extends this tstructure to the noncellular part of the category which holds over any field, and its potential applications in computations.
This talk is based on several joint projects involving Tom Bachmann, Robert Burklund, Bogdan Gheorghe, Dan Isaksen, Hana Jia Kong and Guozhen Wang.
$\endgroup$ 
Michael Ching (Amherst College)
$\begingroup $(Joint with Kristine Bauer and Matthew Burke.) Lurie defines the “tangent bundle” to an $\infty$category C to be the $\infty$category of excisive functors from finite pointed spaces to C. In this talk, I will describe an abstract framework which includes both this construction and the ordinary tangent bundle functor on the category of smooth manifolds (as well as many other examples). That framework is an extension to $\infty$categories of the “tangent categories” of Cockett and Cruttwell (based on earlier work of Rosický).
Those authors and others have explored the extent to which various concepts from differential geometry, such as connections, curvature and cohomology, can be developed abstractly within a tangent category. Thus our result provides a framework for “doing” differential geometry in the context of Goodwillie’s calculus of functors. For example, we show that Goodwillie’s notion of nexcisive functor can be recovered from the general notion of “njet” in a tangent category.
$\endgroup$ 
Jonathan Campbell (Duke)
$\begingroup $Talk cancelled.
In this talk I'll explain how one might attack Hilbert's Generalized Third Problem via homotopy theory, and describe recent progress in this direction. Two ndimensional polytopes, $P$, $Q$ are said to be scissors congruent if one can cut $P$ along a finite number of hyperplanes, and reassemble the pieces into $Q$. The scissors congruence problem, aka Hilbert's Generalized Third Problem, asks: when can we do this? what obstructs this? In two dimensions, two polygons are scissors congruent if and only if they have the same area. In three dimensions, there is volume and another invariant, the Dehn Invariant. In higher dimensions, very little is known — but the problem is known to have deep connections to motives, values of zeta functions, the weight filtration in algebraic Ktheory, and regulator maps. I'll give a leisurely introduction to this very classical problem, and explain some new results obtained via homotopy theoretic techniques. This is joint work with Inna Zakharevich.
$\endgroup$ 
Hood Chatham (MIT)
$\begingroup $The real $\mathrm{K}$theory spectrum $\mathrm{KO}$ is "almost complex oriented". Here are a collection of properties that demonstrate this:
(1) $\mathrm{KO}$ is the $C_2$ fixed points of a complex oriented cohomology theory $\mathrm{KU}$.
(2) Complex oriented cohomology theories have trivial Hurewicz image, whereas $\mathrm{KO}$ has a small Hurewicz image – it detects $\eta$ and $\eta^2$.
(3) Complex oriented cohomology theories receive a ring map from $\mathrm{MU}$. $\mathrm{KO}$ receives no ring map from $\mathrm{MU}$ but it receives one from $\mathrm{MSU}$.
(4) If $E$ is a complex orientable cohomology theory, every complex vector bundle $V$ is $E$orientable. Not every complex vector bundle $V$ is $\mathrm{KO}$orientable, but $V\oplus V$ and $V^{\otimes 2}$ are.
Because this is an electronic talk, I will focus on spectral sequence demonstrations using my inprogress spectral sequence software.
$\endgroup$ 
Yuri Sulyma (Brown University)
$\begingroup $Much interaction between homotopy theory and $p$adic geometry has been spurred in recent years by the work of BhattMorrowScholze, who built a universal $p$adic cohomology theory by constructing a novel filtration on topological Hochschild homology. Their construction works by flat descent to the case of perfectoid rings, where it is given by the usual Postnikov filtration. Here, work of Hesselholt (generalized by BMS) shows that THH is even polynomial on a degree 2 generator, generalizing classical Bökstedt periodicity for the case of $\mathbb F_p$. We study a variant, the \emph{regular slice filtration} from equivariant stable homotopy theory. The slice filtration is again concentrated in even degrees, generated by $RO(\mathbb T)$graded classes which can loosely be thought of as \emph{norms} of the Bökstedt generator. The slices themselves are $RO(\mathbb T)$graded suspensions of certain Mackey functors. When $R$ is $p$torsionfree, the $E_2$ page of the slice spectral sequence is concentrated in even degrees.
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Ben Knudsen (Northeastern University)
$\begingroup $Talk cancelled due to MIT closure in response to corona virus.
We ask when embedding calculus can distinguish pairs of smooth manifolds that are homeomorphic but not diffeomorphic. We prove that, in dimension 4, the answer is “almost never.” In contrast, we exhibit an infinite list of highdimensional exotic spheres detected by embedding calculus. The former result implies that the algebraic topology of knot spaces is insensitive to smooth structure in dimension 4, answering a question of Viro. The latter result gives a partial answer to a question of Francis and hints at the possibility of a new classification of exotic spheres in terms of a stratified obstruction theory applied to compactified configuration spaces. This talk is based on joint work in progress with Alexander Kupers.
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Gabe AngeliniKnoll (Freie Universität Berlin)
$\begingroup $The seminar will meet at 3:15 PM in 2131.
The chromatic redshift program of AusoniRognes suggests that algebraic Ktheory shifts chromatic height by one. In my talk, I will describe a computational approach to this program where chromatic height is measured by vanishing of Morava Ktheory. In particular, we see that the vanishing range of Morava Ktheory of topological periodic cyclic homology of a certain family of Thom spectra y(n) increases by one. We also prove that algebraic Ktheory preserves vanishing of Morava Ktheory for y(n), a result recently proven in parallel by LandMeierTamme by entirely different methods. Our theorem relies on a technical result about when commuting Morava Ktheory with a sequential limit is possible, which I will discuss. As second application of this technical result, we prove a higher chromatic height analogue of Mitchell’s theorem for truncated BrownPeterson spectra associated to a prime p and an integer n, which remains conditional for large primes p and integers n. This is based on joint work with J.D. Quigley and joint work with A. Salch.
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Allen Yuan (MIT)
$\begingroup $Algebraic topology is the study of spaces via algebraic invariants. Given such an invariant of a space $X$, one can ask: how much of $X$ is captured by that invariant? For instance, can one recover $X$ itself (up to homotopy)?
This question was first addressed in work of Quillen and Sullivan on rational homotopy theory in the 1960's and in work of DwyerHopkins and Mandell on $p$adic homotopy theory in the 1990's. They showed that various algebraic enhancements of the notion of \emph{cohomology} allow one to recover various approximations to a space $X$, such as its \emph{rationalization} or \emph{$p$completion}.
In this thesis, we describe how to unify these ideas and recover a space in its entirety, rather than up to an approximation, using deeper invariants. The approach is centered around an insight of Nikolaus and Scholze, who demonstrate that the classical Frobenius endomorphism for rings in characteristic $p$ naturally generalizes to a phenomenon in higher algebra (more precisely, for $\E_{\infty}$ring spectra), which we call the \emph{higher Frobenius}. Our main result is that there is an action of the circle group on (a certain subcategory of) $p$complete $\E_{\infty}$rings whose monodromy is the higher Frobenius. Using this higher Frobenius action, we give a fully faithful model for a simply connected finite complex $X$ in terms of Frobeniusfixed $\E_{\infty}$rings.
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Florian Naef (MIT)
$\begingroup $Given a manifold M, Chas and Sullivan construct a Lie bialgebra structure on the homology of the space of \(unparametrized\) loops using intresections and selfintersections of loops. We give an algebraic description of this structure under Chen's isomorphism identifying loop space homology with cyclic homology. More precisely, we construct a homotopy involutive Lie bialgebra structure on cyclic cochains that depends on the partition function of a ChernSimons type field theory. Moreover, we discuss the (non)homotopy invariance of that structure and its relation to the configuration space of two points.
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Kathryn Hess (École Polytechnique Fédérale de Lausanne)
$\begingroup $(Joint work with Magdalena Kedziorek.) In earlier work with Beaudry, Merling, and Stojanoska, we established a formal framework for homotopical Galois theory of commutative algebras in a monoidal model category, generalizing the Galois theory developed by Rognes for commutative ring spectra, which was itself inspired by that for commutative rings. We showed that it applied, in particular, to motivic spaces and spectra, providing examples of motivic Galois extensions with no classical counterpart.
In this talk I will first recall this general framework and its key properties (invariance under base change, one direction of the Galois correspondence) and establish its invariance under reasonable changes of underlying monoidal category. I will then explain how to combine this formal homotopical Galois theory with Sagave and Schlichtkrull’s $I$space approach to $E_{\infty}$algebras, establishing a Galois theory of extensions of $E_{\infty}$algebras. I will describe the properties of this theory and provide examples of Galois extensions of $E_{\infty}$algebras.
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Agnes Beaudry (University of Colorado Boulder)
$\begingroup $In this talk, I will present certain Real oriented models of LubinTate theories at p=2 and arbitrary heights. For these models, we give explicit formulas for the action of some finite subgroups of the Morava stabilizer groups on the coefficient rings. This is an input necessary for future computations of the homotopy fixed point spectral sequences for the associated higher real Ktheories. The Real orientations will provide information about the differentials. The construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory, and are based on techniques introduced by HillHopkinsRavenel. This is joint work with Hill, Shi and Zeng.
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Sander Kupers (Harvard University)
$\begingroup $I will explain joint work with Oscar RandalWilliams on recent progress in computing the rational homotopy type of the topological group of diffeomorphisms of evendimensional disks, and situate these results in the context of previously known results and what we expect the answer to be.
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Dylan Wilson (Harvard University)
$\begingroup $We'll discuss joint work with Jeremy Hahn surrounding the 'norm' construction of HillHopkinsRavenel in equivariant homotopy theory. In particular, we'll talk about how to use variants of Hochschild homology to make some computations of homotopy groups of norms. In the case of the $C_2$ equivariant norm of the mod 2 EilenbergMacLane spectrum this relates to some classical questions, and we'll show how our method gives a new proof of the Segal conjecture for the group $C_2$.
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Nir Gadish (MIT)
$\begingroup $Hyperplane arrangements are a classical meeting point of topology, combinatorics and representation theory. Generalizing to arrangements of linear subspaces of arbitrary codimension, the theory becomes much more complicated. However, a crucial observation is that many natural sequences of arrangements seem to be defined using a finite amount of data. In this talk I will describe a notion of 'finitely generation' for diagrams of arrangements, unifying the treatment of known examples. Such collections turn out to exhibit strong forms of stability, both in their combinatorics and in their cohomology representation. This structure makes the appearance of 'representation stability' transparent and opens the door to generalizations.
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Peter Patzt (Purdue University)
$\begingroup $The level $p$ principal congruence subgroup of $SL_n(Z)$ is defined to be the subgroup of matrices congruent to the identity matrix mod $p$. These groups have trivial cohomology in high enough degrees. In the 1970s, Lee and Szczarba gave a conjectural description of the top dimensional cohomology groups of these congruence subgroups. In joint work with Miller and Putman, we resolve this conjecture by proving it for $p=2,3,5$ ($p=3$ was known) and disproving it for larger primes by finding more cohomology than conjectured. In particular, we compute the top dimensional cohomology of these groups for $p=2,3,5$ and we find new exotic cohomology classes for $p$ at least 7 coming from the first homology group of the associated compactified modular curve.
I will also discuss joint work with Miller and Nagpal on a stability pattern in the high dimensional cohomology of congruence subgroups.
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Inbar Klang (Columbia)
$\begingroup $Let $C_n$ denote the cyclic group of order $n$. Given a $C_n$ring spectrum, Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell defined its $C_n$relative topological Hochschild homology. Just as Hochschild homology is an algebraic approximation to topological Hochschild homology, this has an algebraic approximation in the form of Hochschild homology for Green functors, defined by Blumberg, Gerhardt, Hill, and Lawson. I will introduce these concepts and discuss joint work with Adamyk, Gerhardt, Hess, and Kong in which we develop a theoretical framework and computational tools for these Hochschild homology theories.
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Paolo Salvatore (Università di Roma Tor Vergata)
$\begingroup $We construct a cellular decomposition of the AxelrodSingerFultonMacPherson compactification of the configuration spaces in the plane, that is compatible with the operad composition. Cells are indexed by trees with bicoloured edges, and vertices are labelled by cells of the cacti operad. This answers positively a conjecture by Kontsevich and Soibelman.
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Kyle Ormsby (Reed College)
$\begingroup $The seminar will meet at 3:15 PM in 2131.
Infamously, the motivic Hopf map $\eta$ is nonnilpotent, and the $\eta$periodic motivic sphere spectrum detects this phenomenon. I will describe a slicetheoretic approach to this object, resulting in a computation of the $E_2$page of the $\eta$periodic slice spectral sequence. This permits a complete computation of the homotopy groups of the $\eta$periodic sphere when $1$ is a sum of four squares in the base field. We also prove that, for a general field $k$, this spectral sequence is determined by the $\eta$periodic slice spectral sequence over $R$. Some delicate convergence problems obstruct the final resolution of this problem, and the audience is invited to solve these. Our computations lead to a conjecture about the $\eta$periodic sphere as a "connective Witttheoretic $J$spectrum," and also promise to help with slice computations of the motivic sphere. This is joint work with Oliver Röndigs.
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J.D. Quigley (Cornell)
$\begingroup $(Joint work with Dominic Culver) Let $kq$ denote the very effective cover of Hermitian Ktheory. The $kq$based motivic Adams spectral sequence, or $kq$resolution, is a motivic analog of Mahowald’s $bo$resolution. We applied the $kq$resolution in the $C$motivic setting to calculate the $\eta$periodic stable stems (recovering results of Andrews, Guillou, Isaksen, and Miller) and $v_1$periodic stable stems. I will summarize these calculations and discuss work in progress towards analogous results over general base fields and in the $C_2$equivariant setting. In addition to largescale calculations of periodic phenomena, the $kq$resolution can be used to calculate lowdimensional $2$complete MilnorWitt stems. I will present some calculations in the $C$ and $R$motivic settings (recovering results of DuggerIsaksen), then discuss work in progress towards analogous results over general base fields and the $2$complete version of calculations of RöndigsSpitzweckØstvaer.
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Elden Elmanto (Harvard University)
$\begingroup $A universal homeomorphism of schemes is a morphism of schemes (frighteningly, with no finiteness conditions) which is a homeomorphism on underlying topological spaces (!) and remains so after arbitrary pullback. In some sense, beginning with at least Grothendieck, one should view a universal homeomorphism as analogous to the topologists’ notion of a "weak equivalence." In this talk, I will explain this philosophy, give examples, and explain two results about the behavior of algebraic Ktheory under universal homeomorphisms. The first is an equicharacteristic result: if $k$ is a field of exponential characteristic $c$, then homotopy Ktheory is invariant under universal homeomorphisms of kschemes, after inverting $c$; this is a consequence of joint work with Adeel Khan and implies the statement for usual Ktheory in positive characteristics. The second is a mixed characteristic result for usual (but $p$inverted) Ktheory: over $Z_p$ schemes, universal homeomorphisms are completely controlled by its "characteristic zero part." The formulation of the theorem, and its proof, is inspired by some results in mixed characteristic birational geometry. This is joint work in progress with Akhil Mathew and Jakub Witaszek.
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Dev Sinha (University of Oregon)
$\begingroup $(joint work with Lorenzo Guerra and Paolo Salvatore) Unstably, the extended powers on a space $D_n X$ are the homotopy orbits of its cartesian product $X^n$ with respect to their standard symmetric group actions. Extended powers have played a prominent role in work of Steenrod, Nishida and the Chicago school, and are essential in defining derived commutativity. By the BarrattPriddy(QuillenSegal) theorem, they are intimately related to $QX$, the free infinite loopspace on a space $X$.
While the modp homology of these constructions as well as a formulae for coproduct have been known since work of Nakaoka, KudoArakai, DyerLashof, and CohenLadaMay, calculations require application of Adem relations as well as dualization, so applications of explicit calculations in ring structure have been few. But we have found that a Hopf ring structure developed by Strickland and Turner provides the right framework for understanding cohomology rings of symmetric groups (the case when $X$ is a point).
In this talk, I will review this previous work on cohomology of symmetric groups, and then discuss the general case, where we also develop a divided powers structure. Just as the homology is a free object, namely the free algebra over the DyerLashof algebra on the homology of $X$, the cohomology is the free divided powers Hopf ring on the cohomology of $X$. A twist in the story is that at odd primes one must take coefficients in the direct representation and the sign representation to obtain a free answer. From extended powers we calculate ring structure for $QX$, and we understand Steenrod action from this viewpoint as well.
We envision many applications, ranging from cohomology of stable mapping class groups to calculations in Goodwillie calculus to revisiting Wellington’s calculation of the unstable Adams spectral sequence for $QS^0$ to making chromatic analogues of these calculations. While there will not be time for such musings and small bits of progress, audience members are invited to chat with me over the coming year.
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Tom Bachmann (MIT)
$\begingroup $(Report on work in progress, joint with Mike Hopkins.)
The geometric Hopf map $A^2\setminus 0\rightarrow P^1$ induces (after desuspension) a wellknown stable map $\eta\colon S^{1,1}\rightarrow S^0$. Contrary to the classical situation, in motivic homotopy theory, this map is *not* nilpotent. It is thus natural to study the etaperiodic motivic stable homotopy category; this is particularly interesting/difficult locally at the prime 2.
We show that the 2local, $\eta$periodic sphere is the fiber of an Adams operation in the 2local, connective Witt theory spectrum. As a consequence we recover and extend computations of Andrews, Guillou, Isaksen, Ormsby, Miller, Röndigs, and Wilson, and also compute the homotopy groups of etaperiodic algebraic special linear cobordism.
I will provide an overview of these ideas and results.
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Jeremy Hahn (MIT)
$\begingroup $I will survey the problem of classifying smooth, $(n1)$connected, closed ($2n$)manifolds, at least in large dimensions $2n>248$. The classification up to diffeomorphism was first attempted in a 1962 paper of C.T.C. Wall, where it was related to questions about the boundaries of "almost closed" manifolds. Several of these questions were answered in the 70s and 80s, most notably by Stephan Stolz, but for example the Kervaire Invariant 1 question remained unresolved until 2009.
I will explain work in progress, joint with Robert Burklund, Tyler Lawson, and Andrew Senger, proving for $n>124$ that the boundary of any ($n1$)connected, almost closed ($2n$)manifold also bounds a parallelizable manifold. In large dimensions this solves the last of Wall's original questions about his boundary homomorphism, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and RandalWilliams.
Work of Galatius, RandalWilliams, and Krannich relates our theorem not just to the classification of ($n1$)connected ($2n$)manifolds, but also to the calculation of their mapping class groups.
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Jay Shah (University of Notre Dame)
$\begingroup $Let G be a finite group and X a topos with homotopy coherent Gaction. We construct a stable homotopy theory $Sp^G(X)$ which recovers and extends the theory of genuine Gspectra. We explain what our construction yields when: (i) X is the topos of sheaves on a topological space with Gaction (ii) X is the etale $C_2$topos of a scheme S adjoined a square root of 1. We conclude with an application to realization functors out of the stable motivic homotopy category of a scheme. This is joint work with Elden Elmanto.
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Daniel Ruberman (Brandeis University)
$\begingroup $Tools of gauge theory have been used since the 1980s to produce exotic 4manifolds—smooth manifolds that are homeomorphic but not diffeomorphic. In principle, the same circle of ideas can be used to study the diffeomorphism group of a 4manifold, but concrete examples and calculations are hard to come by. I will give examples of elements of infinite order in the fundamental group of the diffeomorphism group of some basic simply connected 4manifolds. This is joint work with Dave Auckly.
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Lyuboslav Panchev (MIT)
$\begingroup $The seminar will meet at 3:15 PM in 2131.
We present progress in trying to verify a longstanding conjecture by Mark Mahowald on the $v_1$periodic component of the classical Adams spectral sequence for a Moore space M. The approach we follow was proposed by John Palmieri in his work on the stable category of Acomodules. We improve on Palmieri's work by working with the endomorphism ring of M  End(M)  thus resolving some of the initial difficulties of his approach and formulating a conjecture of our own that would lead to Mahowald's formulation. We further improve upon a method for calculating differentials via double filtration first used by Miller and apply it to our problem.
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Lukas Brantner (Oxford/MSRI)
$\begingroup $If k is a field of characteristic zero, a theorem of Lurie and Pridham (based on previous work of Deligne, Drinfel'd, Feigin, Hinich, and others) establishes an equivalence between formal moduli problems and differential graded Lie algebras over k. We generalise this equivalence to finite and mixed characteristic by using “partition Lie algebras”. These mysterious new gadgets are intimately related to the genuine equivariant topology of the partition complex, which allows us to access the operations acting on their homotopy groups (relying on earlier work of DyerLashof, Priddy, Goerss, and AroneB.). This is joint work with Akhil Mathew.
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John Berman (UT Austin)
$\begingroup $The Euler characteristic is a beloved invariant of spaces which are finite in homology. On the other hand, Baez and Dolan’s homotopy cardinality is an invariant of spaces which are finite in homotopy, with applications from group theory to mathematical physics. Baez asks whether these invariants are two faces of the same coin. We will answer Baez’s question by constructing an invariant on a class of pprofinite spaces which unifies the Euler characteristic and homotopy cardinality, and we also show that there can be no such invariant without restricting attention to a single prime. The construction uses the Sullivan Conjecture to compute the algebraic Ktheory of a category of ‘spaces with rational Euler characteristic.’
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Donald Yau (Ohio State University)
$\begingroup $I will discuss a homotopical version of the adjoint lifting theorem in category theory, allowing for Quillen equivalences to be lifted from monoidal model categories to categories of algebras over colored operads. The generality of this approach allows us to simultaneously answer questions of rectification and of changing the base model category to a Quillen equivalent one. Special cases of our main theorem recover many known results regarding rectification and change of model category, as well as other new results. In particular, we recover a recent result of RichterShipley about a zigzag of Quillen equivalences between commutative HQalgebra spectra and commutative differential graded Qalgebras, but our version involves only three Quillen equivalences instead of six. This is based on joint work with David White.
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Mona Merling (University of Pennsylvania)
$\begingroup $The stable parametrized hcobordism theorem provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. For a compact smooth manifold M it gives a decomposition of Waldhausen's A(M) into $QM_+$ and a delooping of the stable hcobordism space of M. I will talk about joint work with Malkiewich on this story when M is a smooth compact Gmanifold.
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Manuel Krannich (University of Cambridge)
$\begingroup $The mapping class group of a highly connected almost parallelisable manifold of dimension $2n>4$ was computed by Kreck in the 70’s. His answer, however, left open two extension problems which were later understood in some dimensions, but remained unsettled in most cases. Motivated by renewed interest in these groups in relation to moduli spaces of manifolds, I will explain how to resolve the remaining extension problems for n odd, resulting in a complete description of these mapping class groups in terms of an arithmetic group and the group of exotic spheres. This involves a certain homotopy sphere which might be of particular interest to stable homotopy theorists as its image in coker(J) is still not understood, although omnipresent in the study of highly connected manifolds.
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Andrew Putman (University of Notre Dame)
$\begingroup $The Johnson filtration is an important and mysterious sequence of subgroups of the mapping class group. I will prove that each term is finitely generated once the genus is sufficiently large. The main tool is the BieriNeumannStrebel invariant. This is joint work with Tom Church and Mikhail Ershov.
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Alexander Kupers (Harvard University)
$\begingroup $I will discuss joint work with Oscar RandalWilliams, in which we compute the rational cohomology of the Torelli spaces of highlyconnected highdimensional manifolds in a stable range. These Torelli spaces are the classifying spaces of the subgroup of the topological group of diffeomorphisms consisting of those path components which act by the identity on homology, and bring us one step closer to understanding the rational homotopy type of diffeomorphism groups.
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Kathryn Lesh (Union College)
$\begingroup $I will discuss our current state of knowledge about the U(n)equivariant homotopy type of $L_{n}$, the space of proper decompositions of complex nspace. In particular, I will describe a recent result (joint with Arone and Dwyer) describing the Bredon homology and cohomology of $L_{n}$ with coefficients in a plocal Mackey functor. It turns out to be the same as a point when n is not a power of p, and is approximated by the symplectic Tits building when n is a power of p. The method uses a general approximation theory for a space with an action of a compact Lie group, by ptoral subgroups. The space $L_{n}$ first arose in the Weiss tower for the functor taking a vector space V to BAut(V), but has made other appearances, and is an important ingredient in the bu Whitehead Conjecture.
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Andrew Blumberg (UT Austin)
$\begingroup $The seminar will meet at 4:30 PM in 2190.
This talk is part of the Brandeis–Harvard–MIT–Northeastern joint Mathematics Colloquium
Nearly 40 years ago, Bousfield and Ravenel noticed that the operations on padic complex Ktheory are abstractly closely related to the Iwasawa algebra. In the mid90's, Dwyer and Mitchell used this observation to describe the topological Ktheory of the algebraic Ktheory of Z in terms of padic Lfunctions. In this talk, I will give a gentle introduction to this amazing story and then explain work with Mike Mandell which extends these results and suggests a program for using topological information to answer numbertheoretic questions.
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Irina Bobkova (Institute for Advanced Study)
$\begingroup $For Morava $E$theory $E_n$ and a finite subgroup $F$ of the Morava stabilizer group, the spectrum $E_n^{hF}$ is periodic and the Picard group of the category of modules over the ring spectrum $E_n^{hF}$ contains the cyclic subgroup generated by $\Sigma E_n^{hF}$. In most known examples, the Picard group is found to be precisely this cyclic group. However, at chromatic height $n=2$ and $p=2$, the Picard group of the category of $E_2^{hC_4}$modules is not cyclic and contains an extra element of order 2. I will describe the tools we use to compute this Picard group: a group homomorphism from $RO(C_4)$ to it and the Picard spectral sequence. This talk is based on joint work with Agnes Beaudry, Mike Hill and Vesna Stojanoska.
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Arnav Tripathy (Harvard University)
$\begingroup $A longstanding question in the study of elliptic cohomology or topological modular forms has been the search for geometric cocycles. I will explain joint work with D. BerwickEvans which turns Segal's physicallyinspired suggestions into rigorous cocycles for the case of equivariant elliptic cohomology over the complex numbers, with some focus on the role of supersymmetry on allowing for the possibility of rigorous mathematical definition. As time permits, I hope to indicate towards the end how one might naturally extend these ideas to higher genus.
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Nathaniel Stapleton (University of Kentucky)
$\begingroup $We will discuss a question about the functoriality of certain evaluation maps for classifying spectra of finite groups that arose when thinking about questions related to chromatic homotopy theory. I will describe an algebraic solution to this problem found in joint work with Reeh, Schlank. If time permits, I hope to describe a related question regarding idempotents of $p$groups in the $K(n)$local category.
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Emily Riehl (Johns Hopkins University)
$\begingroup $The pioneering work of Joyal, Lurie, et al to extend ordinary category theory to the setting of $\infty$categories is "analytic", with the precise statements of theorems given in reference to a particular model (quasicategories) and proofs drawing on the combinatorics of simplicial sets. This talk will describe joint work with Dominic Verity that reveals that much of that theory can be redeveloped "synthetically" in an axiomatic framework that is natively "modelindependent", casting new light on the theory of quasicategories while simultaneously generalizing it to other models. At the conclusion, we consider the question of the modelinvariance of $\infty$category theory, proving that $\infty$categorical structures are preserved, reflected, and created by a number of "changeofmodel" functors. As we explain, it follows that even the "analyticallyproven" theorems that exploit the combinatorics of one particular model remain valid in the other "biequivalent" models.
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Dan Isaksen (Wayne State University)
$\begingroup $I will describe various computational aspects of motivic and equivariant stable homotopy groups. The discussion will include some techniques for brute force computations, as well as some larger structural phenomena uncovered through these calculations.
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Jianfeng Lin and Zhouli Xu (MIT)
$\begingroup $The seminar will meet at 3:30 PM in 4153.
A fundamental problem in $4$dimensional topology is the following geography question: "which simply connected topological $4$manifolds admit a smooth structure?" After the celebrated work of KirbySiebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "$11/8$Conjecture". This conjecture, proposed by Matsumoto, states that for any smooth spin $4$manifold, the ratio of its secondBetti number and signature is least $11/8$.
Furuta proved the "$10/8+2$"Theorem by studying the existence of certain $Pin(2)$equivariant stable maps between representation spheres. In this talk, we will present a complete solution to this problem by analyzing the $Pin(2)$equivariant Mahowald invariants. In particular, we improve Furuta's result into a "$10/8+4$"Theorem. Furthermore, we show that within the current existing framework, this is the limit. This is joint work with Mike Hopkins and XiaoLin Danny Shi.
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Jianfeng Lin and Zhouli Xu (MIT)
$\begingroup $The seminar will meet at 4:30 PM in 4153.
The existence of $Pin(2)$equivariant stable maps between representation spheres has deep applications in $4$dimensional topology. We will sketch the proof for our main result on the $Pin(2)$equivariant Mahowald invariants.
More specifically, we will discuss the $Pin(2)$equivariant Mahowald invariants of powers of certain Euler classes in the $RO(Pin(2))$graded equivariant stable homotopy groups of spheres. The proof analyzes maps between certain finite spectra arising from $BPin(2)$ and various Thom spectra associated with it. To analyze these maps, we use the technique of celldiagrams, known results on the stable homotopy groups of spheres, and the $j$based AtiyahHirzebruch spectral sequence.
This is joint work with Mike Hopkins and XiaoLin Danny Shi. This talk provides the homotopy theory required by the first talk but will not depend upon it.
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Ben Knudsen (Harvard University)
$\begingroup $I will report on a program of research aimed at computing stable multiplicities of symmetric group representations in the cohomology of the ordered configuration spaces of the torus. Our approach is premised on a multiplicative decomposition of configuration spaces of product manifolds in terms of a Boardman–Vogt tensor product of operadic modules. The decomposition gives rise to a "Kuenneth" spectral sequence with second page identifiable in purely algebraic terms. The spectral sequence collapses in characteristic zero, reducing such computations to problems in the complex representation theory of certain combinatorial categories. This work is joint with W. Dwyer and K. Hess.
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Dylan Wilson (University of Chicago)
$\begingroup $Complex conjugation gives rise to many natural $C_2$ actions on objects in homotopy theory (projective spaces, $K$theory, complex cobordism, etc.) Recent work on homotopy theory at the prime $2$ has exploited these actions to great effect. We discuss work in progress, joint with Jeremy Hahn, that tries to answer the following questions: Is there an odd primary analogue of complex conjugation? If there is, can we use it? We describe some potential answers to this question as well as the current status of a program, initiated by HillHopkinsRavenel, for applying $C_p$ equivariant homotopy theory to the study of Morava $E$theory.
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Prasit Bhattacharya (University of Virginia)
$\begingroup $Constructing type n spectrum and understanding its periodic $v_n$selfmap is an integral part chromatic homotopy theory, potentially can produce infinite families in chromatic layer $n$ of the stable homotopy groups of spheres. Recently, in a joint work with Philip Egger, we produce $2$local type $2$ spectra $Z$ which admit $v_2^1$selfmap. In fact, $Z$ in many ways the height 2 analogue of the spectrum $Y:= M_2(1) \wedge C\eta$. In a joint work with Nicolas Ricka, we extend the family consisting of $Y$ and $Z$ to chromatic height 3, by producing a type $3$ spectrum at the prime $2$. In this talk, I will explain multiple ways of constructing this spectrum, one of them uses the technique of Jeff Smith that is used in the proof of the famous 'thick subcategory theorem’ due to Hopkins and Smith. Then I will outline the proof of $v_3^4$selfmap and give evidences in favor of this spectrum admitting $v_3^1$selfmap. Time permitting, I will also talk about the action of height $3$ Morava stabilizer group on Morava Etheory of this spectrum and give a complete answer for the $E_2$page of the descent spectral sequence which computes $K(3)$local homotopy groups of this spectrum.
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Tom Bachmann (MIT)
$\begingroup $In joint work with M. Hoyois, we established (the beginnings of) a theory of "normed motivic spectra". These are motivic spectra with some extra structure, enhancing the standard notion of a motivic $E_\infty$ring spectrum (this is similar to the notion of $G$commutative ring spectra in equivariant stable homotopy theory). It was clear from the beginning that the homotopy groups of such normed spectra afford interesting power operations. In ongoing joint work with E. Elmanto and J. Heller, we attempt to establish a theory of these operations and exploit them calculationally. I will report on this, and more specifically on our proof of a weak motivic analog of the following classical result of Würgler: any homotopy commutative ring spectrum with $2=0$ is generalized EilenbergMacLane.
$\endgroup$ 
Jeremy Hahn (MIT)
$\begingroup $The space BSU admits two infinite loop space structures, one arising from addition of vector bundles and the other from tensor product. A surprising fact, due to Adams and Priddy, is that these two infinite loop spaces become equivalent after $p$completion. I will explain how the AdamsPriddy theorem allows for an identification of $sl_1(ku_p)$, the spectrum of units of $p$complete complex $K$theory. I will then describe work, joint with Andrew Senger, that attempts to similarly understand the spectrum of units of the 2completion of $tmf_1(3)$. Our computations seem suggestive of broader phenomena, and I hope to end with several open questions.
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Akhil Mathew (University of Chicago)
$\begingroup $The cyclotomic trace from algebraic $K$theory to topological cyclic homology (TC) is an important tool in describing the $p$adic $K$theory of $p$adic rings, and has been used in several computations (such as HesselholtMadsen's computation of the $K$theory of local fields). TC, while more complicated to define, has somewhat simpler formal properties than $K$theory and is as a result often easier to compute. I will describe some new results to the effect that TC is a strong approximation to $p$adic $K$theory, and some resulting structural consequences for $p$adic $K$theory. This is joint work with Dustin Clausen and Matthew Morrow.
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Hiro Tanaka (Harvard University)
$\begingroup $I'll talk about my attempts at applying modern homotopytheoretic techniques to study (i) the theory of Lagrangian cobordisms, and (ii) its applications to Lagrangian Floer theory and Fukaya categories. It's a tantalizing world where Thom spaces aren't the obvious doorway to cobordism spectra, where there seem to be geometric explanations for really rich algebraic structures (like the paracyclic action on the sdot construction), and where (frankly) not nearly enough is known.
$\endgroup$ 
Jeremiah Heller (University of Illinois UrbanaChampaign)
$\begingroup $The seminar will meet at 3:00 PM in 2236.
One formulation of Segal's conjecture is as a comparison between fixed points and homotopy fixed points of the sphere, for a finite group. I'll discuss a generalization for the motivic sphere, based on joint work with several collaborators.
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Vesna Stojanska (University of Illinois UrbanaChampaign)
$\begingroup $I will discuss a formal framework for homotopical Galois extensions, and consider a couple of different adaptations in the context of motivic homotopy theory. Each will be accompanied with interesting basic examples, including EilenbergMacLane and Ktheory spectra. This is based on work with Beaudry, Heller, Hess, Kedziorek, and Merling.
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Eva Belmont (MIT)
$\begingroup $Chromatic localization can be seen as a way to calculate a particular infinite piece of the homotopy of a spectrum. For example, the (finite) chromatic localization of a $p$local sphere is its rationalization, and the corresponding chromatic localization of its Adams $E_2$ page recovers just the zerostem. We study a different localization of Adams $E_2$ pages for spectra, which recovers more information than the chromatic localization. This approach can be seen as the analogue of chromatic localization in a category related to the derived category of comodules over the dual Steenrod algebra, a setting in which Palmieri has developed an analogue of chromatic homotopy theory. We work at $p=3$ and compute the $E_2$ page and first nontrivial differential of a spectral sequence converging to $b_{10}^{1}Ext_P^*(\mathbb{F}_3,\mathbb{F}_3)$ (where $P$ is the Steenrod reduced powers), and give a complete calculation of other localized Ext groups, including $b_{10}^{1}Ext_P^*(\mathbb{F}_3,\mathbb{F}_3[\xi_1^3])$.
$\endgroup$ 
David Gepner (Purdue University)
$\begingroup $No abstract
$\endgroup$ 
Gijs Heuts (Utrecht University)
$\begingroup $I will discuss a homotopy theory obtained from that of pointed spaces by inverting the maps inducing isomorphisms in $v_n$periodic homotopy groups. The case $n = 0$ corresponds to rational homotopy theory. In analogy with Quillen’s results in the rational case, I will outline how this $v_n$periodic homotopy theory is equivalent to the homotopy theory of Lie algebras in $T(n)$local spectra. One can also compare it to the homotopy theory of cocommutative coalgebras in $T(n)$local spectra. For $n > 0$ these theories are no longer equivalent; the failure can be expressed in terms of the convergence of the Goodwillie tower of the identity in periodic homotopy.
$\endgroup$ 
Sam Nariman (Northwestern University)
$\begingroup $There are at least three different approaches to construct characteristic invariants of flat symplectic bundle. Reznikov generalized ChernWeil theory for finite dimension Lie groups to the infinite dimensional group of symplectomorphisms. He constructed nontrivial invariants of symplectic bundles whose fibers are diffeomorphic to complex projective spaces. Kontsevich used formal symplectic geometry to build interesting classes that are not yet known to be nontrivial. For surface bundles whose holonomy groups preserve the symplectic form, Kotschick and Morita used the flux homomorphism to construct many nontrivial stable classes.
In this talk, we introduce infinite loop spaces whose cohomolgy groups describe the stable characteristic invariants of symplectic surface bundles. As an application, we give a homotopy theoretic description of Kotschick and Morita's classes and prove a result about codimension 2 foliations that implies the nontriviality of KM classes.
$\endgroup$ 
Maria Basterra (University of New Hampshire)
$\begingroup $An earlier result of Tillmann shows that a carefully constructed surface operad detects infinite loop spaces in the sense that its algebras group complete to infinite loop spaces. An important ingredient in this result is Harer's homological stability theorem for mapping class groups of surfaces.
Motivated by this example, we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions (OHS). We prove that such operads detect infinite loop spaces in the sense above. In fact, we show that for these operads group completion provides a functor from their category of algebras to the category of infinite loop spaces.
I will sketch the proof of this result and, as time permits, present some examples and applications.
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Mark Behrens (University of Notre Dame)
$\begingroup $I will show that the $C_2$equivariant EilenbergMacLane spectrum associated to the constant mod 2 Mackey functor can be realized as a Thom spectrum. This is joint work with Dylan Wilson.
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Alex Suciu (Northeastern University)
$\begingroup $I will discuss some of the interplay between duality and finiteness properties of spaces and groups, the structure of differential graded algebra models associated to them, and the geometry of the corresponding cohomology jump loci.
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Bena Tshishiku (Harvard University)
$\begingroup $A basic problem in the study of fiber bundles is to compute the ring $H^*(BDiff(M))$ of characteristic classes of bundles with fiber a smooth manifold M. When M is a surface, this problem has ties to algebraic topology, geometric group theory, and algebraic geometry. Currently, we know only a very small percentage of the total cohomology. In this talk I will explain some of what is known and discuss some new characteristic classes (in the case dim M >>0) that come from the unstable cohomology of arithmetic groups.
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Julie Bergner (University of Virginia)
$\begingroup $The notion of unital $2$Segal space was defined independently by DyckerhoffKapranov and GalvezCarrilloKockTonks as a generalization of a category up to homotopy. The notion of unital 2Segal space was defined independently by DyckerhoffKapranov and GalvezCarrilloKockTonks as a generalization of a category up to homotopy. A key example of both sets of authors is that the output of applying Waldhausen's $S$construction to an exact category is a unital $2$Segal space. In joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we expand the input of this construction to augmented stable double Segal spaces and prove that it induces an equivalence on the level of homotopy theories. Furthermore, we prove that exact categories and their homotopical counterparts can be recovered as special cases of augmented stable double Segal spaces.
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Michael Ching (Amherst College)
$\begingroup $It is a wellknown heuristic that the Goodwillie derivatives of the identity functor on a (pointed compactlygenerated) $∞$category should form an operad. In this talk I will give a simple construction of such an operad structure that provides new insight even in the familiar case of based spaces. It is easy to see too that the derivatives of an arbitrary functor form a bimodule over the relevant operads. Moreover, these models allow us to express the derivatives of the identity on an $∞$category $C$ as the inverse limit of a "prooperad" and we conjecture that the Taylor towers of functors from $C$ to spectra are classified by "modules" over that prooperad, generalizing results with Arone in the cases where $C$ is based spaces or spectra.
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Kathryn Mann (Brown University)
$\begingroup \def\Homeo{\operatorname{Homeo}} \def\Diff{\operatorname{Diff}} $A flat $M$bundle is a topological (or smooth) $M$bundle with a foliation transverse to the fibers; these are classified by the classifying spaces for $\Homeo(M)$ or $\Diff(M)$ with the discrete topology. In this talk, I will describe an alternative approach to their study, introducing nonlinear analogs of character varieties for representations of discrete groups to groups of homeomorphisms or diffeomorphisms. Our motivation comes from the case of $M=S^1$, where character spaces have a natural interpretation through dynamical invariants of group actions.
In new joint work with M. Wolff, we use this perspective to characterize the isolated points of the character space for representations of surface groups into $\Homeo(S^1)$. Remarkably, these points are precisely the geometric representations, coming from an embedding of the group into a lattice in a Lie subgroup of $\Homeo(S^1)$. This gives a new instance of the classical dynamical theme of rigidity of lattice actions, and progress towards an $h$principle for transversely projective codimension $1$ foliations.
$\endgroup$ 
Doug Ravenel (University of Rochester)
$\begingroup $Reporting on joint work with Zhouli Xu and 4 others, I will talk about how equivariant techniques MIGHT be used to settle the telescope conjecture at the prime $2$.
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Sander Kupers (Harvard University)
$\begingroup $We discuss joint work with Søren Galatius and Oscar RandalWilliams on the application of higheralgebraic techniques to classical questions about the homology of automorphism groups such as mapping class groups and general linear groups. This uses a new "multiplicative" approach to homological stability – in contrast to the "additive" one due to Quillen – which has the advantage of providing information outside of the stable range.
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Ben Knudsen (Harvard University)
$\begingroup $We develop an approach to the study of the configuration spaces of a cell complex $X$ that is both flexible and suitable for computation. We proceed by viewing $X$, together with its subdivisions, as a "subdivisional space," a kind of diagram object, which has associated to it certain diagrammatic versions of configuration spaces. These objects, which model the correct homotopy types, mix the discrete and the continuous, and they may be attacked by combining techniques drawn from discrete Morse theory and factorization homology. We apply our theory in the $1$dimensional example of a graph, obtaining an enhanced version of a family of chain models for graph braid groups originally studied by Swiatkowski. These complexes come equipped with a robust computational toolkit, which we exploit in numerous calculations, old and new. This is joint work with Byung Hee An and Gabriel DrummondCole.
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Dylan Wilson (University of Chicago)
$\begingroup $Motivated by a program of HillHopkinsRavenel to prove the $3$primary Kervaire invariant problem, we are led to study a slight generalization of (induced) representation spheres called slice spheres. These are simple to define: a compact $G$spectrum is a slice sphere if each of its geometric fixed points is a finite wedge of spheres. These objects appear fundamental in carrying over techniques from the case $G=C_2$ to other groups, such as $C_p$, and the goal of the talk is to give various examples of how they show up in interesting settings. As a general application, we describe how one may compute the $n$slice of a spectrum using slice spheres. Then we specialize to the case $G=C_p$ and discuss ongoing work on the $3$primary Kervaire invariant problem, following a program of HillHopkinsRavenel. We describe several methods of attack, and our current partial results. These include an equivariant refinement of $\alpha_1$, some information about the $C_p$equivariant dual Steenrod algebra, and a computation of the $C_3$slices of topological modular forms with full level $2$ structure.
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David Ayala (Montana State University)
$\begingroup $The majority of this talk will examine the Bruhat stratified orthogonal group:
 The Bruhat cells of the general linear group assemble as a combinatorial stratification of the orthogonal group.
 Compatibility of this stratification with matrix multiplication can be articulated as an associative algebra structure on its exitpath category in a certain Morita category of categories.
 Articulated as so, there is an action of this Bruhat stratified orthogonal group $O(n)$ on the category of $n$categories; this action is given by adjoining adjoints.
 This results in a continuous action of the topological group $O(n)$ on the category of $n$categories with adjoints.
The last point is a ket input into a proof of the Cobordism Hypothesis using factorization homology – this context will be discussed.
This is joint work with John Francis.
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Craig Westerland (University of Minnesota)
$\begingroup $In 2002, Malle formulated a conjecture regarding the distribution of number fields with specified Galois group. The conjecture is an enormous strengthening of the inverse Galois problem; it is known to hold for abelian Galois groups, but for very few nonabelian groups.
We may reformulate Malle's conjecture in the function field setting, where it becomes a question about the number of branched covers of the affine line (over a finite field) with specified Galois group. In joint work with Jordan Ellenberg and TriThang Tran, we have shown that the upper bound in Malle's conjecture does hold in this setting.
The techniques used involve a computation of the cohomology of the (complex points of the) Hurwitz moduli spaces of these branched covers. Surprisingly (at least to me), these cohomology computations can be rephrased in terms of the homological algebra of certain braided Hopf algebras arising in combinatorial representation theory and the classification of Hopf algebras. This relationship can be leveraged to provide the upper bound in Malle's conjecture.
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Lars Hesselholt (Nagoya University and University of Copenhagen)
$\begingroup $The seminar will meet at 4:30 PM in 2190.
This talk is part of the Brandeis–Harvard–MIT–Northeastern joint Mathematics Colloquium
This talk concerns a twentythousandyear old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the BökstedtHsiangMadsen topological cyclic homology, which receives a denominatorfree Chern character, and the related BhattMorrowScholze integral $p$adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to the cohomological interpretation of the HasseWeil zeta function by regularized determinants envisioned by Deninger.
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Nora Ganter (University of Melbourne)
$\begingroup $The notion of categorical groups, also known as $2$groups is increasingly recognised as the formalism governing symmetries in the context of string theory. The talk aims to fill this formalism with life by providing an interesting set of examples, together with a link to the stable three stem.
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Nitu Kitchloo (Johns Hopkins University)
$\begingroup $Real JohnsonWilson theories, $ER(n)$ are a family of cohomology theories generalizing ($2$local) real $K$theory, $KO$, which is $ER(1)$. They were first studied by HuKriz and later by KitchlooWilson. Real JohnsonWilson theories are defined as fixed points of an involution acting on the complexoriented JohnsonWilson theories $E(n)$, but they are themselves not complex oriented.
Real JohnsonWilson theories have proven to be remarkably useful, as well as computationally amenable. For example, their properties were exploited to demonstrate some new nonimmersions of real projective spaces into euclidean space. The main tool for computing real JohnsonWilson cohomology is a (Bocksteintype) spectral sequence that begins with $E(n)$cohomology and converges to $ER(n)$cohomology. We take advantage of the internal algebraic structure of this spectral sequence converging to $ER(n)^*(\operatorname{pt})$, to prove that for certain spaces $Z$ with Landweberflat $E(n)$cohomology, the cohomology ring $ER(n)^*(Z)$ can be obtained from $E(n)^*(Z)$ by a somewhat subtle base change. In particular, our results allow us to compute the Real JohnsonWilson cohomology of the EilenbergMacLane spaces $Z=K(\mathbb{Z},2m+1), K(\mathbb{Z}/2,m), K(\mathbb{Z}/2^q, 2m)$ for any integers $m$ and $q$, as well as connective covers of $ BO $: $ BO $, $ BSO $, $ BSpin $ and $ BO\langle 8 \rangle $.
This is joint work with Stephen W. Wilson and Vitaly Lorman.
$\endgroup$ 
Zhouli Xu (MIT)
$\begingroup $In this talk, I will review some classical methods computing stable homotopy groups of spheres, and report some recent progress using motivic homotopy theory.
Motivated by Dan Isaksen's work, Bogdan Gheorghe, Guozhen Wang and myself show that the category of motivic modules over the cofiber of $ \tau $ and the derived category of $BP_*BP$ comodules are equivalent as stable infinity categories. As a consequence, there is an isomorphism between the motivic Adams spectral sequence of the cofiber of tau and the algebraic Novikov spectral sequence, so one could prove classical Adams differentials through motivic methods. In joint work with Dan Isaksen and Guozhen Wang, we extend the $2$primary computation of stable stems to the 90's. If time permits, I will mention connections of this project to the Kervaire invariant problem and the new Doomsday conjecture.
$\endgroup$

John Francis (Northwestern University)
$\begingroup \def\TQFT{\mathop{\mathrm{TQFT}}}\def\obj{\mathop{\mathrm{obj}}} $The cobordism hypothesis—after Baez–Dolan, Costello, Hopkins–Lurie, and Lurie—asserts that for a symmetric monoidal $(\infty,n)$category $C$ in which every object has a dual and every $k$morphism has a left and right adjoint for $k<n$, there is an equivalence $\TQFT(C) = \obj(C)$ between $C$valued framed topological field theories and objects of $C$. This is the formulation due to Lurie. I'll give a proof of the cobordism hypothesis based on factorization homology. Factorization homology is a multiplicative analogue of ordinary homology. Usual homology integrates an abelian group, chain complex, or spectrum over a manifold $M$, which one can think as the moduli space of points in $M$ itself. The result takes disjoint unions of manifolds to direct sums. The alpha version of factorization homology integrates an $E_n$algebra over a moduli space of finite subsets of a manifold $M$. The beta version of factorization homology integrates an $(\infty, n)$category over the moduli space of stratifications of $M$. The result takes disjoint unions to tensor products. I'll define this beta version of factorization homology. It satisfies a version of the Eilenberg–Steenrod axioms—this part is work in progress. These Eilenberg–Steenrod axioms together with an argument in the spirit of Pontryagin–Thom theory implies the cobordism hypothesis. This is joint work with David Ayala.
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Hiro Tanaka (Harvard University)
$\begingroup $In Morse theory, it's important to understand how a gradient trajectory can break into multiple gradient trajectories–for instance, this is how one proves the Morse chain complex is a chain complex. In this talk, I'll define a stack classifying families of domains of such broken and breaking trajectories. I'll then sketch a proof that the $\infty$category of factorizable sheaves on this stack is equivalent to the $\infty$category of nonunital $A_\infty$ algebras. Much of this is motivated by a strategy to formulate Morse theory and Floer theory as a deformation problem (similar to CohenJonesSegal), but I may not talk too much about that because the details are not worked out yet. This is joint work with Jacob Lurie.
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Jay Shah (MIT)
$\begingroup $Parametrized or indexed $\infty$category theory studies $\infty$categories fibered over a given base $\infty$category. This theory can be harnessed for the purposes of equivariant homotopy theory when one specializes to the case where is the base is the orbit category of a finite group. In this talk, we present a theory of parametrized homotopy limits and colimits that recovers and extends the DottoMoi theory of $G$colimits. We apply this theory to prove that the $G$$\infty$category of $G$spaces is freely generated under $G$colimits by the contractible $G$space, thereby affirming a conjecture of Mike Hill.
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Angelica Osorno (Reed College)
$\begingroup $Due to the Simons lectures, the seminar will meet at 3:00 PM in 2135.
The notion of 2Segal spaces was introduced by Dyckerhoff and Kapranov as a higher dimensional version of Rezk's Segal spaces. In this talk we will explore the motivation for this notion, give examples, and show that it is related to a certain class of double categories via a version of Waldhausen's construction. This is joint work with J. Bergner, V. Ozornova, M. Rovelli, and C. Scheimbauer.
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Dan BerwickEvans (University of Illinois UrbanaChampaign)
$\begingroup $The seminar will meet at 3:00 PM in 2131.
A basic way to probe a cohomology theory is through its complexification and the resulting Chern character map. Features of the cohomology theory can then be transferred to this complexification, e.g., pushforwards determine RiemannRoch factors. Field theories also have a good notion of character theory via their partition functions, which are functions on certain moduli spaces. Constructions in field theories similarly endow partition functions with additional structure. I’ll apply these ideas to elliptic cohomology and 2dimensional supersymmetric field theories, showing how the character theories of these objects line up with one another. Refining this comparison points toward a connection between field theories and topological modular forms.
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Denis Nardin (MIT)
$\begingroup $The category of $G$spectra plays an important role in homotopy theory, even when a group action is not obviously involved. In this talk we will present a universal property for it providing an alternate proof of the GuillouMay theorem. If time permits, we will use it to provide a uniqueness theorem for the norm functors.
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Clark Barwick (MIT)
The geometry of orbital $\infty$categories

Bena Tshishiku (Harvard University)
$\begingroup \def\Mod{\mathop{\mathrm{Mod}}}\def\Diff{\mathop{\mathrm{Diff}}} $Let $M$ be a manifold, and let $\Mod(M)$ be the group of diffeomorphisms of $M$ modulo isotopy (the mapping class group). The Nielsen realization problem for diffeomorphisms asks, "Can a given subgroup $G<\Mod(M)$ be lifted to the diffeomorphism group $\Diff(M)$?" This question about group actions is related to a question about flat connections on fiber bundles with fiber $M$. In the case $M$ is a closed surface, the answer is "yes" for finite $G$ (by work of Kerckhoff) and "no" for $G=\Mod(M)$ (by work of Morita). For most infinite $G<\Mod(M)$, we don't know. I will discuss some obstructions that can be used to show that certain groups don’t lift. Some of this work is joint with Nick Salter.
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Mike Mandell (Indiana University)
$\begingroup $Hesselholt has recently been advertising "periodic topological cyclic homology" ($TP$) as potentially filling some of the same roles for finite primes as periodic cyclic homology plays rationally. It is constructed from topological Hochschild homology ($THH$) analogously to the way periodic cyclic homology is constructed from Hochschild homology. In joint work with Andrew Blumberg, we prove a strong Kunneth theorem for the periodic topological cyclic homology of smooth and proper dg categories over a finite field $k$, namely, the derived smash product $TP(X) \wedge_{TP(k)} TP(Y)$ is weakly equivalent to $TP(X \otimes_k Y)$.
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Jack Morava (Johns Hopkins University)
$\begingroup $A LubinTate group for a local field L defines a maximal totally ramified abelian extension of L, whose completion is perfectoid; Artin reciprocity identifies the Galois group of this extension with the group of units of L. Scholze and Nikolaus, following work of Hesselholt on the algebraic closure of $Q_p$, have calculated the topological Hochschild homology of such fields; these are complex orientable GEM spectra, whose associated formal groups seem to be rigid analytic versions of the LubinTate group defining the extension, with the Galois group acting as generalized Adams operations. These spectra may have interesting connections with chromatic homotopy theory.
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Ben Antieau (University of Illinois at Chicago)
$\begingroup $Barwick proved that the Ktheory of a stable infinitycategory with a bounded $t$structure agrees with the Ktheory of its heart in nonnegative degrees. Joint work with David Gepner and Jeremiah Heller extends this to an equivalence of nonconnective Ktheory spectra when the heart satisfies certain finiteness conditions such as noetherianity. Applications to negative Ktheory and homotopy Ktheory of ring spectra are provided, which were the original motivation for our work.
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Sune Precht Reeh (MIT)
$\begingroup $Given a representation $V$ of a finite group $G$ we can associate a dimension function that to each subgroup $H$ of $G$ assigns the dimension of the fixed point space $V^H$. The dimension functions are "super class functions" that are constant on the conjugacy classes of subgroups in $G$. For a pgroup the list of BorelSmith conditions characterizes the super class functions that come from real representations.
In a joint paper with Ergün Yalcin we show that though we cannot lift BorelSmith functions to real representations for a general group $G$, we can lift a multiple of any Borelsmith function to an action of $G$ on a finite homotopy sphere (which would be the unit sphere if we had a representation).
To solve the problem we localize at each prime p, and solve it in general for saturated fusion systems. That is, we give a list of BorelSmith conditions for a fusion system that characterize the dimension functions of the fusion stable real representations. The proof for fusion systems involves biset functors and characteristic bisets for saturated fusion systems.
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Ismar Volic (Wellesley College)
$\begingroup $Manifold calculus of functors has in recent years been applied with great success to various spaces of embeddings. In this talk, I will present some new applications and situations where this theory is proving useful. One application is to spaces of homotopy string links where manifold calculus provides a connection to certain spaces of diagrams and trees, as well as to classical objects like Milnor invariants. Another application is to spaces of immersions with a finite number of selfintersections where manifold calculus supplies a new understanding of some classical problems in combinatorial topology. A common theme (and difficulty) in these seemingly unrelated topics is a certain subspace arrangement, and some time will be devoted to the discussion of its structure.
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Paul Goerss (Northwestern University)
$\begingroup $The chromatic view of stable homotopy theory uses the geometry of formal groups to organize calculations. There are two steps; getting local information and then reassembly. Twentyfive years ago, Mike Hopkins made a remarkable conjecture for reassembly – remarkable partly because it grew out a close analysis of local data. Recently this conjecture has been tested, modified, and ratified in various ways. I will report on joint work with Agnès Beaudry and HansWerner Henn exploring this circle of ideas.
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Teena Gerhardt (Michigan State University)
$\begingroup $The algebraic Ktheory of integral group rings has been of interest since the early days of Ktheory. Computations, however, have proven difficult. In this talk I will describe joint work with Vigleik Angeltveit on a strategy for studying the algebraic Ktheory of the group ring $\Z[C_2]$. In particular I will discuss how methods and computations from equivariant stable homotopy theory yield new information about these algebraic Ktheory groups.
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Gabriel DrummondCole (Center for Geometry and Physics)
$\begingroup $Homotopy probability theory is a homological enrichment of algebraic probability theory, a toy model for the algebra of observables in a quantum field theory. I will introduce the basics of the theory and use it to describe a reformulation of fluid flow equations on a compact Riemannian manifold.
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Ben Knudsen (Harvard University)
$\begingroup $I will describe a construction providing Lie algebras with enveloping algebras over the operad of little ndimensional disks for any n. These algebras enjoy a combination of good formal properties and computability, the latter afforded by a PoincareBirkhoffWitt type result. The main application pairs this theory with the theory of factorization homology in a study of the rational homology of configuration spaces, leading to a wealth of computations, improvements of classical results, and a combinatorial proof of homological stability.
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Rade Zivaljevic (Mathematical Institute SANU)
$\begingroup $Special classes of simplicial complexes (chessboard, 'unavoidable’, threshold, 'simple games’, etc.) frequently appear in applications of algebraic topology in discrete geometry and combinatorics. We outline some of these applications, including the proof of a new theorem of Van KamprenFlores type (arXiv:1502.05290 [math.CO], Theorem 1.2), which confirms a conjecture of Blagojevic, Frick, and Ziegler (Tverberg plus constraints). The lecture is based on a joint work with Dusko Jojic (University of Banja Luka) and Sinisa Vrecica (University of Belgrade).
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Chris Kottke (New College of Florida)
$\begingroup $In the geometry and analysis of loop spaces, a fundamental question is how to characterize geometric objects (functions, resp. line bundles, etc.) on the loop space of a manifold which are related by transgression to objects one degree higher (line bundles, resp. gerbes, etc.) on the manifold itself. There are various geometric results of this sort for low degrees, in which a key role is played by 'fusion' – a (partially defined) binary operation on loop space which was first introduced by Stolz and Teichner and further developed by Waldorf.
I will present a joint result with Richard Melrose which answers this question for arbitrary degree in cohomology. More precisely, we define an enhanced version of Cech cohomology of the continuous loop space of M in terms of fusion and a second 'figureofeight' product, through which transgression from M factors as an isomorphism, and which can be iterated to higher loop spaces.
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Kyle Ormsby (Reed College)
$\begingroup $Recent work of RöndigsSpitzweckØstvær sharpens the connection between the slice and Novikov spectral sequences. Using classical vanishing lines for the $E_2$page of the AdamsNovikov spectral sequence and the work of AndrewsMiller on the $\eta$periodic ANSS, I will deduce some new vanishing theorems in the bigraded homotopy groups of the $\eta$complete motivic sphere spectrum. In particular, I will show that the $m$th $\eta$complete MilnorWitt stem is bounded above (by an explicit piecewise linear function) when $m = 1$ or $2$ (mod $4$). This is joint work with Oliver Röndigs and Paul Arne Østvær.
$\endgroup$ 
Emmanuel Dror Farjoun (Hebrew University)
$\begingroup $It is common practice to simplify an object of a given "complicated" category $X\in C$ via the application of collection of functors $F_\alpha: C\to C_\alpha$. One then hopes to reconstruct $X$ via a (co)limit from $(F_\alpha X)_{\alpha \in I}$. Examples include restricting a bundle over a space $X$ to subspaces $X_\alpha$, extending a ring of scalars, truncating a homotopy type $X$ via the Postnikov tower $P_n X$, or forming polynomial approximations of a functor $F$.
Here the question of conjugates arises: For a given $X$, find (the groupoid of) all objects $W$ in the category $C$ with the same given values $F_\alpha W\cong F_\alpha X$.
It turns out that under adjoint conditions one can recover the category $C$ from the diagram of categories $C_\alpha$, this leads to standard expressions for conjugates.
This approach puts on the same footing the classification of forms via Galois cohomology, the Mislin genus and Postnikov conjugates, the Taylor tower for functors and vectors bundles as well as usual descent to subschemes.
For example, the equivalence of $\infty$categories $Top \cong \lim_n Top_n$, where $Top_n$ is the category of $n$truncated spaces is an immediate consequence. (Work by Assaf Horev and Lior Yanovski, HU, Jerusalem.)
$\endgroup$ 
Tyler Lawson (University of Minnesota)
$\begingroup $The Steenrod operations act on the cohomology of a space, but this algebraic data has a more refined lift. There is a ring spectrum $A$ so that every space has an associated $A$module, and on taking homotopy groups this recovers the action of the Steenrod algebra. The $A$module structure, however, also contains the data of secondary and higher operations. In this talk we'll discuss the DyerLashof operations which act on the homology of infinite loop spaces and $E_\infinity$ ring spectra, and how these operations can be realized by the action of a ring spectrum $R$ with a concrete relationship to $A$.
$\endgroup$ 
Marc Hoyois (MIT)
$\begingroup $If $X$ and $Y$ are varieties over a field, there are two interesting candidates for the set of homotopy classes of maps from $X$ to $Y$: the first is simply the quotient of the set of maps by the relation generated by $\Abb^1$homotopies, and the second is the set of maps in the $\Abb^1$homotopy category of MorelVoevodsky. They are different in general. The former often carries relevant algebrogeometric information, while the latter is more mysterious but also more computable. Comparing them is therefore an essential step to apply $\Abb^1$homotopy theory to concrete classification problems in algebraic geometry. In this talk I will survey existing comparison results as well as some recent work with Aravind Asok and Matthias Wendt.
$\endgroup$

Nat Stapleton (Max Planck Institute)
$\begingroup $The seminar will meet at 3:30 PM in 2131.
Nat will speak at 3:30 pm, followed by Saul at 4:30 pm
The behavior of the category of $E_n$local spectra simplifies in various ways as $p \to \infty$. For a collection of categories indexed by the prime numbers we construct a category 'at the infinite prime' that captures behavior of all but finitely many of the input categories. We then apply this construction to the $E_n$local and $K(n)$local situations and analyze the resulting categories. This talk represents joint work with Barthel and Schlank.
$\endgroup$ 
Saul Glasman (Institute for Advanced Study)
$\begingroup $I'll give an exposition of the stratification viewpoint on equivariant homotopy theory, and describe how this helps inspire new approaches to equivariant algebraic Ktheory. I'll propose a categorification of the classical norm cofibration sequence for a cyclic group of prime order and the equivariant fracture square, with a version of the stable module category standing in for the Tate spectrum, leading to a Ktheoretic approximation to the HillHopkinsRavenel norm. Cyclotomic spectra look enoticely perspicuous in the stratification picture, and if possible I'll begin to describe how it harmonizes with our theory of cyclonic objects to give a natural construction of the THH of a Waldhausen infinitycategory. This is joint work in progress with Clark Barwick.
$\endgroup$ 
Tom Church (Stanford University)
$\begingroup $The seminar will meet at 3:30 PM in 2131.
Time changed to 3:30 pm and room number changed to 2136 due to Simons lectures
Two of the recent successes of representation stability are the description of the stable completed cohomology of arithmetic groups by CalegariEmerton, and the proof of excision in continuous Ktheory by Calegari. I'll explain these theorems, focusing on concrete cases such as H_1, K_1 and K_2 where we can work out explicitly exactly what is going on; I won't say anything fancy unless we can see it in at least one case. No knowledge of representation stability or Ktheory needed; in fact, I'll give an introduction to many aspects of classical Ktheory such as the congruence subgroup property and the Steinberg group.
$\endgroup$ 
Rune Haugseng (University of Copenhagen)
$\begingroup $The AKSZ construction, as implemented by PantevToënVaquiéVezzosi in the context of derived algebraic geometry, gives a symplectic structure on the derived stack of maps from an oriented compact manifold to a symplectic derived stack. I will discuss how this gives rise to a family of extended topological field theories valued in higher categories of symplectic derived stacks, with the higher morphisms given by a notion of higher Lagrangian correspondences. This is ongoing work with Damien Calaque and Claudia Scheimbauer.
$\endgroup$ 
Emanuele Dotto (MIT)
$\begingroup $For every homotopy invariant endofunctor of the category of pointed Gspaces (G a finite group) equivariant calculus produces a 'genuine' tower, which under suitable circumstances converges to the original functor. The talk will focus on the description of the layers of this tower in terms of certain 'graphequivariant' spectra, defined from Goodwillie's crosseffects.
$\endgroup$ 
Gijs Heuts (University of Copenhagen)
$\begingroup $We use the Goodwillie tower of the category of pointed spaces to relate the telescopic homotopy theory of spaces (in the sense of Bousfield) to the homotopy theories of Lie algebras and commutative coalgebras in T(n)local spectra, in analogy with rational homotopy theory. As a consequence, one can determine the Goodwillie tower of the BousfieldKuhn functor in terms of topological AndreQuillen homology, giving a different perspective on recent work of Behrens and Rezk.
$\endgroup$ 
Hiro Tanaka (Harvard University)
$\begingroup $In symplectic geometry, the Fukaya category of a symplectic manifold is about as important as DbCoh is in algebraic geometry. As such, one wouldn't expect a totally topological way to characterize the Fukaya category of a symplectic manifold. However, it turns out that in a large class of noncompact symplectic manifolds, one expects to have a topological characterization; more surprisingly, it seems that the stable homotopy theory of Lagrangian cobordisms may recover the Fukaya category altogether. We'll begin with some basics of Fukaya categories, illustrate how there is a functor of oocategories from a category of Lagrangian cobordisms of M to its Fukaya category, and explain structural theorems that move us closer to proving a conjecture that the Fukaya category can be recovered from the category of Lagrangian cobordisms.
$\endgroup$ 
Inna Zakharevich (University of Chicago)
$\begingroup $The Grothendieck ring of varieties is defined to be the free abelian group generated by varieties, modulo the relation that for a closed subvariety $Y$ of $X$, $[X] = [Y] + [X \backslash Y]$; the ring structure is defined via the Cartesian product. For example, if $X$ and $Y$ are piecewise isomorphic, in the sense that there exist stratifications on $X$ and $Y$ with isomorphic strata, then $[X] = [Y]$ in the Grothendieck ring.
There are two important questions about this ring:
1. What does it mean when two varieties $X$ and $Y$ have equal classes in the Grothendieck ring? Must $X$ and $Y$ be piecewise isomorphic?
2. Is the class of the affine line a zero divisor?
Last December Borisov answered both of these questions with a single example, by constructing an element $[X]  [Y]$ in the kernel of multiplication by the affine line; in a beautiful coincidence, it turned out that $X \times \Abb^1$ and $Y \times \Abb^1$ were not piecewise isomorphic. In this talk we will describe an approach using algebraic Ktheory to construct a topological version of the Grothendieck ring of varieties. We shall prove that $\pi_1$ of this space is generated by birational automorphisms of varieties which extend to piecewise automorphisms, which allows us to construct a group that surjects onto the kernel of multiplication by the affine line. By analyzing this group we will sketch a proof that Borisov's coincidence was not a coincidence at all: any element in the kernel of multiplication by the affine line can be represented as $[X][Y]$, where $X \times \Abb^1$ and $Y \times \Abb^1$ are not piecewise isomorphic.
$\endgroup$ 
Teena Gerhardt (Michigan State University)
cancelled

Bhargav Bhatt (University of Michigan)
$\begingroup $I will describe certain surprising features of algebraic geometry that arise if one works exclusively with perfect rings of positive characteristic $p$; these features are are strongly reminiscent of derived algebraic geometry. When combined with some higher algebraic $K$theory, this will allow us to attach "determinants" to certain mildly nonlinear objects. Time permitting, I will explain why these determinants are useful in constructing an object of interest in arithmetic geometry: an algebraic variety in characteristic $p$ that parametrizes $\Z_p$lattices in a finite dimensional $\Q_p$vector space. This is joint work with Peter Scholze.
$\endgroup$ 
Ian Hambleton (McMaster University and Fields Institute)
$\begingroup $Actions of finite groups on spheres can be studied in various different geometrical settings, such as (A) smooth $G$actions on a (closed manifold) homotopy sphere, (B) finite $G$homotopy representations (as defined by tom Dieck), and (C) finite $G$CW complexes homotopy equivalent to a sphere. These three settings generalize the basic models arising from unit spheres $S(V)$ in orthogonal or unitary $G$representations. In the talk, I will discuss the group theoretic constraints imposed by assuming that the actions have rank 1 isotropy (meaning that the isotropy subgroups of $G$ do not contain $\Z/p \times \Z/p$, for any prime $p$). Motivation for this requirement arises from the work of Adem and Smith (2001) on the existence of free action on products of spheres.
The main results are as follows: we give a complete answer in setting (C), where we prove that a necessary and sufficient group theoretic condition is that certain extensions, called $QD(p)$, of $SL(2,p)$ by $\Z/p \times \Z/p$ are not involved in $G$. In setting (B) we encounter more group theoretic restrictions, and give a complete answer for the finite simple groups $G$ of rank 2. The arguments use chain complexes over the orbit category. This is joint work with Ergun Yalcin.
$\endgroup$ 
David Treumann (Boston College)
$\begingroup $Let $L$ be an exact Lagrangian submanifold of a cotangent bundle $T^*M$. If a topological obstruction vanishes, a Floertheoretic construction of Nadler and Zaslow gives a functor from the category of local systems of $R$modules on $L$ to the category of constructible sheaves of $R$modules on $M$. I will discuss a variation of this construction that allows $R$ to be a ring spectrum. I hope it won't take any knowledge of symplectic geometry to follow the talk. This is joint work with Xin Jin.
$\endgroup$ 
Eric Peterson (Harvard University)
$\begingroup $Chromatic homotopy theory is a divideandconquer program for algebraic topology, where we study an approximating sequence of what we'd first assumed to be "easier" categories. These categories turn out to be very strangely behaved  and further appear to be equipped with intriguing and exciting connections to number theory. To give an appreciation for the subject, I'll describe the most basic of these strange behaviors, then I'll describe an ongoing project which addresses a small part of the "chromatic splitting conjecture".
$\endgroup$ 
Kirsten Wickelgren (Georgia Tech)
$\begingroup $A celebrated result of EisenbudKimshaishviliLevine computes the local degree of a smooth function $f : \R^n \to \R^n$ with an isolated zero at the origin. Given a polynomial function with an isolated zero at the origin, we prove that the local $\A^1$Brouwer degree equals the degree quadratic form of EisenbudKhimshiashviliLevine, answering a question posed by David Eisenbud in 1978. This talk will present this result and then discuss applications to the study of singularities if time permits. This is joint work with Jesse Kass.
$\endgroup$ 
Marc Hoyois (MIT)
$\begingroup $The secondary $K$theory of a derived stack $X$ is the $K$theory of 2vector bundles on $X$, also known as smooth proper $k$linear dgcategories when $X = \mathrm{Spec}(k)$. It receives nontrivial maps from several interesting invariants: the Brauer spectrum of $X$, the iterated $K$theory $K(K(X))$, and the Grothendieck ring of varieties (if $X$ is a field of characteristic zero). Toën and Vezzosi have constructed a character map associating to every 2vector bundle a torusinvariant function on the double free loop space of $X$. I will explain how to refine their construction to obtain a secondary Chern character on secondary $K$theory. This involves a localization theorem for traces in symmetric monoidal $(\infty,2)$categories and a categorified version of the ordinary Chern character, which is a functor from noncommutative mixed motives over $X$ (in the sense of Kontsevich) to $S^1$equivariant perfect complexes on $LX$. This is joint work with Sarah Scherotzke and Nicol\'o Sibilla.
$\endgroup$ 
John Terilla (CUNY)
$\begingroup $Homotopy probability theory is a version of probability theory in which the vector space of random variables is replaced with a chain complex. I'll discuss how to use homotopy algebra (rather than analysis) to extract meaningful expectations and correlations among random variables. I'll give some natural examples, including an example that extends ordinary probability theory on a finite volume Riemannian manifold and has applications to fluid flow.
$\endgroup$ 
Cary Malkiewich (University of Illinois UrbanaChampaign)
$\begingroup $Topological Hochschild homology ($THH$) is a beautiful and computable invariant of rings and ring spectra. In this talk, I will focus on the ring spectrum $DX$, and discuss a few different aspects of $THH(DX)$. For example, it splits when $X$ is a suspension, and we can use this for computations in topological cyclic homology. I will also recall the "Atiyah duality" between $THH(DX)$ and the free loop space $LX$, and prove that this duality preserves the genuine $S^1$structure. This uses the new "norm" model of $THH$, and a surprising technical result about orthogonal $G$spectra. If there is time, I will apply these tools once more and describe an enrichment of the character map from representation theory.
$\endgroup$ 
Sune Reeh (MIT)
$\begingroup $A saturated fusion system associated to a finite group $G$ encodes the $p$structure of the group as the Sylow $p$subgroup enriched with additional conjugation. The fusion system contains just the right amount of algebraic information to for instance reconstruct the $p$completion of $BG$, but not $BG$ itself. Abstract saturated fusion systems $F$ without ambient groups exist, and these have ($p$completed) classifying spaces $BF$ as well. In spectra, the suspension spectrum of $BF$ becomes a retract of the suspension spectrum of $BS$, for the Sylow $p$subgroup $S$, so $BF$ gets encoded as a characteristic idempotent in the double Burnside ring of $S$. This way of looking as fusion systems as stable retracts of their Sylow $p$subgroups is a very useful tool for generalizing theorems from groups or $p$groups to saturated fusion systems. In joint work with Tomer Schlank and Nat Stapleton, we use this retract approach to do HopkinsKuhnRavenel character theory for all saturated fusion systems by building on the theorems for finite $p$groups.
$\endgroup$ 
Rob Thompson (CUNY)
$\begingroup $The BousfieldKan (or unstable Adams) spectral sequence can be constructed for various homology theories such as BrownPeterson homology $BP$, JohnsonWilson theory $E(n)$ or Morava $E$theory $E_n$. For nice spaces the $E_2$term is given by $\mathrm{Ext}$ in a category of unstable comodules. We establish an unstable Morava change of rings isomorphism between $\mathrm{Ext}_{\mathcal{U}_{BP_*BP}}(BP_*,M)$ and $\mathrm{Ext}_{\mathcal{U}_{{E_n}_*E_n}}({E_n}_*,{E_n}_*\otimes_{BP_*}M)$ for unstable $BP_*BP$comodules that are $v_n$local and satisfy $I_nM=0$. We show that the latter can be interpreted as Ext in the category of comodules over a certain bialgebra. This has implications for the convergence of the BousfieldKan spectral sequence.
$\endgroup$ 
Martin Gallauer Alves de Souza (University of California, Los Angeles)
$\begingroup $J.P. May proved that traces in stable monoidal homotopy categories are additive in cofiber sequences. It is then natural to ask about the interaction of traces and homotopy colimits in general. In this talk we will give an answer for homotopy colimits indexed over EIcategories. The proof involves generalizing the notion of trace from the realm of categories to derivators. It will be sketched if time permits.
$\endgroup$

Emanuele Dotto (MIT)
$\begingroup $Let $G$ be a finite group. There is a notion of "$J$excision" of functors on pointed $G$spaces, for every finite $G$set $J$. When $J$ is the trivial $G$set with $n$ elements it agrees with Goodwillie's definition of $n$excision. When $J=G$ it recovers Blumberg's notion of equivariant excision.
The talk will focus on the $J$excisive approximations of a homotopy functor, and how they fit together into a "Taylor tree". We will discuss the convergence of the tree, as well as possible classifications of $J$homogeneous functors. Finally, we will relate the layers of the "genuine" tower of the identity functor on pointed $G$spaces to partition complexes, and discuss possible applications of $\mathbb{Z}/2$calculus to Real algebraic $K$theory.
$\endgroup$ 
Agnes Beaudry (University of Chicago)
$\begingroup $In its strongest form, the chromatic splitting conjecture gives a precise description of the homotopy type of $L_{1}L_{K(2)}S$, which has been shown to hold for $p\geq 5$ by Hopkins and for $p=3$ by Goerss, Henn and Mahowald. In this talk, I will explain why this description cannot hold at the prime $p=2$. More precisely, let $V(0)$ be the mod $2$ Moore spectrum. I will give a summary of how one uses the duality resolution techniques to show that $\pi_{k}L_1L_{K(2)}V(0)$ is not zero when $k$ is congruent to $5$ modulo $8$. I will explain how this contradicts the decomposition of $L_1L_{K(2)}S$ predicted by the chromatic splitting conjecture.
$\endgroup$ 
Paul Goerss (Northwestern University)
$\begingroup $The chromatic view of stable homotopy theory assembles a finite spectrum from its $K(n)$localizations, focusing our attention on the $K(n)$local category. This category has a number of interrelated dualities, which together go under the name of GrossHopkins duality. I'd like to explore this in the case $n=2$ using the topological resolutions developed with Henn, Mahowald, Rezk, and others. In particular, I'd like to explain how there is an elegant inevitability to calculations long regarded as impenetrable.
$\endgroup$ 
Guozhen Wang (MIT)
$\begingroup $Using the self maps provided by the HopkinsSmith periodicity theorem, we can decompose the unstable homotopy groups of a space into its periodic parts, except some lower stems. For fixed n, using the BousfieldKuhn functor we can associate to any space a spectrum, which captures the $v_n$periodic part of its homotopy groups.
I will talk about the homotopy type of the BousfieldKuhn functor applied to spheres, which would tell us much about the $v_n$periodic part of the homotopy groups of spheres provided we have a good understanding of the telescope conjecture. I will make use of the Goodwillie tower of the identity functor, which resolves the unstable spheres into spectra which are the Steinberg summands of the classifying spaces of the additive groups of vector spaces over finite fields.
By understanding the attaching maps of the Goodwillie tower after applying the BousfieldKuhn functor, we would be able to determine the homotopy type of its effect on spheres. As an example of how this works in concrete computations, I will compute the K(2)local homotopy groups of the three sphere at primes p>3.
The computations show that the unstable homotopy groups not only have finite ptorsion, their K(2)local parts also have finite $v_1$torsion, which indicates there might be a more general finite $v_n$torsion phenomena in the unstable world, conjectured by many people.
$\endgroup$ 
Michael Andrews (MIT)
$\begingroup $Algebraic topologists are interested in the class of spaces which can be built from spheres. For this reason, when one tries to understand the continuous maps between two spaces up to homotopy, it is natural to restrict attention to the maps between spheres first. The groups of interest are called the homotopy groups of spheres. Topologists soon realized that it is easier to work in a stable setting. Instead, one asks about the stable homotopy groups of spheres or, equivalently, the homotopy groups of the sphere spectrum. Calculating all of these groups is an impossible task but one can ask for partial information.
In particular, one can try to understand the global structure of these groups by proving the existence of recurring patterns; this is analogous to the fact that we cannot find all the prime numbers, but we can prove theorems about their distribution. These patterns are clearly visible in spectral sequence charts for calculating $\pi_*(S^0)$ and my thesis came about because of my desire to understand the mystery behind these powerful dots and lines, which others in the field appeared so in awe of. I will tell the story of the stable homotopy groups of spheres for odd primes at chromatic height 1, through the lens of the Adams spectral sequence.
$\endgroup$ 
Mike Shulman (University of San Diego)
$\begingroup $One of the most important insights of classical topos theory is that a topos (a category of sheaves) has an 'internal language', so that we can reason about its objects roughly 'as if they were sets'. The recent development of 'homotopy type theory' provides a similar internal language for $\infty$toposes ($\infty$categories of stacks), allowing us to reason about its objects 'as if they were spaces'. I will sketch this language and show how to apply it to study sub$\infty$toposes; these are represented internally as 'higher modalities' on a MartinLöfVoevodsky universe, generalizing LawvereTierney operators from classical topos theory.
$\endgroup$ 
David Gepner (Purdue)
$\begingroup $Together with trace methods, the localization sequence comprises one of the only known methods for computing algebraic $K$theory. If $R$ is a ring spectrum and $R[S^{1}]$ is a localization of R, then there is a fiber sequence of $K$theory spectra $K(\text{fiber}) \to K(R) \to K(R[S^{1}])$. In this talk, we will show that (under mild conditions) the fiber term is compactly generated by a Koszultype spectrum formed from $R$ and $S$, which when $R= BP \langle n \rangle$ and $S = \{v_n \}$ differs from $BP \langle n1 \rangle = R/v_n$. We will then apply trace methods to show that their $K$theories differ, answering a question of Rognes. Time permitting, we will sketch how this fits into a general program (primarily due to Waldhausen, Rognes, Ausoni, and others) to understand the $K$theory of the sphere in terms of the chromatic filtration of the stable homotopy category. This is joint work with Benjamin Antieau and Tobias Barthel.
$\endgroup$ 
Emily Riehl (Harvard University)
$\begingroup $Pioneering work of Andre Joyal and Jacob Lurie has shown that ordinary category theory can be extended to quasicategories ("$\infty$categories"), a type of ($\infty$,1)category. In joint work with Verity, we show that the category theory of quasicategories is 2categorical: new definitions of the basic notions  (co)limits, adjunctions, fibrations  equivalent to the JoyalLurie definitions, can be encoded in the homotopy 2category of quasicategories. Our 2categorical proofs in the homotopy 2category restrict to the classical ones in the sub 2category CAT. We give a short list of axioms, satisfied by (iterated) complete Segal spaces, that suffice for this homotopy 2categorical development, so this "formal" approach to the category theory of quasicategories immediately extends to other models of higher homotopical categories.
$\endgroup$ 
Ryan Grady (Boston University)
$\begingroup $In this talk I will highlight how formal moduli problems and derived stacks have a role in constructing quantum field theories. I will then focus on a specific example which allows one to give a new proof of the algebraic index theorem of NestTsygan and Fedosov. If time permits, I will also discuss observables in QFT and their structure as a factorization algebra; giving a further reinterpretation of the algebraic index theorem and some insight into higher analogues of index theory.
$\endgroup$ 
Nicolo Sibilla (University of British Columbia)
$\begingroup $The Fukaya category of open symplectic manifolds is expected to have good localtoglobal properties. Based on this idea several people have developed sheaftheoretic models for the Fukaya category of punctured Riemann surfaces: the name topological Fukaya category appearing in the title refers to the (equivalent) constructions due to DyckerhoffKapranov, Nadler and SibillaTreumannZaslow. The theory involved in setting up the topological Fukaya category has surprising connections with many different areas of geometry and topology, such as for instance the corepresentability of the Waldhausen Sconstruction. In this talk I will focus on defining the topological Fukaya category and explain applications to Homological Mirror Symmetry for toric CalabiYau threefolds. This is work in progress joint with James Pascaleff.
$\endgroup$ 
Mike Hill (University of Virginia)
$\begingroup $Classically, there is essentially only one $E_\infty$ operad, and it parameterizes multiplications commutative up to all higher homotopies. In the $G$equivariant context, the situation is muddied by the possible ways the group can interact with the powers of a space or spectrum. In this talk, I'll discuss the notion of an $N_\infty$ operad, an operad in $G$space which just like the $E_\infty$ operad parameterizes multiplications commutative up to all higher homotopies but which also allows $G$ to permute factors. The use of these allows one to understand operadically the transfer map on equivariant infinite loop spaces, to see what structure is preserved by equivariant Bousfield localization, and to tease apart what sort of additional structure the category of modules over an equivariant commutative ring spectrum has.
$\endgroup$ 
John Harper (Ohio State University)
$\begingroup $Consider a flavor of structured ring spectra that can be described as algebras over an operad $O$ in spectra. A natural question to ask is when the fundamental adjunction comparing $O$algebra spectra with coalgebra spectra over the associated Koszul dual comonad $K$ can be modified to turn it into an equivalence of homotopy theories. In a paper published in 2012, Francis and Gaitsgory conjecture that replacing $O$algebras with the full subcategory of homotopy pronilpotent $O$algebras will do the trick. In joint work with Kathryn Hess we show that every 0connected $O$algebra is homotopy pronilpotent; i.e. is the homotopy limit of a tower of nilpotent $O$algebras.
This talk will describe recent work, joint with Michael Ching, that resolves in the affirmative the 0connected case of the FrancisGaitsgory conjecture; that replacing $O$algebras with 0connected $O$algebras turns the fundamental adjunction into an equivalence of homotopy theories. This can be thought of as a spectral algebra analog of the fundamental work of Quillen and Sullivan on the rational homotopy theory of spaces, the subsequent work of Goerss and Mandell on the padic homotopy theory of spaces, and the work of Mandell on integral cochains and homotopy type. Corollaries include the following: (i) 0connected $O$algebra spectra are weakly equivalent if and only if their $TQ$homology spectra are weakly equivalent as derived $K$coalgebras, and (ii) if a $K$coalgebra spectrum is 0connected and cofibrant, then it comes from the $TQ$homology spectrum of an $O$algebra.
$\endgroup$ 
Nathaniel Stapleton (Universität Bonn)
$\begingroup $We construct a total power operation on nfold class functions compatible with the total power operation in Morava Etheory through the character map of Hopkins, Kuhn, and Ravenel. In essence, this gives a formula for the total power operation in Morava Etheory applied to a finite group. This is joint work with Barthel.
$\endgroup$ 
Emily Riehl (Harvard University)
$\begingroup $Pioneering work of Andre Joyal and Jacob Lurie has shown that ordinary category theory can be extended to quasicategories ("$\infty$categories"), a type of ($\infty$,1)category. In joint work with Verity, we show that the category theory of quasicategories is 2categorical: new definitions of the basic notions  (co)limits, adjunctions, fibrations  equivalent to the JoyalLurie definitions, can be encoded in the homotopy 2category of quasicategories. Our 2categorical proofs in the homotopy 2category restrict to the classical ones in the sub 2category CAT. We give a short list of axioms, satisfied by (iterated) complete Segal spaces, that suffice for this homotopy 2categorical development, so this "formal" approach to the category theory of quasicategories immediately extends to other models of higher homotopical categories.
$\endgroup$

Andrew Ranicki (University of Edinburgh)
$\begingroup $The total surgery obtruction invariant was introduced 35 years ago to unify the two stages of the classical BrowderNovikovSullivanWall surgery theory of topological manifold types in the homotopy type of an ndimensional global Poincare duality space X, with n>4. The space X has local Poincare duality if and only if it is a homology manifold. In effect, a topological manifold in the homotopy type of X is the same as a globally contractible quadratic Poincare nullcobordism of the chain level failure of local Poincare duality. (The talk will explain the terms involved). The invariant is the obstruction to the existence of such a nullcobordism. The talk will review progress in total surgery obstruction theory, which is best understood in terms of a combinatorial analogue of the Verdier duality in sheaf theory.
$\endgroup$ 
Mona Merling (Johns Hopkins University)
$\begingroup $The first definitions of equivariant algebraic Ktheory were given in the early 1980's by Fiedorowicz, Hauschild and May, and by Dress and Kuku; however these early spacelevel definitions only allowed trivial action on the input ring or category. Equivariant infinite loop space theory allows us to define spectrum level generalizations of the early definitions: we can encode a Gaction (not necessarily trivial) on the input as a genuine Gspectrum. I will discuss some of the subtleties involved in turning a ring or category with Gaction into the right input for equivariant algebraic Ktheory, and some of the properties of the resulting equivariant algebraic Ktheory Gspectrum. I will also discuss recent developments in equivariant infinite loop space theory (e.g., multiplicative structures) that should have longrange applications to equivariant algebraic Ktheory.
$\endgroup$ 
Joseph Hirsh (MIT)
$\begingroup $Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.
$\endgroup$ 
Sune Reeh (MIT)
$\begingroup $Every finite group G gives rise to a saturated fusion system consisting of a Sylow psubgroup S plus some additional conjugation structure coming from the larger group G. Instead of having G act on S, we can consider G as an (S,S)biset and ask what properties it has in relation to the fusion system. The resulting notion of characteristic bisets makes sense for abstract fusion systems as well, and such characteristic bisets were originally used by BrotoLeviOliver to define a classifying spectrum for every saturated fusion system. In joint work with Matthew Gelvin we give a classification of all characteristic bisets for a given saturated fusion system F and show that there is a unique minimal one $\lambda_F$ contained in all others. We describe the structure of $\lambda_F$ and the close relation between $\lambda_F$ and other important concepts in the theory of fusion systems, such as for instance the linking system used to construct the classifying space of F.
$\endgroup$ 
Jesse Wolfson (University of Chicago)
$\begingroup $In the 1960s, Atiyah and Janich independently constructed the families index as a natural map from the space of Fredholm operators to the classifying space of topological Ktheory. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic Ktheory. The index map allows us to relate the ContouCarrere symbol, a local analytic invariant of families of schemes, to algebraic Ktheory. Using this, we prove reciprocity laws for ContouCarrere symbols in all dimensions. This extends previous results, of Anderson and Pablos Romo in dimension 1, and of Osipov and Zhu, in dimension 2.
The material for this talk is contained in arXiv:1410.1466 and arXiv:1410.3451, with technical foundations in arXiv:1402.4969.
$\endgroup$ 
Dan Freed (UT Austin)
$\begingroup $One mathematical gateway to field theories in physics is via bordism. There is a unital multiplication on field theories, and so naturally a subset which are invertible. Invertible topological field theories can be realized as infinite loop maps in stable homotopy theory, and the GalatiusMadsenTillmannWeiss theorem identifies the domain. After exposing these ideas, I will indicate two applications. In the first, joint with Hopkins and Teleman, an invertible topological field theory is the obstruction to consistently orienting moduli spaces. In the second, invertible topological field theories approximate the longrange behavior of special condensed matter systems.
$\endgroup$ 
Bertrand Guillou (University of Kentucky)
$\begingroup $The motivic Hopf map $\eta$ is not nilpotent in the motivic stable homotopy groups of spheres, contrary to the situation in the homotopy of spaces. In the motivic Adams spectral sequence computing the motivic stable homotopy groups, there result a number of $h_1$towers. The motivic Adams spectral sequence contains strictly more information than the classical case and is therefore quite complicated, but the $h_1$local part is understood and computes the $\eta$localization of the motivic sphere. I will discuss joint work with Dan Isaksen on the $\eta$local motivic sphere and related topics.
$\endgroup$ 
Joana Cirici (Freie Universität Berlin)
$\begingroup $I will explain a homotopical treatment of intersection cohomology recently developed by ChataurSaraleguiTanré, which associates a "perverse homotopy type" to every singular space. In this context, there is a notion of "intersectionformality", measuring the vanishing of Massey products in intersection cohomology. The perverse homotopy type of a complex projective variety with isolated singularities can be computed from the morphism of differential graded algebras induced by the inclusion of the link of the singularity into the regular part of the variety. I will show how, in this case, mixed Hodge theory allows us prove some intersectionformality results (work in progress with David Chataur).
$\endgroup$ 
Marc Hoyois (MIT)
$\begingroup $Let $X$ be a smooth projective variety over the real numbers and let $f: X \to X$ be a selfmap. To $X$ one can associate a real manifold $X(R)$ and a complex manifold $X(C)$. $l$adic cohomology gives a purely algebraic description of the Lefschetz number of $f_{X(C)}$, but the Lefschetz number of $f_{X(R)}$ is invisible to $l$adic cohomology. I will explain how the Lefschetz number of $f_{X(R)}$ is a motivic homotopy invariant and how a motivic version of the Lefschetz fixedpoint formula for $f$ subsumes the topological fixedpoint formulas for $f_{X(C)}$ and $f_{X(R)}$. I will then consider the situation over an abstract field and formulate an analogous refinement of the $l$adic GrothendieckLefschetz trace formula.
$\endgroup$ 
Marco Perez (MIT)
$\begingroup $We present a new model category structure on the category of chain complexes over a ring $R$, called the \underline{$n$projective model structure}, whose cofibrant objects are given by the class of chain complexes with projective dimension at most $n$ (or $n$projective complexes). One interesting application of this structure consists in finding another way to compute extension groups ${\rm Ext}^i_R(M,N)$ for every pair of left $R$modules $M$ and $N$, by using certain cofibrant and fibrant replacements of the sphere chain complexes $S^0(M)$ and $S^i(N)$, respectively. Recall that one normally computes ${\rm Ext}^i_R(M,N)$ by using either a left resolution of $M$ by projective modules or a right resolution of $N$ by injective modules. Somewhat surprisingly, there turn out to be many other ways to do it. We prove that one can use a left resolution of $M$ by modules of projective dimension at most $n$. The disadvantage of doing so is that we use right resolutions of $N$ by a class of modules which is hard to describe.
Reference: Pérez, M. {\it Homological dimensions and Abelian model structures on chain complexes} (to appear in {\it Rocky Mountain Journal of Mathematics}).
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Gunnar Carlsson (Stanford University)
$\begingroup $[Note that that the automatically produced topology seminar poster corresponding to this talk is incorrect  this will be a colloquium talk be held in E25111, with tea at 4:00 and the talk commencing at 4:30.] Persistent homology is an invariant of finite metric spaces which is of use in a number of different applications. We will discuss methods of organizing them into a space, which are of use both for coordinatizing data sets whose objects are metric spaces, as well as extending the notion to a usable form of multidimensional persistence.
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Ezra Getzler (Northwestern University)
$\begingroup $We present a simple set of axioms on a category of "spaces" which allows us to show that the category of "fibrant simplicial spaces" is a category of fibrant objects. This provides a general framework for studying higher stack theory (or, equivalently, higher Lie groupoids). As an example, we outline a generalization of Kuranishi theory to the higher stack of perfect complexes.
This is joint work with Kai Behrend.
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Ilya Grigoriev (University of Chicago)
$\begingroup $Much like for vector bundles, one can attempt to study bundles with fiber a manifold using characteristic classes, which are invariants that correspond to elements of the cohomology ring of the classifying space BDiff $M$. The easiest of these to define are the socalled "generalized MillerMoritaMumford classes". In the case when the manifold $M=S_g$ is a surface, the MadsenWeiss theorem together with Harer stability imply that as $g$ grows, a large number of these classes become *nonzero*. On the other hand, relationship between BDiff $S_g$ and the moduli space of Riemann surfaces (which has finite dimension) implies that a large number of these classes *are* zero.
Recently, Galatius and RandallWilliams proved an analogue of the MadsenWeiss theorem and of Harer stability for the case when $M$ is a connected sum of products of spheres $S^d \times S^d$. I will describe the implications of their results on the study of generalized MMMclasses, other vanishing and nonvanishing results about the MMMclasses for such manifolds, and whether BDiff $M$ could be modeled by a finitedimensional space.
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Emmanuel Farjoun (Hebrew University)
$\begingroup $In this joint work with Segev and other collaborators we try to look at (discrete) group maps from a homotopy point of view, by e.g.\ taking homotopy quotient spaces rather than usual quotients as sets. This puts seemingly distinct concepts on a common ground and yields results such as the finiteness of higher $H_i(G)$ for certain infinite groups $G$, as well as a relative version of Schur extensions for general, nonperfect groups. A relative version of the stability of the repeated automorphism group $\textup{aut}(\textup{aut}(\cdots \textup{aut}(G))..)$ will be presented, somewhat related to finiteness of $H_i(G)$ as above.
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Mark Behrens (MIT)
$\begingroup $Mahowald used the bobased Adams spectral sequence to compute the 2primary v1periodic stable homotopy groups of spheres, and from this he deduced the v1periodic telescope conjecture for p = 2. I will discuss what I know about the tmfresolution at p = 2, incorporating work of many collaborators over the years, most significantly Tyler Lawson, Kyle Ormsby, Vesna Stojanoska, and Nat Stapleton.
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Lars Hesselholt (Nagoya University and University of Copenhagen)
$\begingroup $An old result of Gillet and Grayson shows that, to calculate the simplicial loop space of Waldhausen's Sconstruction of an exact category, it suffices to apply Kan's Exfunctor just once instead of infinitely many times. In this talk, I will explain that, in another instance, Waldhausen's construction turns out to be unreasonably fibrant, namely, that for every exact category with duality $(C,D,\eta)$, there is a canonical isomorphism $\Omega^{1,1}NiS^{1,1}({C},D,\eta) \simeq \Omega^{2,1}NiS^{2,1}({C},D,\eta)$ in the homotopy category of pointed real spaces. The proof uses the surprising fact, proved by Schlichting, that, on the set of components of the subspace of the lefthand side consisting of the points fixed by the canonical involution, the abelian monoid structure induced by orthogonal sum is an abelian group structure. It further uses that the real additivity theorem holds for both sides, as proved by Schlichting and by myself and Madsen, respectively, along with a new groupcompletion result due to Moi.
$\endgroup$ 
Nat Stapleton (MIT)
$\begingroup $Strickland proved that the Morava Etheory of the symmetric group corepresents the scheme classifying subgroups of the formal group associated to Etheory after taking the quotient by a certain transfer ideal. In this talk I will discuss a new proof of this result using character maps from height n to height 1. I will emphasize different parts of the proof than were discussed in the Thursday seminar talk. This talk includes joint work with Tomer Schlank and Tobias Barthel.
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Matilde Marcolli (CalTech)
$\begingroup $I will describe different recent approaches to the renormalization and computation of Feynman integrals in perturbative quantum field theory via differential forms on the complement of singular hypersurfaces and periods of motives.
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Ismar Volic (Wellesley College)
$\begingroup $I will survey the ways in which some homotopytheoretic methods, manifold calculus of functors main among them, have in recent years been used for extracting information about the topology of spaces of knots and links. Cosimplicial and operadic models for these spaces will also be featured. I will also mention with some recent results about spaces of homotopy string links and in particular about how one can use functor calculus in combination with configuration space integrals to extract information about Milnor invariants as well as derive higherorder asymptotic invariants of vector fields.
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Jonathan Campbell (UT Austin)
$\begingroup $In this talk I'll indicate how to prove a duality result relating the topological Hochschild homology (THH) of a ring spectrum with the THH of its Koszul dual. Both THH and Koszul duality will be defined, and a proof will be sketched.
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Andre Joyal (Université du Québec à Montréal)
$\begingroup $We show that Homotopy Type Theory can be formulated in the language of category theory. We will address the problem of finding an elementary notion of higher topos.
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Hakon Bergsaker (University of Bergen and MIT)
$\begingroup $Given a ring spectrum R, there is an associated algebraic Ktheory spectrum K(R). In general K(R) is very hard to compute; one method for approaching it is to use the cyclotomic trace map to topological cyclic homology, TC(R). This map turns out to be a good approximation in many cases, and TC(R) can be calculated provided one has a good grasp on the various cyclic fixed points of the topological Hochschild homology spectrum, THH(R).
In this talk I will focus on the case where R is the complex cobordism spectrum MU. In this case computing TC(MU) essentially reduces to computing the circleTate construction on THH(MU). I will describe and build on previous homological computations to study the Adams spectral sequence of the circleTate construction on THH(MU). This is work in progress.
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Tyler Lawson (University of Minnesota)
$\begingroup $In this talk I'll discuss comodules in homotopy theory, specifically aimed at lifting $(MU_*, MU_*MU)$comodules up to a homotopical notion. I'll then describe how Goodwillie calculus to give an iterative sequence of approximations to this structure.
$\endgroup$ 
Emanuele Dotto (MIT)
$\begingroup $In a nonequivariant setting, a functor is excisive if it takes homotopy pushout squares to homotopy pullback squares. Given a finite group G and a functor from Gspaces to Gspaces (or Gspectra), this definition of excision does not 'capture enough equivariancy'. For example the category of endofunctors of Gspaces with this property does not model Gspectra. One solution is to replace squares by 'cubes with action', where the group is allowed to act on the whole diagram by permuting its vertices.
I will talk about the homotopy theory of these equivariant diagrams and relate the resulting notion of equivariant excision to previous work of Blumberg.
As an application of this theory, I will give a proof of the Wirthmuller isomorphism that uses only the equivariant suspension theorem and formal manipulations of limits and colimits.
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Tom Church (Stanford University)
$\begingroup $I will give a gentle survey of the theory of representation stability, viewed through the lens of its applications. These applications include: homological stability for configuration spaces of manifolds; understanding the stable (and unstable) homology of arithmetic lattices; uniform generators for congruence subgroups and congruence subgroups; and distributional stability for random squarefree polynomials over finite fields.
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Vesna Stojanoska (MIT)
$\begingroup $PoitouTate duality is a duality for the Galois cohomology of finite modules over the absolute Galois group of a global field. This arithmetic duality is reminiscent of Poincaré duality for manifolds familiar to topologists. In joint work with Tomer Schlank we upgrade this to a duality for spectra with action by such an absolute Galois group, arriving at a Galoisequivariant BrownComenetz duality. We believe this upgraded duality should lead to a better understanding of rational points on algebraic varieties.
$\endgroup$ 
Kyle Ormsby (MIT)
$\begingroup $Let $L/k$ be a finite Galois extension with Galois group $G$. In joint work with Jeremiah Heller, I construct and analyze a functor $F_{L/k}$ from genuine $G$spectra to $P^1$spectra over $\mathrm{Spec}(k)$ which agrees with the constant presheaf functor $c$ when $G = e$. Marc Levine has recently proven that when $k$ is algebraically closed of characteristic $0$, (the left derived functor of) $c$ is full and faithful on homotopy categories. I will show that when $k$ is real closed, $F_{k[i]/k}$ induces a full and faithful embedding of the $C_2$equivariant stable homotopy category into the stable motivic homotopy category of $k$. In particular, there is an isomorphism between the integergraded stable homotopy groups of the $C_2$equivariant sphere spectrum and the motivic sphere spectrum over $k$.
$\endgroup$ 
Emily Riehl (Harvard University)
$\begingroup $Reedy categories come with a degree filtration on objects, enabling the inductive definitions of diagrams and natural transformations. We show that the axioms supply a canonical cell complex presentation for the hom bifunctor with cells defined to be pushoutproducts of "boundary inclusions". This translates to a canonical presentation of any diagram or natural transformation as a (relative) cell complex and as a (relative) Postnikov tower whose cells are built from the latching or matching maps. This work, joint with Dominic Verity, makes the proof of the Reedy model structure essentially trivial and leads to a geometric criterion characterizing the Reedy categories which give formulae for homotopy (co)limits. Work in progress extends these results to generalized Reedy categories, where algebraic weak factorization systems provide a natural tool to define the equivariant factorizations required to extend diagrams from one degree to the next.
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David Ayala (University of Southern California)
$\begingroup $Factorization homology is an invariant of an $n$manifold M together with an $n$disk algebra $A$. Should $M$ be a circle, this recovers the Hochschild complex of $A$; should $A$ be an abelian group, this recovers the homology of $M$ with coefficients in $A$. In general, factorization homology retains more information about a manifold than its underlying homotopy type, and can be interpreted as the global observables of a perturbative TQFT. In this talk we will lift Poincaré duality to factorization homology as it intertwines with Koszul duality for $n$disk algebras  all terms will be explained. We will point out a number of consequences of this duality, which concern manifold invariants, algebra invariants, and TQFT's.
This is a report on joint work with John Francis.
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Dan Isaksen (Wayne State University)
$\begingroup $I will present the results of a detailed computational analysis of the motivic and classical Adams spectral sequences at the prime 2. Some highlights include:
1) corrections to previously published results about stable homotopy groups beyond the 50stem.
2) a brute force approach to the existence of the 62dimensional Kervaire class.
3) a conjectural description of the homotopy groups of the etalocal motivic sphere.
4) an outline of a program to compute new stable stems by combining motivic Adams E2term data with classical AdamsNovikov E2term data.
$\endgroup$ 
Lennart Meier (University of Virginia)
$\begingroup $For an affine (derived) scheme, the global sections functor from quasicoherent sheaves to modules over the global sections of the structure sheaf is an equivalence. We will report on joint work with Akhil Mathew that the same is actually true for many derived stacks occuring in chromatic homotopy theory, such as the derived (compactified) moduli stack of elliptic curves. This and similar techniques allow to show the norm map from homotopy orbits to homotopy fixed points to be an equivalence in many cases (like the $GL_2(Z/n)$action on Tmf$(n)$). Such equivalences have been useful in Stojanoska's work on the Anderson selfduality of Tmf. At the end, we will report on work in progress to extend these results to topological automorphic forms.
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Craig Westerland (University of Minnesota)
$\begingroup $AtiyahSegal and others define a twisted form of $K$theory associated to classes in $H^3(X)$. Their method is geometric, using the Fredholm operator model for the spaces which define $K$theory. Homotopically, this amounts to a multiplicative map from $K(Z,2)$ to the space of units of $K$theory, $GL_1(K)$. In joint work with Hisham Sati, we extended this construction to higherchromatic versions of $K$theory, Morava's $E$theories, $E_n$. We computed the space of $E$infinity maps from $K(Z,n+1)$ to $GL_1(E_n)$, thereby introducing a natural form of twisted $E$theory. I will talk about these constructions and subsequent work which applies them to the study of the stable homotopy groups of the ($K(n)$local) sphere.
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Hiro Tanaka (Harvard University)
$\begingroup $Consider any stable $\infty$category $\mathcal{C}$: Examples include $\text{DbCoh}(X)$, or the category of modules over some ring spectrum. We generalize the notion of a Bridgeland stability condition for a triangulated category to one for $\mathcal{C}$, and under some assumptions, the space of stability conditions is a complex manifold. For every stability condition $\sigma$, one can obtain a filtration on the algebraic $K$theory of $\mathcal{C}$. These filtrations vary on the complex manifold only along real codimension $1$ "walls" inside the complex manifold, and there should be a "wallcrossing formula" relating the $E_2$ pages of the spectral sequence associated to a filtration. I started looking into this because I wanted to encode Hallalgebralike structures on the Ran space of the circle, so I will discuss that as motivation first.
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David Carchedi (Max Planck Institute)
$\begingroup $Given two smooth maps of manifolds $f:M \to L$ and $g:N \to L,$ if they are not transverse, the fibered product $M \times_L N$ may not exist, or may not have the expected dimension. In the world of derived manifolds, such a fibered product always exists as a smooth object, regardless of transversality. In fact, every derived manifold is locally of this form. In this talk, we briefly explain what derived manifolds ought to be, why one should care about them, and how one can describe them. We end by explaining a bit of our joint work with Dmitry Roytenberg in which we make rigorous some ideas of Kontsevich to give a model for derived intersections as certain differential graded manifolds.
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Christian Haesemeyer (University of California, Los Angeles)
$\begingroup $We will explain our recent joint work with G. Cortinas, M. Walker and C. Weibel concerning properties of the algebraic Ktheory of toric varieties in positive characteristic. The results are proved using trace methods and a variant of the cyclic nerve construction that provides a homotopy theoretical model of the socalled Danilov sheaves of differentials. Most of the technical work happens completely within the world of monoid schemes (which are a particular manifestation of what goes by the name of "schemes over the field with one element").
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S\o ren Galatius (Stanford University)
$\begingroup $For an inclusion $S \subset S'$ of connected orientable surfaces, J. Harer proved in 1985 that the map $H_k(BDiff(S)) \rightarrow H_k(BDiff(S'))$, induced by extending orientation preserving diffeomorphisms of $S$ by the identity map of $S'S$, is an isomorphism when $k$ is small compared to the genus of $S$. I will discuss a generalization of this statement to higherdimensional manifolds. As a consequence, we prove that if $M$ is a closed smooth simply connected manifold of dimension $2n > 4$, such that $M$ is diffeomorphic to the connected sum of $g$ copies of $S^n \times S^n$ and some other manifold, then the cohomology of $BDiff(M)$ in the range $* \leq (g4)/2$ is described in terms of a single characteristic class in a twisted cobordism group. This is joint work with O. RandalWilliams.
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Anna Marie Bohmann (Northwestern University)
$\begingroup $An equivariant infinite loop space machine should turn categorical or algebraic data into genuine spectra. While infinite loop space machines have been a crucial part of homotopy theory for decades, equivariant versions are in early stages of development.
I will describe joint work with A. Osorno in which we build an equivariant infinite loop space machine that starts with diagrams of categories on the Burnside category and produces a genuine Gspectrum via the work of GuillouMay. This machine readily applies to produce EilenbergMacLane spectra for Mackey functors and topological Ktheory.
$\endgroup$

Marco Varisco (University at Albany, SUNY)
$\begingroup $The FarrellJones Conjecture predicts the structure of the algebraic Ktheory K(ZG) of the integral group ring of an arbitrary discrete group G. It asserts that a socalled assembly map (whose target is the spectrum K(ZG) and whose source is the homotopy colimit of K(ZH) over all virtually cyclic subgroups H of G) is an equivalence. I will describe joint work with Wolfgang Lück, Holger Reich, and John Rognes, in which we prove partial injectivity results about the rationalized assembly map under finiteness assumptions on the group G, generalizing a theorem of BökstedtHsiangMadsen. The main tool is the cyclotomic trace map from algebraic Ktheory to topological cyclic homology.
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Rune Haugseng (MIT)
$\begingroup $Categories enriched in symmetric monoidal categories such as spectra turn up in various places in algebraic topology. Unfortunately these can be difficult to work with in a homotopically meaningful way, which suggests that for many purposes it would be better to work with less rigid structures, where composition is only associative up to coherent homotopy. In this talk I will introduce a general theory of such weak or homotopycoherent enrichment, built using a nonsymmetric variant of Lurie's infinityoperads. I will then describe how the correct homotopy theory of these enriched infinitycategories can be constructed as a localization of a homotopy theory defined using infinityoperads; this is joint work with David Gepner. In addition, I will discuss some comparison results and, time permitting, mention analogues of natural transformations and correspondences in this setting.
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Luis Pereira (MIT)
$\begingroup $The overall goal of this talk is to apply the theory of Goodwillie calculus to the category $Alg_{\mathcal{O}}$ of algebras over a spectral operad. Its first part will deal with generalizing many of the original results of Goodwillie so that they apply to a larger class of model categories and hence be applicable to $Alg_{\mathcal{O}}$. The second part will apply that generalized theory to the $Alg_{\mathcal{O}}$ categories. The main results here are: an understanding of finitary homogeneous functors between such categories; identifying the Taylor tower of the identity in those categories; showing that finitary nexcisive functors can not distinguish between $Alg_{\mathcal{O}}$ and $Alg_{\mathcal{O_{\leq n}}}$, the category of algebras over the truncated $O_{\leq n}$; and a weak form of the chain rule between such algebra categories, analogous to the one studied by Arone and Ching in the case of Spaces and Spectra.
$\endgroup$ 
Geoffroy Horel (MIT)
$\begingroup $In this talk I will describe a general theory of modules over an algebra over an operad. Specializing to the operad Ed of little ddimensional disks, I will show that each d1 manifold gives rise to a theory of modules. I will then describe a geometric construction of the homomorphisms objects in these categories of modules inspired by factorization homology (also called chiral homology). A particular case of this construction is higher Hochschild cohomology (i.e. Hochschild cohomology for Edalgebras). This construction enlightens the relationship between Hocshchild cohomology and geometric objects like the cobordism category and the spaces of long knots.
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John Lind (Johns Hopkins University)
$\begingroup $Twisted Ktheory is a cohomology theory whose cocycles are like vector bundles but with locally twisted transition functions. If we instead consider twisted vector bundles with a symmetry encoded by the action of a compact Lie group, the resulting theory is equivariant twisted Ktheory. This subject has garnered much attention for its connections to conformal field theory and representations of loop groups. While twisted Ktheory can be defined entirely in terms of the geometry of vector bundles, there is a homotopytheoretic formulation using the language of parametrized spectra. In fact, from this point of view we can define twists of any multiplicative generalized cohomology theory, not just Ktheory. The aim of this talk is to explain how this works, and then to propose a definition of equivariant twisted cohomology theories using a similar framework. The main ingredient is a structured approach to multiplicative homotopy theory that allows for the notion of a Gtorsor where G is a grouplike A_{\infty} space.
$\endgroup$ 
Jack Ullman (MIT)
$\begingroup $The equivariant slice spectral sequence was introduced by Hill, Hopkins and Ravenel in their solution of the Kervaire invariant problem, and is rapidly becoming an important computational tool in equivariant stable homotopy theory. In this talk, I will describe new results on a variant called the regular slice spectral sequence (or RSSS). I will explain how geometric fixed point and norm functors interact with the slice filtration, giving a Leibniz formula for the latter. I will then use BrownComenetz duality to relate the RSSS to the homotopy orbit and homotopy fixed point spectral sequences. Next, I will use model theory to obtain Toda bracket operations in the RSSS. Finally, I will use some of these tools to obtain a formula for the slice tower of a cofree spectrum, prove real Bott periodicity and prove a special case of the AtiyahSegal completion theorem.
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Nitu Kitchloo (Johns Hopkins University)
$\begingroup $The Stable Symplectic category can the thought of as a category of Symplectic Motives. The objects in this topological category are symplectic manifolds, and the space of morphisms is an infinite loop space obtained by stabilizing the space of immersed totallyreal correspondences between the source and target. A variant of this category can be traced back almost 30 years to early work of A. Weinstein on geometric quantization. In my talk, I will motivate the definition of the Stable Symplectic category. This will lead us to the construction of a canonical fiber functor F, on this category with values in the monoidal category of modules over a commutative ring spectrum Omega. The main aim of my talk is to explore the Motivic Galois group Aut(F) (i.e. the group of monoidal automorphisms of F). This group will be shown to be the abelian quotient of the GrothendieckTeichmuller group as described by Kontsevich. Extending this observation along the lines of homotopy theory, we will motivate the topological hochschild homology of Omega:THH(Omega), as an integral candidate for Aut(F). If time permits, I would like to formulate some natural geometric questions in symplectic topology in terms of THH(Omega) and the Waldhausen Ktheory K(Omega). This is joint work in part with Jack Morava.
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Dustin Clausen (MIT)
$\begingroup $We will discuss an analog in algebraic Ktheory of the PoitouTate global duality in Galois cohomology. A key point is that the use of algebraic Ktheory instead of Galois cohomology allows to give a direct and pictorial construction of the fundamental class which is at the base of these dualities. It also allows to connect this arithmetic theory with some classical and modern work in homotopy theory, such as Quillen's on the Jhomomorphism and Rezk's on logarithmic cohomology operations.
$\endgroup$ 
Ethan Devinatz (University of Washington)
$\begingroup $Given a finite spectrum of type $n$, explicit $v_n$ selfmaps are more easily constructed if that spectrum is a ring spectrum, by which I mean the spectrum is provided with a pairing which has a twosided unit but is not necessarily homotopy commutative or homotopy associative. If in addition, the spectrum is homotopy associative and homotopy commutative, one can sometimes say more.
Twenty five years ago I proved that if $X$ is a finite ring spectrum of type n, then there exists a $v_n$ selfmap f such that the cofiber $X(f^i)$ of the selfmap $f^i$ is a ring spectrum for any $i$, and the pairing on $X(f^i)$ extends the pairing on $X$. In this talk, I will discuss my recent result that if $X$ is higher homotopy commutative up to some finite order, then $f$ may be chosen so that this higher homotopy commutative structure may be extended to such a structure on $X(f^i)$.
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Matthew Gelvin (University of Copenhagen)
$\begingroup $For $G$ a finite group with Sylow subgroup $S$, the conjugation action of $G$ on the subgroups of $S$ gives rise to the data of a \emph{saturated fusion system} $\mathcal{F_S(G)}$ on $S$. On the other hand, $S$ acts on $G$ by left and right multiplication. The resulting $(S,S)$biset $_SG_S$ turns out to contain much of the same information as $\mathcal{F_S(G)}$, in that the biset determines the fusion system, but not conversely.
These notions can be abstracted to make no reference to the ambient group $G$, resulting in an \emph{abstract saturated fusion system} $\mathcal{F}$ on $S$ and a \emph{characteristic biset} $\Omega$ for $\mathcal{F}$. Again, $\Omega$ determines $\mathcal{F}$, but each $\mathcal{F}$ has many associated characteristic bisets.
This talk will focus on the failure of a saturated fusion system to uniquely determine a characteristic biset. We will show that there is a parametrization of all characteristic bisets for a fixed $\mathcal{F}$, which will have as a consequence the surprising result that each saturated fusion system has a unique \emph{minimal} associated characteristic biset.
$\endgroup$ 
Michael Ching (Amherst College)
$\begingroup $I will describe a collection of theorems that exemplify homotopic descent. Each of these theorems says that a certain Quillen adjunction is 'comonadic' in a homotopical sense: that is, it identifies the homotopy theory on one side of the adjunction with the homotopy theory of coalgebras over a certain comonad that acts on the other side. I will say what I mean by the homotopy theory of such coalgebras and give a BarrBeck comonadicity condition.
The examples concern operad theory and Goodwillie calculus. One result identifies the homotopy theory of 0connected algebras over an operad of spectra with that of 0connected divided power coalgebras over the Koszul dual operad. (This is joint work with John Harper.) Another describes the homotopy theory of nexcisive homotopy functors (between categories of spaces and/or spectra) in terms of appropriate comonads. (This is joint work with Greg Arone.) In the case of functors from spaces to spectra, and algebras over the commutative operad, there is a close connection between these two examples, which I shall describe.
$\endgroup$ 
André Henriques (University of Utrecht)
$\begingroup $Roughly ten years ago, Stephan Stolz and Peter Teichner have set up a detailed plan for constructing TMF geometrically. Unfortunately, their idea of definition is still incomplete. A couple of months ago, I had an idea (which fits into the StolzTeichner program) about which I am quite excited: There should be a universal CFT, which I'll call $U$. The CFT $U$ should bear with respect to other CFTs a relationship that is analogous to the relationship that an infinite dimensional Hilbert space bears with respect to other finite dimensional vector spaces. Moreover, there should exist a property of quantum fields of $U$, which I'll call 'Fredholm' such that the space of Fredholm quantum fields of $U$ is a classifying space for the cohomology theory TMF. I'll explain what the theory $U$ is, and what it means for a quantum field to be 'Fredholm'. Disclaimer: this is all very speculative, and I don't think that, in its current form, this will yield TMF.
$\endgroup$

Steve Awodey (Carnegie Melon University)
$\begingroup $I will give a survey of the recently discovered connection between constructive logic and homotopy theory. This forms the basis of Voevodsky's Univalent Foundations program, a new approach to foundations with intrinsic geometric content and a computational implementation. Time permitting, I will explain the Univalence axiom.
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Angélica Osorno (University of Chicago)
$\begingroup $An equivariant infinite loop space machine is a functor that constructs genuine equivariant spectra out of simpler categorical or space level data. In the late 80's LewisMaySteinberger and Shimakawa developed generalizations of the operadic approach and the $\Gamma$space approach respectively. In this talk I will describe work in progress that aims to understand these machines conceptually, relate them to each other, and develop new machines that are more suitable for certain kinds of input. This work is joint with Anna Marie Bohmann, Peter May and Mona Merling.
$\endgroup$ 
Kirsten Wickelgren (Harvard University)
$\begingroup $The Picard scheme $\text{Pic}^0$ representing invertible sheaves can be compactified by a moduli space $J$bar of rank 1, torsionfree sheaves called the compactified Jacobian. For a smooth algebraic curve $X$ over a field $k$ with boundary $\partial X$, applying $H_1$ to the AbelJacobi map $X \to \text{Pic}^0 (X/ \partial X)$ gives the Poincaré duality isomorphism $H_1(X, Z/\ell) \to H^1_c(X, Z/\ell(1)) = H^1(X, \partial X, Z/\ell(1))$. We show the analogous result for the compactified Jacobian that applying $H_1$ to the AbelJacobi map $X/\partial X \to J$bar gives the Poincaré duality isomorphism $H_1(X, \partial X, Z/\ell) \to H^1(X, Z/\ell(1))$. In particular, $H_1(X/ \partial X \to J$bar$)$ is an isomorphism. This is joint work with Jesse Kass.
$\endgroup$ 
Emily Riehl (Harvard University)
$\begingroup $Quasicategories (aka $\infty$categories) are convenient models of categories weakly enriched in spaces. Analogs of the standard categorical theorems involving limits and colimits, adjunctions, equivalences, monads and so forth have been proven by Joyal, Lurie and others. The goal of this talk is to describe a new groundlevel approach that allows for 'formal' reproofs of these facts that requires only very mild model category prerequisites and hence generalizes. A highlight will be the construction and characterization of the quasicategory of algebras associated to a homotopy coherent monad. This is a progress report on ongoing joint work with Dominic Verity.
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Tomer Schlank (MIT)
$\begingroup $Given a fibration $f:X \to S$ of CWcomplexes one can use Eilenberg obstruction theory to study the spaces of sections of $f$. These obstruction theory give rise to obstructions to the existence of a section lying in the groups $H^{s+1}(S, \pi_s(F))$ where $F$ is the fibre of $f$. A topos is a generalization of the concept of topological space which is ubiquitous in algebraic geometry. In the talk I shall present joint work with I. Barnea generalizing Eilenberg obstruction theory for sections of maps of topoi $f:X \to S$. If time permits I will describe applications to Galois theory of number fields.
$\endgroup$ 
Matthew Gelvin (University of Copenhagen)
$\begingroup $For $G$ a finite group with Sylow subgroup $S$, the conjugation action of $G$ on the subgroups of $S$ gives rise to the data of a \emph{saturated fusion system} $\mathcal{F_S(G)}$ on $S$. On the other hand, $S$ acts on $G$ by left and right multiplication. The resulting $(S,S)$biset $_SG_S$ turns out to contain much of the same information as $\mathcal{F_S(G)}$, in that the biset determines the fusion system, but not conversely. These notions can be abstracted to make no reference to the ambient group $G$, resulting in an \emph{abstract saturated fusion system} $\mathcal{F}$ on $S$ and a \emph{characteristic biset} $\Omega$ for $\mathcal{F}$. Again, $\Omega$ determines $\mathcal{F}$, but each $\mathcal{F}$ has many associated characteristic bisets. This talk will focus on the failure of a saturated fusion system to uniquely determine a characteristic biset. We will show that there is a parametrization of all characteristic bisets for a fixed $\mathcal{F}$, which will have as a consequence the surprising result that each saturated fusion system has a unique \emph{minimal} associated characteristic biset.
$\endgroup$ 
Vesna Stojanoska (MIT)
$\begingroup $We combine three strategies to studying cooperations in connective topological modular forms: the Adams spectral sequence and its relation to BrownGitler modules following Mahowald's approach to cooperations in connective real K theory, Laures's theory of qexpansions of multivariable modular forms, as well as level structure approximations. As a result, we obtain an algorithmic procedure for determining the structure of the smash product of tmf with itself. This is a report on joint work in progress with Behrens, Ormsby, and Stapleton.
$\endgroup$ 
Nathaniel Stapleton (MIT)
$\begingroup $The generalized character theory of Hopkins, Kuhn and Ravenel has proved to be a very useful tool in the study of Morava $E_n$. In this talk, I will outline a compact construction of the transchromatic generalized character maps. The Morava $E$theory of cyclic groups and symmetric groups have well known algebrogeometric interpretations. Using the relationship between the character maps and the transfer maps for Morava $E$theory, I will provide algebrogeometric interpretations of the cohomology of some finite groups other than symmetric groups and cyclic groups.
$\endgroup$ 
Kyle Ormsby (MIT)
$\begingroup $The motivic truncated BrownPeterson spectra BP$<$n$>$ interpolate between motivic cohomology (BP$<$0$>$), algebraic Ktheory (BP$<$1$>$), and the motivic BrownPeterson spectrum itself, a close relative of algebraic cobordism. We use the motivic Adams spectral sequence and globaltolocal comparison maps to compute the BP$<$n$>$homology of the rational numbers. Along the way, we prove a Hasse principle for the motivic BP$<$n$>$ and deduce several classical and recent theorems about the algebraic $K$theory of particular fields. This is joint work with Paul Arne Østvær.
$\endgroup$ 
Larry Guth (MIT)
$\begingroup $I'm going to talk about connections between the geometry of a map and its homotopy type. Suppose we have a maps from the unit $m$sphere to the unit $n$sphere. We say that the $k$dilation of the map is $< L$ if each $k$dimensional surface with $k$dim volume $V$ is mapped to an image with $k$dim volume at most $LV$. Informally, if the $k$dilation of a map is less than a small $\epsilon$, it means that the map strongly shrinks each $k$dimensional surface. Our main question is: can a map with very small $k$dilation still be homotopically nontrivial? Here are the main results. If $k > (m+1)/2$, then there are homotopically nontrivial maps from $S^m$ to $S^{m1}$ with arbitrarily small $k$dilation. But if $k \leq (m + 1)/2$, then every homotopically nontrivial map from $S^m$ to $S^{m  1}$ has $k$dilation at least $c(m) > 0$.
$\endgroup$ 
Paul Arne Østvær (University of Oslo)
$\begingroup $We compute the motivic slices of hermitian $K$theory and higher Witttheory. The corresponding slice spectral sequences relate motivic cohomology to hermitian $K$groups and Witt groups, respectively. Using this we compute the hermitian $K$groups of number fields, and (re)prove Milnor's conjecture on quadratic forms for fields of characteristic different from 2. Joint work with Oliver Röndigs.
$\endgroup$ 
Søren Galatius (Stanford University)
$\begingroup $I will discuss recent joint work with Oscar RandalWilliams concerning the manifolds $W_g^{2n}$ obtained as the connected sum of $g$ copies of $S^n \times S^n$. For $n=1$ this is a genus $g$ surface, and there is a moduli space $M_g$ parametrizing smooth surface bundles with genus $g$ fibers. For higher $n$ there is an analogous moduli space $M_g^n$ parametrizing smooth fiber bundles with fibers $W_g$ (although for $n > 1$ it is no longer finite dimensional). We prove that for $n > 2$ the cohomology groups $H^k(M_g^n)$ are independent of $g$ as long as $g >> k$, generalizing a result of John Harer and others for $n=1$.
$\endgroup$

Man Chueng Cheng (MIT)
$\begingroup $It was shown by GreenleesSadofsky that classifying spaces of finite groups satisfy a Morava Ktheory version of Poincare duality. This duality map can be viewed as coming from a SpanierWhitehead type construction for differentiable stacks. In this talk I will define differentiable stacks and explain this construction. I will also discuss the generalization of the above result to a more general class of stacks and give some examples.
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Marcy Robertson (University of Western Ontario)
$\begingroup $A multicategory, also known as a colored operad, is simply a generalized noncommutative algebra. In this talk we focus on studying maps between multicategories enriched in simplicial sets. We show that the homotopy function complex of maps between any two multicategories can be computed as the moduli space of a certain small category of (operatic) bimodules. As an application, we show how this description leads to several important decompositions which allow one to compute various geometric invariants.
$\endgroup$ 
Robin Koytcheff (Brown University)
$\begingroup $Budney recently constructed an operad that encodes splicing of knots and extends his little 2cubes operad action on the space of (long) knots. He further decomposed the space of knots as the space freely generated over the splicing operad by the subspace of torus and hyperbolic knots. Infection of knots (or links) by string links is a generalization of splicing from knots to links and is useful for studying concordance of knots. In joint work with John Burke, we have constructed a colored operad that encodes this infection operation. This suggests looking for other ways to decomposes spaces of knots and links, which is a main direction of our work in progress.
$\endgroup$ 
Charles Rezk (University of Illinois UrbanaChampaign)
Power operations in elliptic cohomology

Bertrand Toën (Université de Montpellier)
$\begingroup $I will report on a work in progress in collaboration with PantevVaquiéVezzosi. The central notion will be of a shifted symplectic structure on a derived scheme or derived stack, which I will explain in details. The main result is an existence statement of shifted symplectic structures on derived mapping spaces toward a symplectic target, that will be used to construct many examples (moduli of sheaves on $CY$ varieties, moduli of representation of $\pi_1$ of a compact oriented manifold, Lagrangian intersections ...). Finally, I will explain what quantization means in this context as well the general strategy to prove existence of canonical quantizations.
$\endgroup$ 
Burt Totaro (University of Cambridge)
$\begingroup $Symonds (2010) showed that the cohomology ring of a finite group $G$ with a faithful complex representation of dimension $n$ is generated by elements of degree at most $n^2$. This was a remarkable advance, since no bound was known before. Symonds's proof combined equivariant cohomology with commutative algebra (CastelnuovoMumford regularity). We give better bounds for the cohomology ring of a pgroup. The methods also apply to the Chow ring of algebraic cycles on $BG$.
$\endgroup$ 
David Ayala (Harvard University)
$\begingroup $Many proposed higher categories come from geometric situations. This talk will demonstrate a constructive connection between a homotopy theory of local invariants of nmanifolds and that of weak ncategories in the sense of Rezk. Connections to specific topological field theories will be discussed. This is a report on joint work with Nick Rozenblyum.
$\endgroup$ 
Inna Zakharevich (MIT)
$\begingroup $Hilbert's third problem asks the following question: given two polyhedra with the same volume, is it possible to dissect one into finitely many polyhedra and rearrange it into the other one? The answer (due to Dehn in 1901) is no: there is another invariant that must also be the same. Further work in the 60s and 70s generalized this to other geometries by constructing groups which encode scissors congruence data. Though most of the computational techniques used with these groups related to group homology, the algebraic Ktheory of various fields appears in some very unexpected places in the computations. We will give a different perspective on this problem by examining it from the perspective of algebraic Ktheory: we construct the Ktheory spectrum of a scissors congruence problem and relate some of the classical structures on scissors congruence groups to structures on this spectrum.
$\endgroup$ 
Po Hu (Wayne State University)
$\begingroup $I will discuss topological Hermitian cobordism, an RO(G)graded Z/2 x Z/2 spectrum constructed in a joint paper with Igor Kriz. I will also talk about the method for calculating its RO(G)graded coefficients completely.
$\endgroup$ 
Igor Kriz (University of Michigan)
$\begingroup $I plan to discuss my recent joint work with Po Hu and Daniel Kriz on a lifting of Khovanov homology to connective ktheory. I will also talk about its relation to recent work by Lipshitz and Sarkar. As a technical tool, our approach exhibits a curious link between modular functors and the ElmendorfMandell approach to multiplicative infinite loop space theory.
$\endgroup$ 
David Gepner (Universität Regensburg and MIT)
$\begingroup $The motivic perspective on algebraic Ktheory is a useful means of constructing trace maps as well as other additive or localizing invariants which are often easier to compute. For instance, a homotopy invariant form of Ktheory is obtained by forming the $A^1$localization of noncommutative motives before passing to homotopy groups. In order to relate this to the usual notion of homotopy Ktheory it is useful to have a direct construction of this localization, and this involves understanding the additive group $A^1$ in the noncommutative context. This is joint work with Andrew Blumberg and Goncalo Tabuada.
$\endgroup$ 
Dominic Verity (Center of Australian Category Theory)
$\begingroup $The aim of this talk is to discuss the homotopy coherence properties of adjunctions between quasicategories.
Taking as our lead the theory of the "walking adjunction" A of 2category theory, we generalise to categories enriched in quasicategories and show that this same 2category plays a similar role in this new context. Specifically, using insights drawn from the calculus of string diagrams we give an explicit presentation of A as a simplicially enriched category. We then use this to show that if C is any quasicategory enriched category and u is a right adjont 0arrow in C, in some suitable sense to be discussed, then this data may be completed to give a simplicially enriched functor A>C. Furthermore, we show that the space of all such exensions is contractible.
That adjunctions of quasicategories may be completed up to enriched functors on A in this way contains, in its very essence, the adjunction data discussed by Jacob Lurie. Such enriched functors encapsulate both the coherent monad and the coherent comonad generated by such an adjunction and provide the building blocks upon which to found a formal theory of such things along the lines established by Street in the 2categorical context.
$\endgroup$ 
Stefan Schwede (Universität Bonn)
$\begingroup $The filtration on the infinite symmetric product of spheres by number of factors provides a sequence of spectra between the sphere spectrum and the integral EilenbergMac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. In this talk I will discuss the equivariant stable homotopy types, for finite groups, obtained from this filtration for the infinite symmetric product of representation spheres. The filtration is more complicated than in the nonequivariant case, and already on the zeroth homotopy groups an interesting filtration of the augmentation ideal of the Burnside ring functor arises. Our method is by 'global' homotopy theory, i.e., we study the simultaneous behaviour for all finite groups at once. The equivariant subquotients are no longer rationally trivial, nor even concentrated in dimension 0.
$\endgroup$ 
Roman Mikhailov (Steklov Mathematical Institute and Institute for Advanced Study)
$\begingroup $The talk is based on recent results obtained jointly with Jie Wu. For all $n>k$, we construct a finitely generated group by explicit generators and relations such that its center is the $n$th homotopy group of the $k$th sphere.
$\endgroup$

Emily Riehl (Harvard University)
$\begingroup $Algebraic model categories are a variant of Quillen's classical notion in which the (co)fibrations are equipped with extra structure witnessing their defining lifting properties. Many ordinary model categories admit this extra structure, giving rise to a plethora of examples. In this talk we present several theorems illustrating various features of this theory. In particular, we focus on a series of results that guarantee the existence of algebraic Quillen adjunctions and algebraic monoidal model structures just when particular cofibrations are cellular: eg, relative cell complexes, not mere retracts of such. On account of these results, the algebraic theory places great emphasis on a distinction that is also present in expository accounts of the classical theory, where its role is less transparent.
$\endgroup$ 
Andrew Blumberg (UT Austin)
$\begingroup $The algebraic Ktheory of the complex cobordism spectrum is an object of basic interest, both because it provides an interesting example of Ktheory of a nonclassical ring and because it should shed light on K(S). There is reason to believe that K(MU) should be approachable via trace methods, which focuses attention on understanding THH(MU) and TC(MU). This talk describes work in progress to describe the equivariant homotopy type of THH of a Thom spectrum as an equivariant Thom spectrum. The ingredients for this description include the HillHopkinsRavenel norm and a modernized view of equivariant infinite loop space theory. This is joint work with Angeltveit, Gerhardt, and Hill.
$\endgroup$ 
John Francis (Northwestern University)
$\begingroup $Factorization homology, or the topological chiral homology of Lurie, is a homology theory for manifolds conceived as a topological analogue of the chiral homology of Beilinson and Drinfeld. I'll describe an axiomatic characterization of factorization homology, generalizing the EilenbergSteenrod axioms for usual homology. The use of excision for factorization homology facilitates a short proof of the nonabelian Poincare duality of Salvatore and Lurie; this proof generalizes to give a nonabelian Poincare duality for stratified manifolds, joint work with David Ayala and Hiro Tanaka. Work in progress with Kevin Costello aims to express quantum invariants of knots and 3manifolds in factorization terms, which, time permitting, I'll outline.
$\endgroup$ 
Gonçalo Tabuada (MIT)
$\begingroup $We prove that every functor defined on dg categories which is derived Morita invariant, localizing, and $\mathbb{A}^{1}$homotopy invariant, satisfies the fundamental theorem. As an application, we recover, in a unified and conceptual way, Weibel and Kassel's fundamental theorems in homotopy algebraic Ktheory, and periodic cyclic homology, respectively.
$\endgroup$ 
Ricardo Andrade (Stanford University)
$\begingroup $The topological chiral homology of $E_n$algebras can be calculated using certain categories of configurations on manifolds. These are obtainable from a construction associating to each filtered space a category whose morphisms are paths which respect the filtration. In this context, DwyerKan localizations arise from forgetting stages of the filtration. Such a result recovers elementary properties of chiral homology.
$\endgroup$ 
Kirsten Wickelgren (Harvard University)
$\begingroup $The Milnor conjecture identifies the cohomology ring $H^{*}(\text{Gal}(\bar{k}/k), \mathbb{Z}/2)$ with the tensor algebra of $k^{*}$ mod the ideal generated by $x \otimes 1x$ for $x$ in $k  \{0,1\}$ mod 2. In particular, $x \cup 1x$ vanishes, where $x$ in $k^{*}$ is identified with an element of $H^{1}$. We show that order $n$ Massey products of $n1$ factors of $x$ and one factor of $1x$ vanish by embedding $\mathbb{P}^{1}  \{0,1,\infty\}$ into its Picard variety and constructing $\text{Gal}(\bar{k}/k)$equivariant maps from $\pi_{1}^{\text{et}}$ applied to this embedding to unipotent matrix groups. This also identifies Massey products of the form $\langle 1x, x, \ldots , x , 1x\rangle$ with $f \cup 1x$, where $f$ is a certain cohomology class which arises in the description of the action of $\text{Gal}(\bar{k}/k)$ on $\pi_{1}^{\text{et}}(\mathbb{P}^1  \{0,1,\infty\})$.
$\endgroup$ 
Kyle Ormsby (MIT)
$\begingroup $The $K(2)$localization of the sphere spectrum admits a conjectural small resolution built from TMF and "TMF with level structures"  the evaluation of the TMF sheaf on the stack of elliptic curves equipped with an order $l$ subgroup. In this talk, I will use variations on Tate normal form to describe several Hopf algebroids that stackify to elliptic curves with level structure. These Hopf algebroids lead to computations of the BehrensLawson spectrum $Q(l)$. This is current work with Mark Behrens, Nat Stapleton, and Vesna Stojanoska.
$\endgroup$ 
Nathaniel Stapleton (MIT)
$\begingroup $In the 80's Hopkins, Kuhn, and Ravenel developed a way to study cohomology rings of the form $E^{*}(\mathrm{B}G)$ in terms of a character map. Their map can be interpreted as a map of cohomology theories beginning with a height $n$ cohomology theory $E$ and landing in a height 0 cohomology theory with a rational algebra of coefficients that they construct out of $E$. In this talk we will use the language of $p$divisible groups to discuss various ways of generalizing their map to every height between 0 and $n$.
$\endgroup$ 
Thomas Kragh (MIT)
In this talk I will start by describing what Hamiltonian Floer homology is and how it relates to 1periodic orbits of a Hamiltonian flow. Then I will consider the case $\mathbb{R}^{2n}$ and describe how finite dimensional approximations lead to considering periodic cobordism theories (complex and real) as "coefficient rings". I will then in the more general case of a Liouville domain sketch how to define a spectrummodule over these coefficient rings with a set of generators in 11 correspondence with periodic orbits.
$\endgroup$ 
Chris SchommerPries (MIT)
$\begingroup $In this talk we report on joint work with Clark Barwick. We give a short list of axioms that a quasicategory should satisfy to be considered a reasonable homotopy theory of $(\infty,n)$categories. We show that the space of such quasicategories is homotopy equivalent to $\mathrm{B}(\mathbb{Z}/2)^{n}$, generalizing a theorem of Toën when $n=1$, and verifying two conjectures of Simpson. In particular, any two such quasicategories are equivalent. We also provide a large class of examples satisfying our axioms, including those of Joyal, Kan, Lurie, Simpson, and Rezk.
$\endgroup$ 
Oscar RandalWilliams (University of Copenhagen)
$\begingroup $Madsen and Weiss' theorem  identifying the stable homology of moduli spaces of Riemann surfaces  and the theorem of BarrattPriddy, Quillen, and Segal  identifying the stable homology of classifying spaces of symmetric groups  fit into a natural hierarchy of statements concerning the homology of moduli spaces of $(n1)$connected $2n$manifolds. I will discuss recent joint work with Søren Galatius which as a special case proves these statements for all $n = 3, 4, \ldots$
$\endgroup$ 
Vesna Stojanoska (MIT)
$\begingroup $It has been observed that certain localizations of the spectrum of topological modular forms tmf are selfdual (MahowaldRezk, GrossHopkins). We provide an integral explanation of these results that is internal to the geometry of the (compactified) moduli stack of elliptic curves $\mathcal{M}_{\text{ell}}$ yet is only true in the derived setting. When $p$ is inverted, choice of level$p$structure for an elliptic curve provides a geometrically wellbehaved cover of $\mathcal{M}_{\text{ell}}$, which allows one to consider tmf as the homotopy fixed points of tmf($p$), topological modular forms with level$p$structure, under a natural action by $\text{GL}_{2}(\mathbb{Z}/p)$. Specializing to $p=2$ or $p=3$ we obtain that as a result of GrothendieckSerre duality, tmf($p$) is self dual. The vanishing of the associated Tate spectra then makes tmf itself Anderson selfdual.
$\endgroup$

Matthew Gelvin (University of Copenhagen)
$\begingroup $Fusion systems are an algebraic description of the $p$local structure of a finite group. A centric linking system is the algebraic data needed to construct a topological classifying space for a fusion system; recent work of Andy Chermak shows that in fact, every saturated fusion system has a unique associated centric linking system. The collection of these data is what is known as a $p$local finite group.
In this talk, I will try to give some insight about the structure of $p$local finite groups. I will also give a brief discussion of the more topological notion of retractive transfer triple, which appears to be equivalent to the data of a $p$local finite group.
$\endgroup$ 
Gabriel Katz (MIT)
$\begingroup $Let $f: X \rightarrow \mathbb{R}$ be a Morse function on a manifold $X$ and $v$ its gradientlike vector field. Classically, the topology of a closed $X$ can be described in terms of the spaces of vtrajectories that link the singular points of $f$. On manifolds with boundary, the situation is somewhat different: there, a massive set of nonsingular functions is available. For such Morse data $(f, v)$, the interactions of the gradient flow with the boundary dX take central stage. We will introduce and measure the convexity and concavity of a $v$flow relative to $dX$. "Some manifolds are intrinsically more concave than others with respect to any gradient flow" is the main slogan of the talk. Stated differently, the intrinsic concavity of $X$ is a reflection of its complexity. We will explain how this approach leads to new topological invariants, both of the flow $v$ and of the manifold $X$. In 3D, we have a good grasp of these invariants and their connection to the classification of 3folds.
$\endgroup$ 
John Harper (University of Western Ontario)
$\begingroup $Quillen's derived functor notion of homology provides interesting and useful invariants in a variety of homotopical contexts, and includes as special cases (i) singular homology of spaces, (ii) homology of groups, and (iii) AndreQuillen homology of commutative rings. Working in the topological context of symmetric spectra, we study topological Quillen homology of commutative ring spectra, $E_n$ ring spectra, and more generally, algebras over any operad $O$ in spectra. Using a QHcompletion constructionanalogous to the BousfieldKan Rcompletion of spaceswe prove under appropriate conditions (a) strong convergence of the associated homotopy spectral sequence, and (b) that connected Oalgebras are QHcompletethus recovering the Oalgebra from its topological Quillen homology plus extra structure. A key problem in usefully describing this extra structure was solved recently using homotopical ideas in joint work with Kathryn Hess that describes a rigidification of the derived comonad that coacts on the object underlying topological Quillen homology, and plays the analogous role (in symmetric spectra) of the Koszul cooperad associated to a Koszul operad in chain complexes. This talk is an introduction to these results with an emphasis on proving (a) and (b) which is joint work with Michael Ching.
$\endgroup$ 
Michael Ching (Amherst College)
$\begingroup $Greg Arone and I have been trying to understand the structure that exists on the derivatives of a functor in the sense of Goodwillie calculus. Previously we have shown that these derivatives possess the structure of a coalgebra over a certain comonad on the category of symmetric sequences. In this talk I'll try to describe in more depth what this structure amounts to for functors from based spaces to spectra. Specifically I'll relate these coalgebras to right modules over the (Koszul duals of) the little disc operads. This is all joint work with Greg, with substantial input also from Bill Dwyer.
$\endgroup$ 
Clemens Berger (Université de Nice)
The Lattice Path Operad and Hochschild Cochains

Craig Westerland (University of Melbourne)
Hochschild homology of structured algebras: Vanishing results associated with the Costello construction of TCFT's on Hochschild homology

Aravind Asok (University of Southern California)
$\begingroup $We will discuss connectedness in unstable $A^1$homotopy theory, focusing on some examples. We will also discuss the classification of low dimensional $A^1$connected smooth proper varieties and its some similarities to and differences from the corresponding topological classifications.
$\endgroup$ 
Gereon Quick (Harvard University)
Homotopy fixed points of profinite spectra

Nick Rozenblyum (MIT)
$\begingroup $For an algebraic group $G$ and a projective curve $X$, we study the category of $D$modules on the moduli space $Bun_G$ of principal $G$bundles on $X$ using ideas from conformal field theory. We describe this category in terms of the action of infinitesimal Hecke functors on the category of quasicoherent sheaves on $Bun_G$. This family of functors, parametrized by the Ran space of $X$, act by averaging a quasicoherent sheaf over infinitesimal modifications of $G$bundles at prescribed points of $X$. We show that sheaves which are, in a certain sense, equivariant with respect to infinitesimal Hecke functors are exactly $D$modules, i.e. quasicoherent sheaves with a flat connection. This gives a description of flat connections on a quasicoherent sheaf on $Bun_G$ which is local on the Ran space.
$\endgroup$ 
Aravind Asok (University of Southern California)
$\begingroup $We will discuss connectedness in unstable $A^1$homotopy theory, focusing on some examples. We will also discuss the classification of low dimensional $A^1$connected smooth proper varieties and its some similarities to and differences from the corresponding topological classifications.
$\endgroup$ 
Don Davis (Lehigh University)
$\begingroup $We use the spectrum tmf to obtain new nonimmersion results for many real projective spaces $RP^n$ for n as small as 113. The only new ingredient is some new calculations of tmfcohomology groups. We present an expanded table of nonimmersion results. We also present several questions about tmf.
$\endgroup$ 
Justin Thomas (University of Notre Dame)
$\begingroup $We prove a conjecture of Kontsevich which states that if $A$ is an $E_{d1}$ algebra then the Hochschild cohomology object of $A$ is the universal $E_d$ algebra acting on $A$. The notion of an $E_d$ algebra acting on an $E_{d1}$ algebra was defined by Kontsevich using the swiss cheese operad of Voronov. We prove a homotopical property of the swiss cheese operad from which the conjecture follows.
$\endgroup$ 
Teena Gerhardt (Michigan State University)
$\begingroup $In this talk I will describe joint work with Vigleik Angeltveit, Mike Hill, and Ayelet Lindenstrauss, yielding new computations of algebraic Ktheory groups. In particular, we consider the Ktheory of truncated polynomial algebras in several variables. Techniques from equivariant stable homotopy theory are often key to algebraic Ktheory computations. In this case we use ncubes of cyclotomic spectra to compute the topological cyclic homology, and hence Ktheory, of the rings in question.
$\endgroup$ 
Tyler Lawson (University of Minnesota)
$\begingroup $Picard and Brauer groupoids, which capture information about invertible modules and central simple algebras, are objects of classical interest in algebra and number theory. The calculation of these objects is often aided by the use of Galois cohomology. We will discuss the generalizations to ring spectra, due to HopkinsMahowaldSadofsky and BakerRichter respectively. We then discuss how to compute these using Galois cohomology in higher category theory. (This talk based on joint work with David Gepner.)
$\endgroup$ 
Michael Ching (University of Georgia)
$\begingroup $I'll talk about joint work with Greg Arone to describe the data needed to reconstruct the Taylor tower of a functor from its layers. That data consists of a bimodule over the derivatives of the relevant identity functors, together with a coaction of a particular cotriple on the category of bimodules. I'll describe what we know about this cotriple for functors of based spaces and/or spectra.
$\endgroup$ 
John Harper (University of Western Ontario)
$\begingroup $We prove a finiteness theorem relating finiteness properties of topological Quillen homology groups and homotopy groups  this result should be thought of as an algebras over operads in spectra analog of Serre's finiteness theorem for the homotopy groups of spheres. We describe a rigidification of the derived cosimplicial resolution with respect to topological Quillen homology, and use this to define Quillen homology completion  in the sense of BousfieldKan  for algebras over operads in symmetric spectra. We prove that under appropriate connectivity conditions, the coaugmentation into Quillen homology completion is a weak equivalence  in particular, such algebras over operads can be recovered from their topological Quillen homology. Many of the results described are joint with K. Hess.
$\endgroup$

Kevin Costello (Northwestern University)
Some remarks on elliptic genera

Mark Walker (University of Nebraska)
$\begingroup $This is joint work with Guillermo Cortinas, Christian Hassemeyer and Chuck Weibel. Let $A$ be a commutative monoid and $k$ a field. What part of the algebraic Ktheory of the monoidring $k[A]$ comes from just the monoid and is independent of the field $k$? I describe a partial answer to this question, one which involves topological cyclic homology, toric varieties, and Voevodsky's cdh topology. I will also explain how our answer leads to a proof of Gubeladze's "nilpotence" conjecture for the algebraic Ktheory of toric varieties in arbitrary characteristic.
$\endgroup$ 
Robin Koytcheff (Brown University)
$\begingroup $Bott and Taubes considered a bundle over the space of knots whose fiber is a compactified configuration space, and they constructed knot invariants by performing integration along the fiber of this bundle. Their method was subsequently used to construct real cohomology classes in spaces of knots in $R^n$, $n>3$. Replacing integration of differential forms by a PontrjaginThom construction, I have constructed cohomology classes with arbitrary coefficients. Motivated by work of Budney and F. Cohen on the homology of the space of long knots in $R^3$, I have also proven a product formula for these classes with respect to connectsum. If time permits, I will mention some progress towards further understanding these classes using a cosimplicial model for knot spaces coming from the GoodwillieWeiss embedding calculus.
$\endgroup$ 
David Ayala (Harvard University)
$\begingroup $I will propose a simple and combinatorial $E_n$operad which is built out of finite posets indexing a stratification of configuration spaces of points in an $n$disk. This poset is constructed from a category $\theta_n$ which has recently become an important player for modeling weak $n$categories (Joyal, Berger, Rezk). The techniques involved use the formalism of quasicategories (Lurie). This project is joint with Richard Hepworth (Copenhagen) and is a work in progress.
One (nearly achieved) goal is to directly and geometrically relate three welldeveloped methods for recognizing $n$fold loop spaces: as certain algebras over the little $n$disk operad, as certain algebras over the BarrattEccles $E_n$ operad (via the Smith filtration), and as certain presheaves on $\theta_n$ (Berger). A version of Dunn's additivity theorem becomes a formal consequence of the setup. A farther away goal is to imitate the construction of topological chiral homology using this proposed $E_n$operad. This should have the benefit of making topological chiral homology (and possibly other field theories) more prepared for computations.
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Srinikanth Iyengar (University of Nebraska)
$\begingroup $I will describe recent work with Dave Benson and Henning Krause wherein we classify colocalizing subcategories of StMod(kG), for a finite group G, in the spirit of earlier work by Hopkins, Neeman, and Benson, Carlson, and Rickard on thick subcategories and localizing subcategories in various contexts. A central result in our work is a criteria for the vanishing of a function object, $Hom_k(M,N)$, in StMod(kG), in terms of geometric data (to be precise: support and cosupport) associated to M and N, via local cohomology and completion functors on the stable module category.
$\endgroup$ 
Kirsten Wickelgren (Harvard University)
$\begingroup $We show a 2nilpotent section conjecture over $R$: for a smooth curve $X$ over $R$ with negative Euler characteristic, $\pi_0(X(R))$ is determined by the maximal 2nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that the set of real points equipped with a real tangent direction of the smooth compactification of $X$ is determined by the maximal 2nilpotent quotient of $Gal(C(X))$ with its $Gal(R)$ action, showing a 2nilpotent birational real section conjecture.
$\endgroup$ 
Kate Ponto (University of Kentucky)
$\begingroup $The Lefschetz fixed point theorem gives a sufficient condition for a continuous endomorphism to have a fixed point: If the Lefschetz number of a continuous endomorphism of a closed smooth manifold is nonzero, that endomorphism has a fixed point. Usually no conclusions can be drawn if the Lefschetz number is zero, but with some (restrictive) hypotheses, there is a converse. I will describe an approach to the converse of the Lefschetz fixed point theorem using traces that also gives converses to the equivariant and fiverwise Lefschetz fixed point theorems.
$\endgroup$ 
Kyle Ormsby (MIT)
$\begingroup $I will discuss the (motivic, or $A^1$) homotopy theory of $G$equivariant schemes, $G$ a finite group. Stabilizing with respect to regular representation of spheres produces a good stable theory which, in the case $G=Z/2$, contains motivic analogues of Atiyah's Real Ktheory and Araki's Real cobordism over arbitrary characteristic $0$ base fields. The algebraic Real $K$theory spectrum is closely related to Hermitian $K$theory (a.k.a. higher GrothendieckWitt theory). Tools from stable equivariant topology like the Tate diagram and slice spectral sequence allow us to resolve the completion (or homotopy limit) problem for the Hermitian $K$theory of fields.
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Ieke Moerdijk (University of Utrecht)
$\begingroup $Lie groupoids (and Lie algebroids) play an increasingly important role in foliation theory, symplectic and Poisson geometry, and noncommutative geometry. In this lecture, I will explain how some basic properties of Lie groups extend to groupoids, and how some other properties don't.
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Chris SchommerPries (MIT)
$\begingroup $In this talk I will report on recent work, joint with Christopher Douglas and Noah Snyder, on understanding the nature of fully extended (a.k.a. local) 3dimensional topological quantum field theories. Specifically, we show that fusion categories are fullydualizable objects in the 3category of tensor categories, a natural categorification of the bicategory of algebras, bimodules, and bimodule maps. Fusion categories themselves are wellknown are arise in several areas of mathematics and physics  conformal field theory, operator algebras, representation theory of quantum groups, and others. In light of Hopkins and Lurie's work on the cobordism hypothesis, this provides a fully local TQFT for arbitrary fusion categories. Moreover, we will discuss how many familiar structures from the theory of fusion categories are given a natural explanation from this point of view.
$\endgroup$ 
David Spivak (MIT)
$\begingroup $In this talk I will give a brief survey of what I call "categorical informatics," which is the study of information and communication from a categorytheoretic perspective. In order to begin such a study, one must ground it in something concrete. To that end, I'll explain a simple categorical model of databases, which are realworld storehouses of organized information.
I will then discuss information transfer between databases and move on to define a more general notion of communication networks. Briefly, a communication network is a simplicial complex (of interacting groups) equipped with a sheaf of databases (common languages) on its simplices. I will show how this same structure can be pared down to give a categorical model for combining information obtained from various sources. I'll end by showing some interesting consequences of the topological nature of this model.
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Thomas Kragh (University of Oslo and MIT)
$\begingroup $The first part of this talk is basic definitions and statement of the Nearby Lagrangian Conjecture (NLC). I will then go on describing previous results relating to my talk especially that of Viterbo in 1998 and that of Fukaya, Seidel and Smith in 2007 (refined this year by Abouzaid to a very strong result in the case of vanishing Maslov class). I will then describe an approach using fibered spectra which partly unifies the two rather different approaches and then state new results in the general case (nonvanishing Maslov class) following from a product structure on a fibered spectrum combined with the intersection product on the base manifold (probably relatable to the pair of pants product and a twisted version of the ChasSullivan product).
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John Palmieri (University of Washington)
$\begingroup $The computer software package Sage (\texttt{www.sagemath.org}) understands some basic constructions from algebraic topology: it understands simplicial complexes, cubical complexes, and it seems to be the only major mathematical software package with an implementation of $\Delta$complexes. It can perform basic operations like joins, products, and connected sums. It can compute homology and cohomology over the integers or over a field. In this talk, we will discuss and demonstrate some of these capabilities, present some related unsolved problems, and discuss future directions.
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David Gepner (University of Illinois at Chicago)
Delooping the space of units of a commutative ring spectrum

Martin Frankland (MIT)
$\begingroup $A $\Pi$algebra is a graded group with additional structure that makes it look like the homotopy groups of a space. Given one such object $A$, one may ask if it can be realized topologically: Is there a space $X$ such that $\pi_*X$ is isomorphic to $A$ as a $\Pi$algebra, and if so, can we classify them?
Work of BlancDwyerGoerss provided an obstruction theory to realizing a $\Pi$algebra $A$, where the obstructions (to existence and uniqueness) live in certain Quillen cohomology groups of $A$. What do these groups look like, and can we compute them?
We will tackle this question from the algebraic side, focusing on Quillen cohomology of truncated $\Pi$algebras. We will then use the obstruction theory to obtain results on the classification of certain 2stage homotopy types, and compare them to what is known from other approaches.
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Martin Frankland (MIT)
$\begingroup $A $\Pi$algebra is a graded group with additional structure that makes it look like the homotopy groups of a space. Given one such object $A$, one may ask if it can be realized topologically: Is there a space $X$ such that $\pi_*X$ is isomorphic to $A$ as a $\Pi$algebra, and if so, can we classify them?
Work of BlancDwyerGoerss provided an obstruction theory to realizing a $\Pi$algebra $A$, where the obstructions (to existence and uniqueness) live in certain Quillen cohomology groups of $A$. What do these groups look like, and can we compute them?
We will tackle this question from the algebraic side, focusing on Quillen cohomology of truncated $\Pi$algebras. We will then use the obstruction theory to obtain results on the classification of certain 2stage homotopy types, and compare them to what is known from other approaches.
$\endgroup$ 
Angélica Osorno (MIT)
$\begingroup $In recent work of BaasDundasRichterRognes, the authors introduce the notion of the $K$theory of a bimonoidal category $R$, and show that it is equivalent to the algebraic $K$theory space of the ring spectrum $HR$. In this thesis we show that $K(R)$ is the group completion of the classifying space of the 2category $Mod_R$ of modules over $R$, and show that $Mod_R$ is a symmetric monoidal 2category. We explain how to use this symmetric monoidal structure to produce a $\Gamma$(2category), which gives an infinite loop space structure on $K(R)$. We show that the equivalence mentioned above is an equivalence of infinite loop spaces.
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Fabien Morel (LudwigMaximiliansUniversität)
$\begingroup $This is one of the main technical step in the proof of the FriedlanderMilnor conjecture, and uses a lot of "classical" topological arguments (lower central series, $A^1$derived functors of the free Lie algebra functors, devissage of the Lie algebra functor in terms of other polynomial functors.
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Gonçalo Tabuada (Universidade Nova de Lisboa)
$\begingroup $In this talk I will describe the construction of the category of noncommutative motives $[1,2,3]$ in DrinfeldKontsevich's noncommutative algebraic geometry program. In the process, I will present the first conceptual characterization of Quillen's higher Ktheory since Quillen's foundational work in the 70's. As an application, I will show how these results allow us to obtain for free the higher Chern character from Ktheory to cyclic homology.
References:
[1] D.C. Cisinski and G. Tabuada, Symmetric monoidal structure on Noncommutative motives. Available at arXiv:1001.0228.
[2] D.C. Cisinski and G. Tabuada, Nonconnective Ktheory via universal invariants. Available at arXiv:0903.3717.
[3] G. Tabuada, Higher Ktheory via universal invariants. Duke Math. Journal, 145 (2008), no.1, 121206. 2010/03/30,,,,Matthew Gelvin"
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Rob Thompson (CUNY)
$\begingroup $A $\Pi$algebra is a sequence of groups, along with an action of the primary homotopy operations, which are indexed by homotopy groups of wedges of spheres. It is natural to view the homotopy groups of a space as an object in the category of $\Pi$algebras. The $v_n$periodic homotopy groups of a space are obtained by inverting nonnilpotent self maps of finite complexes. In this talk I will investigate $v_n$periodic $\Pi$algebras.
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Andrew Salch (Johns Hopkins University)
$\begingroup $We discuss some new methods for computing the cohomology of Morava stabilizer groups at large heights. We apply these methods to compute a good approximation to the cohomology of the 5primary height 4 stabilizer group, giving us some interesting information about $v_4$periodic stable homotopy groups of spheres, and we discuss what is involved in computing the entire cohomology of the Morava stabilizer group for heights 4 and up.
$\endgroup$ 
Mark Behrens (MIT)
$\begingroup $Shimomura and Yabe computed the homotopy groups of the $E(2)$local sphere at primes greater than or equal to $5$. However, their computation is difficult to understand. I will describe a conceptual way to understand the answer, and the computation. The goal is to make the answer as concrete as the image of the $J$ homomorphism. We'll see of this goal is achieved...
$\endgroup$ 
Simona Paoli (Penn State Altoona)
$\begingroup $Homotopy $n$types are an important class of topological spaces: they amount to CW complexes whose homotopy groups vanish in dimension higher than $n$. The problem of modeling homotopy types is relevant both in higher category theory and homotopy theory and received contributions from both areas. There is a particularly simple model of homotopy types in the path connected case, consisting of nfold categories internal to groups, also called $cat^n$groups. This model, however, has the disadvantage that is it does not have an algebraic description of the Postnikov decomposition nor it is easy to establish algebraically when a map of $cat^n$groups is a weak equivalence.
In this talk we introduce a new model of connected $n$types through a subcategory of $cat^n$groups, which we call weakly globular, for which the above issues are resolved in transparent way. We also describe other homotopical properties of this model, and discuss the relevance of these structures for higher category theory.
$\endgroup$

Clark Barwick (Harvard University)
Equivariant derived algebraic geometry and Ktheory

Barry Walker (Northwestern University)
$\begingroup $Ando constructed power operations for the LubinTate cohomology theories using the theory of finite subgroups of a formal group. Moreover, he was able to produce a necessary and sufficient condition for a complex orientation of these cohomology theories to be compatible with the power operations. This result concerns the stable homotopy category of spectra. However, the LubinTate spectra of Morava are very rigid objects. Using ideas of Ando, Hopkins and Rezk, we can classify those orientations of complex KTheory that are compatible with Ando's power operations, but on the point set level. In this talk, we will show the equivalence of these two descriptions for complex padic KTheory. To achieve this goal, we use the language of Bernoulli numbers attached to a formal group law and their relationship with distributions on a $p$adic Lie group. Many of these tools were developed by N. Katz and J. Tate.
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Charles Rezk (University of Illinois UrbanaChampaign)
$\begingroup $This will be a Tuesday talk, at 4pm in 2132.
Morava Etheory (the complex oriented cohomology theories associated to deformations of formal groups) are structured commutative ring spectra, and so support a wellbehaved theory of power operations. We describe what is know about this theory, and we prove a conjecture of Ando, Hopkins, and Strickland, that the ring of power operations for such theories is Koszul.
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Rekha Santhanam (Johns Hopkins University)
$\begingroup $Costenoble and Waner showed that grouplike equivariant $E_\infty$ spaces model equivariant infinite loop spaces. Shimakawa gave an equivariant analog of $\Gamma$spaces to model equivariant infinite loop spaces. We describe equivariant $\Gamma$ spaces as defined by Shimakawa. We show that the categories of equivariant $E_\infty$spaces and equivariant $\Gamma$spaces are Quillen equivalent with appropriate model categories. Following Segal's work, we give a construction of equivariant $\Gamma$ spaces (and hence of equivariant infinite loop spaces) from symmetric monoidal $G$categories for finite group $G$.
$\endgroup$ 
Martin Bendersky (CUNY)
$\begingroup $The (stable) chromatic spectral sequence has had a significant impact on our understanding of the stable homotopy groups of the spheres. I will talk about preliminary attempts to construct an unstable version. I will try to describe a filtration of the stable chromatic spectral sequence induced by the Hopf rings for the odd spheres. There are natural questions that arise in the unstable world (e.g. an unstable version of the Morava stabilizer algebra) and a chromatic interpretation of the Hopf invariant.
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Tyler Lawson (University of Minnesota)
$\begingroup $Lurie's theorem allows the functorial construction of $E_\infty$ ring spectra associated to certain $p$divisible groups. In this talk I will discuss three situations in which we can apply this and attempts to understand the computational results. The first is joint work with Behrens on the relationship between the moduli of elliptic curves and certain moduli of abelian surfaces with complex multiplication. The second is joint work with Hill on Shimura curves that parametrize "false elliptic curves", and in particular trying to obtain computations of the homotopy of the associated spectra without niceties such as $q$expansions and Weierstrass equations. The third is on using Zink's work on displays to produce $E_\infty$ ring spectra from purely algebraic input data, in the form of invertible matrices over Witt rings.
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HansWerner Henn (University of Strasbourg)
The rationalization of the $K(2)$local sphere and Picard groups at chromatic level $2$ for $p=3$.

Kirsten Wickelgren (Harvard University)
$\begingroup $Grothendieck's anabelian conjectures say that hyperbolic algebraic curves over number fields should be $K(\pi,1)$'s in algebraic geometry. It follows that conjecturally the rational points on such a curve are the sections of étale $\pi_1$ of the structure map. These conjectures are analogous to equivalences between fixed points and homotopy fixed points of Galois actions on related topological spaces. We use cohomological obstructions of Jordan Ellenberg coming from nilpotent approximations to the curve to study the sections of étale $\pi_1$ of the structure map. We will relate Ellenberg's obstructions to Massey products, and explicitly compute mod $2$ versions of the first and second for $P^1\{0,1,\infty\}$ over $\mathbf{Q}$. Over $\mathbf{R}$, we show the first obstruction alone determines the connected components of real points of the curve from those of the Jacobian.
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Emmanuel Farjoun (Hebrew University)
Homotopy normality and homotopy ideals.

Gereon Quick (WWU Muenster)
$\begingroup $The playing field of profinite homotopy theory is provided by the homotopy categories of profinite spaces and profinite spectra. A motivating application is the connection to algebraic geometry. For example the etale fundamental group and continuous etale cohomology of a scheme can be defined in a unified way using a profinite etale realization functor. We will discuss this functor and use it to define etale topological cobordism. But it turned out that profinite structures might be useful in other areas such as LubinTate spectra. If time permits we will discuss this idea in progress as well.
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Søren Galatius (Stanford University)
Monoids of moduli spaces of manifolds.
This will be a Tuesday talk, at 4pm in 2132.

Mike Hill (Harvard University)
Equivariant computations and $\mathrm{RO}(G)$graded spectral sequences.

David Ayala (University of Copenhagen)
$\begingroup $This talk will motivate and develop a bordism category consisting of singular manifolds. Applications will be discussed having relevance to 'stable' characteristic classes of families of smooth manifolds and GromovWitten theory.
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Chris SchommerPries (Harvard University)
$\begingroup $The $6$connected cover of $\mathrm{Spin}(n)$, known as the group $\mathrm{String}(n)$, has fascinating connections with both abstract homotopy theory (through String Bordism and TMF) and with quantum field theory (through the 2D SUSY nonlinear sigma model). A better geometric understanding of String geometry has the potential to offer new interactions between these fields. Unfortunately all previous models of $\mathrm{String}(n)$ are infinite dimensional, making a thorough geometric understanding elusive. In this talk we will construct a finite dimensional model of $\mathrm{String}(n)$ as a higher categorical version of a group (known as a $2$group). In the process, we will "categorify" the classical notions of group cohomology and derived functor. In particular we will categorify Segal's topological group cohomology, thereby obtaining a classification of extensions of topological $2$groups.
$\endgroup$

Mike Hopkins (Harvard University)
The Kervaire Invariant
This talk will take place at the normal time (4:30) in 4270.

José Gómez (University of British Columbia)
$\begingroup $This will be a Tuesday talk, at 4:00 in 2151
In this talk we will show that the space of almost commuting elements in a compact Lie group splits after one suspension.
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Christian Ausoni (Universität Bonn)
Algebraic KTheory of Ktheory spectra.

David Spivak (University of Oregon)
$\begingroup $Given a smooth manifold $M$ and two submanifolds $A$ and $B$, their intersection need not be a smooth manifold. By Thom's transversality theorem, one can deform $A$ to be transverse to $B$ and take the intersection: the result, written $A\cap B$, will be a smooth manifold. Moreover, if $A$ and $B$ are compact, then there is a cup product formula in cobordism, integral cohomology, etc. of the form $[A]\cup[B]=[AB]$, where $[\cdot]$ denotes the cohomology fundamental class. The problem is that $AB$ is not unique, and there is no functorial way to choose transverse intersections for pairs of submanifolds. The goal of the theory of derived manifolds is to correct this defect. The category of derived manifolds contains the category of manifolds as a full subcategory, is closed under taking intersections of manifolds, and yet has enough structure that every compact derived manifold has a fundamental class. Even if the submanifolds $A$ and $B$ of $M$ are not transverse (in which case their intersection can be arbitrarily singular), their intersection $A\times_MB$ will be a derived manifold with $[A\times_MB]= [AB]$, and thus satisfy the above cup product formula. To construct the category of derived manifolds, one imitates the constructions of schemes, but in a smooth and homotopical way. I will begin the talk by explaining this construction. Then I will give some examples and discuss some features of the category of derived manifolds. I will end by sketching the ThomPontrjagin argument which implies that compact derived manifolds have fundamental classes.
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Stefan Hornet (Harvard University)
A generalization of a theorem of Ravenel and Wilson.

Fred Cohen (University of Rochester)
$\begingroup $This will be a Wednesday talk, at 4:00 in 2143.
A second subspace of a product is the generalized momentangle complex first defined in generality by Neil Strickland. Definitions, examples, as well as connections will be addressed.
One notable case is given by subspaces of products of infinite dimensional complex projective space 'indexed by a finite simplicial complex'. These spaces appearing in work of GoreskyMacPherson, DavisJanuskiewicz, BuchstaberPanovRay, DenhamSuciu, Franz as well as many others encode information ranging from the structure of toric varieties in one guise, StanleyReisner rings, as well as 'motions of certain types of robotic legs' in another guise.
What do these spaces have to do with the motions of legs of a cockroach? This feature will be illustrated with slides.
Features of these spaces are developed within the context of classical homotopy theory based on joint work with A. Bahri, M. Bendersky, and S. Gitler.
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Fred Cohen (University of Rochester)
$\begingroup $This will be a Tuesday talk, at 4:00 in 2151
The first basic example here is the configuration space of unordered $k$tuples of distinct points in a space $M$. When specialized to the case of $M$ given by the complex numbers, these spaces can be identified as the space of classical complex, monic polynomials of degree $k$ which have exactly k distinct roots.
Elementary features of these spaces as well as their connections to spaces of knots, links, and homotopy groups of spheres will be addressed. These topics are based on joint work with R. Budney as well as J. Berrick, Y. Wong and J. Wu.
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Fred Cohen (University of Rochester)
$\begingroup $The subject of this talk is the structure of the space of homomorphisms from a free abelian group to a Lie group $G$ as well as quotients spaces given by the associated space of representations. These spaces as well as further spaces of representations admit the structure of a simplicial space at the heart of the work here. Features of geometric realizations will be developed.
What is the fundamental group or the first homology group of the associated space in case $G$ is a finite, discrete group?
This deceptively elementary question as well as more global information given in this talk is based on joint work with A. Adem, E. Torres, and J. Gomez.
$\endgroup$ 
Nitu Kitchloo (University of California, San Diego)
$\begingroup $I will describe the stable (in genus) structure of the universal moduli space of flat connections on riemann surfaces. I will also introduce the category of $1$manifolds and $2$cobordisms endowed with flat connections. Using classical techniques of AtiyahBott, and more recent techniques introduced by MadsenWeiss and coauthors, we will give a complete description of the classifying space of this category. This is joint work with R. Cohen and S. Galatius.
$\endgroup$ 
Sam Isaacson (Harvard University)
$\begingroup $Let $C$ be a model category. In a 2001 paper, Dan Dugger showed that if $C$ is combinatorial, it can be realized as a left Bousfield localization of simplicial presheaves on some small site. I'll describe a variation of this theorem: by replacing simplicial sets with a cubical model for the homotopy category, we can produce a presentation for $C$ when $C$ is symmetric monoidal that retains the monoidal structure of $C$ as the Day convolution product.
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Mike Mandell (Indiana University)
Localization in $THH$ and $TC$.

Douglas Ravenel (University of Rochester)
Homotopy fixed point spectra for finite subgroups of the Morava stabilizer group.

Pascal Lambrechts (University of Louvain)
$\begingroup $This will be a Tuesday talk, at 4:00 in 2151.
.For a given compact smooth manifold $M$ we consider the space $\mathrm{Emb}(M,R^k)$ of smooth embeddings of M into some large Euclidean space $R^k$, or rather some geometric variant of it, which is a homotopy invariant of $M$.
I will explain how Goodwillie's cutting method enables us to understand the homotopy type of this space of emeddings. I will then prove that the rational homology of that space is actually an invariant of the rational homotopy type of $M$. The proof is based on Kontsevich's theorem on the formality of the little cube operad and Arone's description of the layers of Weiss' orthogonal tower for the space of embeddings. This is a joint work with Greg Arone and Ismar Volic.
$\endgroup$ 
John Harper (École Polytechnique Fédérale de Lausanne)
$\begingroup $In Haynes Miller's proof of the Sullivan conjecture on maps from classifying spaces, Quillen's derived functor notion of homology (in the case of commutative algebras) is a critical ingredient. This suggests that homology for the larger class of algebraic structures parametrized by an operad will also provide interesting and useful invariants. Working in the two contexts of symmetric spectra and unbounded chain complexes, we establish a homotopy theory for studying Quillen homology of modules and algebras over operads, and we show that this homology can be calculated using simplicial bar constructions. A key part of the argument is proving that the forgetful functor commutes with certain homotopy colimits. A larger goal is to determine the extra structure that appears on the derived homology and the extent to which the original object can be recovered from its homology when this extra structure is taken into account. This talk is an introduction to these results with an emphasis on several of the motivating ideas.
$\endgroup$ 
Samik Basu (Harvard University)
$\begingroup $Let $R$ be an $E_\infty$ ring spectrum. Given a map $f:X \rightarrow \mathrm{BGL}_1(R)$, we can construct a Thom spectrum $Xf$. If $f$ is a loop map, then there is an $A_\infty$ $R$ module structure on the Thom spectrum. I will consider various examples of these Thom spectra and construct $A_\infty$ structures on them. I will then use this identification to calculate Topological Hochschild Homology.
$\endgroup$ 
Brian Munson (Wellesley College)
$\begingroup $For smooth manifolds $P$, $Q$, and $N$, let $\mathrm{Link}(P,Q;N)$ denote the space of smooth maps of $P$ in $N$ and $Q$ in $N$ such that their images are disjoint. I will discuss the connectivity of a "generalized linking number" from the homotopy fiber of the inclusion of $\mathrm{Link}(P,Q;N)$ into $\mathrm{Map}(P,N)\times\mathrm{Map}(Q,N)$ to a certain cobordism space of manifolds over a space which is a homotopy theoretic model for the intersections of $P$ and $Q$. The proof of the connectivity uses some easy statements about connectivities in the world of smooth manifolds as a guide for obtaining similar estimates in a setting where the tools of differential topolgy do not apply. This is joint work with Tom Goodwillie.
$\endgroup$

Boris Botvinnik (University of Oregon)
Cobordism category of manifolds with positive scalar curvature

Kari Ragnarsson (DePaul University)
$\begingroup $This talk is to be held at 4pm in room 2151.
In this talk I will present recent work, joint with Radu Stancu, in which we obtain a bijection between saturated fusion systems on a finite pgroup S and idempotents in the double Burnside ring of S satisfying a "Frobenius reciprocity relation". (These terms will all be defined in the talk.) The theorem and its proof are purely algebraic, so I will focus attention on implications in algebraic topology, answering longstanding questions on the stable splitting of classifying space and generalizing a variant of the AdamsWilderson theorem, as well as the obvious implications for $p$local finite groups.
$\endgroup$ 
Matthias Kreck (Universität Bonn)
Codes, arithmetic and $3$manifolds.
This talk will begin at 5:00. Please note the time change.

John McCleary (Vassar College)
BorsukUlam phenomena.

Kathryn Lesh (Union College)
$\begingroup $I will discuss connections between the calculus of functors and the Whitehead Conjecture, both for the classical theorem of Kuhn and Priddy for symmetric powers of spheres and for the analogous conjecture in topological Ktheory. It turns out that key constructions in Kuhn and Priddy's proof have buanalogues, and there is a surprising connection to the stable rank filtration of algebraic Ktheory.
$\endgroup$ 
Bertrand Guillou (University of Illinois UrbanaChampaign)
$\begingroup $I will describe some joint work with J.P. May in which we investigate when enriched model categories can be modeled as enriched diagrams on a small (enriched) domain category. As an application, we are able to obtain a new model for the equivariant stable homotopy category of a compact Lie group.
$\endgroup$ 
Dev Sinha (University of Oregon)
$\begingroup $This talk is to be held at 4pm in room 2142.
Carefully developing the homology and cohomology of ordered configuration spaces leads to a pretty model for the Lie cooperad. We use this model to unify the Quillen approach to rational homotopy theory with the theory of Hopf invariants. We will also share progress on a new approach to the cohomology of unordered configurations spaces (i.e. symmetric groups), which are of course relevant to homotopy theory at p.
$\endgroup$ 
CarlFriedrich Bödigheimer (Universität Bonn)
$\begingroup $The symmetric groups $S_p$ are considered with the norm induced by the word length (with respect to transpositions as generators). This gives a filtration of their classifying spaces. Furthermore, using certain deletion functions $S_p \rightarrow S_{p1}$ the family of all symmetric groups can be regarded as filtered simplicial object. we show: in its realization, the stratum for norm equal to $h$ has several components, each being homoemorhic to a vector bundle over the moduli space $M_g,_1^m$ of genus $g$ surfaces with one boundary curve and m punctures (for $h =3D 2g + m$).
$\endgroup$ 
Matthew Ando (University of Illinois UrbanaChampaign)
$\begingroup $Let $R$ be an associative ring spectrum. I shall describe several new constructions of the $R$module Thom spectrum associated to a map $f: X \rightarrow \mathrm{BGL}_1 R$. The space $\mathrm{BGL}_1 R$ classifies the twists of $R$theory, and to a fibration of manifolds $g: Y \rightarrow X$ I shall associated an Umkehr map $g_!$ from the $fg$twisted $R$theory of $Y$ to the $f$twisted $R$ theory of $X$. In the case of Ktheory, this twisted Umkehr map appears in the study of $D$brane charge. I shall review this story, and then discuss the analogous construction for TMF.
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Mark Hovey (Wesleyan University)
$\begingroup $In joint work with Keir Lockridge, we have been developing theories of global and weak dimensions for ring spectra. We have good results for ring spectra of dimension zero, and partial results but good conjectures for the finite dimensional case.
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Larry Smith (GeorgAugustUniversität Göttingen)
Local cohomology, Poincare duality algebras, and Macaulay dual systems.
This is held in room 2142!

Alex Suciu (Northeastern University)
$\begingroup $The cohomology jumping loci of a space $X$ come in two basic flavors: the characteristic varieties (the jump loci for cohomology with coefficients in rank $1$ local systems), and the resonance varieties (the jump loci for the homology of the cochain complexes arising from multiplication by degree $1$ classes in the cohomology ring of $X$). I will discuss various ways in which the geometry of these varieties is related to the formality, quasiprojectivity, and homological finiteness propoerties of the fundamental group of $X$.
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Ismar Volic (Wellesley College)
$\begingroup $We will describe how (multivariable) manifold calculus of functors can be used for studying classical knots and links. In particular, this theory yields a classification of finite type invariants and Milnor invariants of knots, links, homotopy links, and braids. Another novelty is that a certain cosimplicial variant of manifold calculus provides a way for studying knots and links in a homotopytheoretic framework. Higherdimensional analogs will also be discussed. This is joint work with Brian Munson.
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Søren Galatius (Stanford University)
$\begingroup $The moduli space of Riemann surfaces $M$ is a classifying space for families of Riemann surfaces. It has a compactification $\bar M$, which is a classfying space for families of modal Riemann surfaces. A nodal Riemann surface is allowed to have singularities which look like the solutions to $zw=0$ in complex $2$space. I will describe how to decompose $\bar M$ as a homotopy colimit of spaces which look more like $M$. Then I will use this to study part of the homology of $\bar M$, using what is known about the homology of $M$.
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Birgit Richter (Universität Hamburg)
$\begingroup $On every bimonoidal category with antiinvolution, R, there is an involution on the associated Ktheory. This Ktheory is the algebraic Ktheory of the spectrum associated to R. In the talk I will construct this involution, discuss examples and indicate why the involution is nontrivial in several examples.
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