Cohomology of a smooth projective variety over the complex numbers admits a Hodge decomposition, and the de Rham algebra of smooth differential forms on it is canonically formal, as a commutative differential graded algebra. For a variety over a field $k$ of characteristic $p$ the analogous statements are no longer true. The de Rham cohomology algebra is in general not formal as an $\mathbb{E}_{\infty}$ algebra over $k$ as can be seen by considering the Frobenius action on de Rham cohomology. I will discuss a recipe for constructing examples of situations where (logarithmic) de Rham cohomology fails to have a Hodge decomposition, based on the discrepancy between the $\mathbb{E}_{\infty}$ algebra structures on de Rham and Hodge cohomology. Analogous construction also produces examples of smooth schemes over $\mathbb{Z}_p$ whose (logarithmic) prismatic cohomology has non-zero $u$-torsion.

Solid modules over $\mathbb{Q}$ or $\mathbb{F}_p$, introduced by Clausen and Scholze, are a well-behaved variant of complete topological vector spaces that forms a symmetric monoidal Grothendieck abelian category. For a discrete field k, we construct the category of ultrasolid k-modules, which specialises to solid modules over $\mathbb{Q}$ or $\mathbb{F}_p$. In this setting, we show some commutative algebra results like an ultrasolid variant of Nakayama's lemma. We also explore higher algebra in the form of animated and E8 ultrasolid k-algebras, and their deformation theory. We focus on the subcategory of complete profinite k-algebras, which we prove is contravariantly equivalent to equal characteristic formal moduli problems with coconnective tangent complex, and interpret this result in terms of Koszul duality.

Topological Hochschild homology (THH) is an invariant of rings closely related to algebraic K-theory. This talk will be about a new description of THH($\mathbb{Z}$) after completion at an odd prime, where $\mathbb{Z}$ denotes the ring of integers; the description is in terms of the image of J spectrum, a classical object in algebraic topology. I will also discuss how this reframes earlier work on this object and a new consequence for $K(1)$-localized algebraic K-theory. This is based on joint work with Sanath Devalapurkar.

In this talk we'll review how an Euler class valued in equivariant homotopical bordism can reveal conserved symmetries in the solutions to equivariant enumerative problems. We apply this idea in joint work with C. Bethea to compute bitangents to symmetric plane quartics, where we will see that homotopical techniques directly reveal patterns which are not obvious from a classical moduli perspective. We will also discuss ongoing work with S. Raman, in which we initiate a study of Galois groups of symmetric enumerative problems, leveraging tools from Hodge theory, hyperbolic geometry, and computational numerical analysis.

One of the key inputs to the proof of the celebrated Dundas–Goodwillie–McCarthy theorem is the computation of the topological cyclic homology of a trivial square zero extension. I will explain how to attach to a ring spectrum A and a nonunital (A,A)-bimodule I, a cyclotomic spectrum T(A,I), which can be used to perform the above calculation in a more or less formal way. Time permitting, I will discuss some further properties and consequences of this construction.

This talk will be a largely expository introduction to the theory of 'prismatic F-gauges,' as developed by Bhatt-Lurie and Drinfeld, intended for algebraic topologists. In particular, I will discuss the close relationship between F-gauges and p-divisible groups, due to Anschutz and Le Bras ('prismatic Dieudonne theory') and explain some extensions and complements based on the Artin-Lurie representability theorem.

I will talk about a 'normalizer decomposition' for the classifying space of a p-local compact group. The decomposition generalizes those of Dwyer for finite groups and of Libman for p-local finite groups and (separately) for compact Lie groups. I'll show how the decomposition gives a homotopy pushout square for the exotic p-compact groups of Aguade and Zabrodsky by building on the example of SU(p). This is joint work with Belmont, Castellana, Grbic, and Strumila.

In 2005, Conrey, Farmer, Keating, Rubinstein, and Snaith posed a conjecture on the asymptotics of moments of quadratic L-functions. 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.

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 Devinatz-Hopkins 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 non-archimedean fields. This is joint work with Tobias Barthel, Tomer Schlank, and Nathaniel Stapleton.

We shall consider the functor $L_{T(n)}K$ of chromatically localized algebraic K-theory. We shall discuss its interaction with pi-finite colimits. This will lead to a possible alternative characterization of this functor as well as results about it's interaction with cyclotomic hyper-descent. This is a key input to the proof of the telescope conjecture. This talk is based on joint works with Shay Ben-Moshe, Shachar Carmeli, and Lior Yanovski, as well as with Robert Burklund, Jeremy Hahn, and Ishan Levy.

The even filtration, introduced by Hahn-Raksit-Wilson, 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 Adams-Novikov filtration of the sphere or the Bhatt-Morrow-Scholze 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.

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 well-defined 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.

Since the mid 1980s, there have been hints of a deep connection between 2-dimensional 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 2-dimensional 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 2-dimensional field theories that realize some of the torsion in TMF.

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 Angelini-Knoll, 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.

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 lambda-ring in which all Adams operations act as the identity. A derived binomial ring is a derived Λ-ring 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.

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 K-theory 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.

A smooth projective variety $Z$ is said to be Calabi-Yau if its canonical bundle is trivial. I will discuss recent joint work with Taelman, in which we use derived algebraic geometry to study how Calabi-Yau varieties in characteristic $p$ deform. More precisely, we show that if $Z$ has degenerating Hodge-de Rham spectral sequence and torsion-free 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 Serre-Tate theory. Our work generalises results of Achinger-Zdanowicz, Bogomolov-Tian-Todorov, Deligne-Nygaard, Ekedahl-Shepherd-Barron, Schröer, Serre-Tate, and Ward.

The little d-discs 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.

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 differential-topological tools in algebraic geometry. I will survey various results in motivic homotopy on oriented intersections, fixed point theorems, framed cobordism, Morse theory, and the Poincaré-Hopf theorem. Time permitting, I will discuss how these tools relate to enumerative geometry over non-closed fields.

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.

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 quasi-isomorphisms of differential graded (dg) coalgebras after applying the normalized chains functor

2. maps of simplicial coalgebras which become quasi-isomorphisms of dg algebras after applying the normalized chains functor followed by the dg cobar construction, and

3. maps of simplicial coalgebras which become quasi-isomorphisms of dg algebras after applying a localized version of the dg cobar construction.

Notion (1) was used by Goerss to provide a fully-faithful model for spaces up to F-homology 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 quasi-categories. I will also explain how (3) leads to a fully-faithful model for the homotopy theory of simplicial sets considered up to maps that induce isomorphisms on fundamental groups and on the F-homology 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.

The study of the action of a finite p-group $G$ 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 K-theories (at the prime p). One such chromatic Smith theorem is proved by Barthel et.al.: if $G$ is a cyclic p-group and X is $K(n)$ acyclic, then $X^G$ is $K(n-1)$ acyclic (and this answers questions like this for all abelian p-groups).

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 p-group, then the dimension over $K(n)_\ast$ of $K(n)_\ast(X)$ will bound the dimension over $K(n-1)_\ast$ of $K(n-1)_\ast(X^G)$.

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 2-groups. In a different direction, at the prime 2, we can show quick collapsing of the AHSS computing the Morava K-theory of some real Grassmanians: this is a non-equivariant result.

In my talk, I'll try to give an overview of some of this.

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.

Syntomic complexes are a form of p-adic motivic cohomology that filter p-adic étale K-theory (or topological cyclic homology), and which are defined in terms of prismatic cohomology. I will explain a description of the syntomic complexes of p-torsionfree regular rings, based on a mixed characteristic analog of the Cartier isomorphism, closely related to the Segal conjecture for THH. (Joint with Bhargav Bhatt.)

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 Lubin-Tate 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.

Trying to do calculus in characteristic p returns strange results like the Cartier isomorphism: the deRham *cohomology groups* of a smooth F_p-variety 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 Cartier-like isomorphism in 'higher chromatic characteristics' (e.g. for connective real K-theory and connective topological modular forms).

Bachmann–Hopkins defines the motivic 'image-of-j' 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 R-motivic stable stems. This is joint work with Eva Belmont and Dan Isaksen.

Beilinson, Macpherson and Schechtman asked us to imagine a world where topological K-theory 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 K-theory, 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 Bhatt-Morrow-Scholze prismatic sheaves.

This is all joint work with Tom Bachmann and Matthew Morrow.

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.

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.

Hermitian K-theory 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 K-theory 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.

Abstract: Factorization homology is a natural invariant of manifolds and En-algebras. 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 E1-algebras. 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 Nikolaus-Scholze. 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.

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 K-theory 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_3-BP-algebra structure for all primes p and heights n. The algebraic K-theory 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.

In the 1970s Quillen proved that algebraic K-theory is homotopy invariant for a regular noetherian base. For a non-regular base ring this is not true. Bass defined the NK-groups in order to study the failure of homotopy invariance in K-theory. In general these groups are not well understood, though they have many interesting properties. Ten years ago, Cortiñas, Haesemeyer, Walker and Weibel used cdh-descent methods to understand the NK-groups 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.

At height $h=2^{n-1}m$, 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 Lubin-Tate spectra which were constructed in recent joint work with Hill-Shi-Zeng. 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 Hill-Shi-Zeng.

In a 2006 article Schlichting conjectured that the negative K-theory of any abelian category must vanish. And in a 2019 article Antieau, Gepner and Heller generalized, conjecturing that the negative K-theory of any infinity-category with a bounded t-structure must vanish.

We will review the history, explain why both conjectures are plausible, and then sketch a counterexample disproving both.

In 1996, Jens Franke conjectured that any stable infinity-category 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 infinity-categories.

Welded tangles are knotted surfaces in R^4. Bar-Natan 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 one-to-one correspondence between circuit algebras and a form of rigid tensor category called "wheeled props." This is a higher dimensional version of the well-known algebraic classification of planar algebras as certain pivotal categories.

This classification allows us to connect these "welded tangled foams," to the Kashiwara-Vergne 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 Kashiwara-Vergne conjecture. Moreover, we explain how this approach illuminates the close relationship between the group KV and the pro-unipotent Grothendieck–Teichmüller group.

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 high-dimensional 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.

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 n-dimensional polytopes, $P$, $Q$ are said to be scissors congruent if one can cut $P$ along a finite number of hyperplanes, and re-assemble 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 K-theory, 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.

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 non-transverse 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.

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.

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.

We'll discuss joint work with Jeremy Hahn surrounding the 'norm' construction of Hill-Hopkins-Ravenel 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 Eilenberg-MacLane 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 $C_2$.

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.

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.

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.

We construct a cellular decomposition of the Axelrod-Singer-Fulton-MacPherson compactification of the configuration spaces in the plane, that is compatible with the operad composition. Cells are indexed by trees with bi-coloured edges, and vertices are labelled by cells of the cacti operad. This answers positively a conjecture by Kontsevich and Soibelman.

Infamously, the motivic Hopf map $\eta$ is non-nilpotent, and the $\eta$-periodic motivic sphere spectrum detects this phenomenon. I will describe a slice-theoretic 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 Witt-theoretic $J$-spectrum," and also promise to help with slice computations of the motivic sphere. This is joint work with Oliver Röndigs.

(Joint work with Dominic Culver) Let $kq$ denote the very effective cover of Hermitian K-theory. 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 large-scale calculations of periodic phenomena, the $kq$-resolution can be used to calculate low-dimensional $2$-complete Milnor-Witt stems. I will present some calculations in the $C$- and $R$-motivic settings (recovering results of Dugger-Isaksen), then discuss work in progress towards analogous results over general base fields and the $2$-complete version of calculations of Röndigs-Spitzweck-Østvaer.

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 K-theory under universal homeomorphisms. The first is an equicharacteristic result: if $k$ is a field of exponential characteristic $c$, then homotopy K-theory is invariant under universal homeomorphisms of k-schemes, after inverting $c$; this is a consequence of joint work with Adeel Khan and implies the statement for usual K-theory in positive characteristics. The second is a mixed characteristic result for usual (but $p$-inverted) K-theory: 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.

(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 Barratt-Priddy(-Quillen-Segal) theorem, they are intimately related to $QX$, the free infinite loopspace on a space $X$.

While the mod-p homology of these constructions as well as a formulae for coproduct have been known since work of Nakaoka, Kudo-Arakai, Dyer-Lashof, and Cohen-Lada-May, 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 Dyer-Lashof 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.

(Report on work in progress, joint with Mike Hopkins.)

The geometric Hopf map $A^2\setminus 0\rightarrow P^1$ induces (after desuspension) a well-known 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 eta-periodic motivic stable homotopy category; this is particularly interesting/difficult locally at the prime 2.

We show that the 2-local, $\eta$-periodic sphere is the fiber of an Adams operation in the 2-local, 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 eta-periodic algebraic special linear cobordism.

I will provide an overview of these ideas and results.

I will survey the problem of classifying smooth, $(n-1)$-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 ($n-1$)-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 Randal-Williams.

Work of Galatius, Randal-Williams, and Krannich relates our theorem not just to the classification of ($n-1$)-connected ($2n$)-manifolds, but also to the calculation of their mapping class groups.

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.

A long-standing 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. Berwick-Evans which turns Segal's physically-inspired 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.

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.

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 (quasi-categories) 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 "model-independent", casting new light on the theory of quasi-categories while simultaneously generalizing it to other models. At the conclusion, we consider the question of the model-invariance of $\infty$-category theory, proving that $\infty$-categorical structures are preserved, reflected, and created by a number of "change-of-model" functors. As we explain, it follows that even the "analytically-proven" theorems that exploit the combinatorics of one particular model remain valid in the other "biequivalent" models.

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.

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 Kirby-Siebenmann, 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 second-Betti 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.

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 cell-diagrams, known results on the stable homotopy groups of spheres, and the $j$-based Atiyah-Hirzebruch 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.

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.

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 Hill-Hopkins-Ravenel, for applying $C_p$ equivariant homotopy theory to the study of Morava $E$-theory.

Constructing type n spectrum and understanding its periodic $v_n$-self-map 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 $v_2^1$-self-map. 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 $v_3^4$-self-map and give evidences in favor of this spectrum admitting $v_3^1$-self-map. Time permitting, I will also talk about the action of height $3$ Morava stabilizer group on Morava E-theory 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.

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 Eilenberg-MacLane.

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 Adams-Priddy 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 2-completion of $tmf_1(3)$. Our computations seem suggestive of broader phenomena, and I hope to end with several open questions.

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 Hesselholt-Madsen'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.

The notion of unital $2$-Segal space was defined independently by Dyckerhoff-Kapranov and Galvez-Carrillo-Kock-Tonks as a generalization of a category up to homotopy. The notion of unital 2-Segal space was defined independently by Dyckerhoff-Kapranov and Galvez-Carrillo-Kock-Tonks 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.

It is a well-known heuristic that the Goodwillie derivatives of the identity functor on a (pointed compactly-generated) $∞$-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 "pro-operad" and we conjecture that the Taylor towers of functors from $C$ to spectra are classified by "modules" over that pro-operad, generalizing results with Arone in the cases where $C$ is based spaces or spectra.

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.

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 $2$.

We discuss joint work with Søren Galatius and Oscar Randal-Williams on the application of higher-algebraic 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.

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 Drummond-Cole.

Motivated by a program of Hill-Hopkins-Ravenel 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 Hill-Hopkins-Ravenel. 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.

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 exit-path 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.

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 non-abelian 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.

This talk concerns a twenty-thousand-year 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 Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze 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 Hasse-Weil zeta function by regularized determinants envisioned by Deninger.

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.

Real Johnson-Wilson 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 Hu-Kriz and later by Kitchloo-Wilson. Real Johnson-Wilson theories are defined as fixed points of an involution acting on the complex-oriented Johnson-Wilson theories $E(n)$, but they are themselves not complex oriented.

Real Johnson-Wilson 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 Johnson-Wilson cohomology is a (Bockstein-type) 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 Landweber-flat $E(n)$-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 Johnson-Wilson cohomology of the Eilenberg-MacLane 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.

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.

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 self-intersections 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.

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. Twenty-five 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 Hans-Werner Henn exploring this circle of ideas.

The algebraic K-theory of integral group rings has been of interest since the early days of K-theory. Computations, however, have proven difficult. In this talk I will describe joint work with Vigleik Angeltveit on a strategy for studying the algebraic K-theory 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 K-theory groups.

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.

I will describe a construction providing Lie algebras with enveloping algebras over the operad of little n-dimensional disks for any n. These algebras enjoy a combination of good formal properties and computability, the latter afforded by a Poincare-Birkhoff-Witt 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.

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 Kampren-Flores 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).

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 'figure-of-eight' product, through which transgression from M factors as an isomorphism, and which can be iterated to higher loop spaces.

Recent work of Röndigs-Spitzweck-Østvær sharpens the connection between the slice and Novikov spectral sequences. Using classical vanishing lines for the $E_2$-page of the Adams-Novikov spectral sequence and the work of Andrews-Miller 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 Milnor-Witt 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.

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 sub-schemes.

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.)

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 Dyer-Lashof 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 $A$.

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 Morel-Voevodsky. They are different in general. The former often carries relevant algebro-geometric 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.

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 non-linear 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.

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.

Let $L$ be an exact Lagrangian submanifold of a cotangent bundle $T^*M$. If a topological obstruction vanishes, a Floer-theoretic 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.

Chromatic homotopy theory is a divide-and-conquer 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".

A celebrated result of Eisenbud--Kimshaishvili--Levine 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 Eisenbud--Khimshiashvili--Levine, 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.

The secondary $K$-theory of a derived stack $X$ is the $K$-theory of 2-vector bundles on $X$, also known as smooth proper $k$-linear dg-categories 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 2-vector bundle a torus-invariant 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.

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.

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.

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 Hopkins-Kuhn-Ravenel character theory for all saturated fusion systems by building on the theorems for finite $p$-groups.

The Bousfield-Kan (or unstable Adams) spectral sequence can be constructed for various homology theories such as Brown-Peterson homology $BP$, Johnson-Wilson 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 Bousfield-Kan spectral sequence.

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 EI-categories. The proof involves generalizing the notion of trace from the realm of categories to derivators. It will be sketched if time permits.

The total surgery obtruction invariant was introduced 35 years ago to unify the two stages of the classical Browder-Novikov-Sullivan-Wall surgery theory of topological manifold types in the homotopy type of an n-dimensional 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 null-cobordism 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 null-cobordism. 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.

The first definitions of equivariant algebraic K-theory were given in the early 1980's by Fiedorowicz, Hauschild and May, and by Dress and Kuku; however these early space-level 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 G-action (not necessarily trivial) on the input as a genuine G-spectrum. I will discuss some of the subtleties involved in turning a ring or category with G-action into the right input for equivariant algebraic K-theory, and some of the properties of the resulting equivariant algebraic K-theory G-spectrum. I will also discuss recent developments in equivariant infinite loop space theory (e.g., multiplicative structures) that should have long-range applications to equivariant algebraic K-theory.

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.

Every finite group G gives rise to a saturated fusion system consisting of a Sylow p-subgroup 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 Broto-Levi-Oliver 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.

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 K-theory. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic K-theory. The index map allows us to relate the Contou-Carrere symbol, a local analytic invariant of families of schemes, to algebraic K-theory. Using this, we prove reciprocity laws for Contou-Carrere 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.

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 Galatius-Madsen-Tillmann-Weiss 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 long-range behavior of special condensed matter systems.

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.

I will explain a homotopical treatment of intersection cohomology recently developed by Chataur-Saralegui-Tanré, which associates a "perverse homotopy type" to every singular space. In this context, there is a notion of "intersection-formality", 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 intersection-formality results (work in progress with David Chataur).

Let $X$ be a smooth projective variety over the real numbers and let $f: X \to X$ be a self-map. 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 fixed-point formula for $f$ subsumes the topological fixed-point 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 Grothendieck-Lefschetz trace formula.

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}).

[Note that that the automatically produced topology seminar poster corresponding to this talk is incorrect - this will be a colloquium talk be held in E25-111, 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.

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.

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 so-called "generalized Miller-Morita-Mumford classes". In the case when the manifold $M=S_g$ is a surface, the Madsen-Weiss theorem together with Harer stability imply that as $g$ grows, a large number of these classes become *non-zero*. 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 Randall-Williams proved an analogue of the Madsen-Weiss 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 MMM-classes, other vanishing and non-vanishing results about the MMM-classes for such manifolds, and whether BDiff $M$ could be modeled by a finite-dimensional space.

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, non-perfect 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.

Poitou-Tate 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 Galois-equivariant Brown-Comenetz duality. We believe this upgraded duality should lead to a better understanding of rational points on algebraic varieties.

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 integer-graded stable homotopy groups of the $C_2$-equivariant sphere spectrum and the motivic sphere spectrum over $k$.

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 pushout-products 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.

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.

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 50-stem.

2) a brute force approach to the existence of the 62-dimensional Kervaire class.

3) a conjectural description of the homotopy groups of the eta-local motivic sphere.

4) an outline of a program to compute new stable stems by combining motivic Adams E2-term data with classical Adams-Novikov E2-term data.

For an affine (derived) scheme, the global sections functor from quasi-coherent 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 self-duality of Tmf. At the end, we will report on work in progress to extend these results to topological automorphic forms.

Atiyah-Segal 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 higher-chromatic 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.

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 "wall-crossing formula" relating the $E_2$ pages of the spectral sequence associated to a filtration. I started looking into this because I wanted to encode Hall-algebra-like structures on the Ran space of the circle, so I will discuss that as motivation first.

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.

We will explain our recent joint work with G. Cortinas, M. Walker and C. Weibel concerning properties of the algebraic K-theory 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 so-called 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").

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 higher-dimensional 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 $* \leq (g-4)/2$ is described in terms of a single characteristic class in a twisted cobordism group. This is joint work with O. Randal-Williams.

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 G-spectrum via the work of Guillou-May. This machine readily applies to produce Eilenberg-MacLane spectra for Mackey functors and topological K-theory.

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.

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 Lewis-May-Steinberger 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.

The Picard scheme $\text{Pic}^0$ representing invertible sheaves can be compactified by a moduli space $J$-bar of rank 1, torsion-free sheaves called the compactified Jacobian. For a smooth algebraic curve $X$ over a field $k$ with boundary $\partial X$, applying $H_1$ to the Abel-Jacobi 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 Abel-Jacobi 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.

Quasi-categories (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 ground-level approach that allows for 'formal' re-proofs of these facts that requires only very mild model category prerequisites and hence generalizes. A highlight will be the construction and characterization of the quasi-category of algebras associated to a homotopy coherent monad. This is a progress report on ongoing joint work with Dominic Verity.

Given a fibration $f:X \to S$ of CW-complexes 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.

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.

We combine three strategies to studying cooperations in connective topological modular forms: the Adams spectral sequence and its relation to Brown-Gitler modules following Mahowald's approach to cooperations in connective real K theory, Laures's theory of q-expansions of multi-variable 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.

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 algebro-geometric interpretations. Using the relationship between the character maps and the transfer maps for Morava $E$-theory, I will provide algebro-geometric interpretations of the cohomology of some finite groups other than symmetric groups and cyclic groups.

The motivic truncated Brown-Peterson spectra BP$<$n$>$ interpolate between motivic cohomology (BP$<$0$>$), algebraic K-theory (BP$<$1$>$), and the motivic Brown-Peterson spectrum itself, a close relative of algebraic cobordism. We use the motivic Adams spectral sequence and global-to-local comparison maps to compute the BP$<$n$>$-homology of the rational numbers. Along the way, we prove a Hasse principle for the motivic BP$<$n$>$ and deduce several classical and recent theorems about the algebraic $K$-theory of particular fields. This is joint work with Paul Arne Østvær.

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 non-trivial? Here are the main results. If $k > (m+1)/2$, then there are homotopically non-trivial maps from $S^m$ to $S^{m-1}$ with arbitrarily small $k$-dilation. But if $k \leq (m + 1)/2$, then every homotopically non-trivial map from $S^m$ to $S^{m - 1}$ has $k$-dilation at least $c(m) > 0$.

We compute the motivic slices of hermitian $K$-theory and higher Witt-theory. 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.

I will discuss recent joint work with Oscar Randal-Williams 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 $n=1$.

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.

The algebraic K-theory of the complex cobordism spectrum is an object of basic interest, both because it provides an interesting example of K-theory of a non-classical 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 Hill-Hopkins-Ravenel norm and a modernized view of equivariant infinite loop space theory. This is joint work with Angeltveit, Gerhardt, and Hill.

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 Eilenberg-Steenrod 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 3-manifolds in factorization terms, which, time permitting, I'll outline.

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 K-theory, and periodic cyclic homology, respectively.

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, Dwyer-Kan localizations arise from forgetting stages of the filtration. Such a result recovers elementary properties of chiral homology.

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 $x \otimes 1-x$ for $x$ in $k - \{0,1\}$ mod 2. In particular, $x \cup 1-x$ vanishes, where $x$ in $k^{*}$ is identified with an element of $H^{1}$. We show that order $n$ Massey products of $n-1$ factors of $x$ and one factor of $1-x$ 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 $\langle 1-x, x, \ldots , x , 1-x\rangle$ with $f \cup 1-x$, where $f$ is a certain cohomology class which arises in the description of the action of $\text{Gal}(\bar{k}/k)$ on $\pi_{1}^{\text{et}}(\mathbb{P}^1 - \{0,1,\infty\})$.

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 Behrens-Lawson spectrum $Q(l)$. This is current work with Mark Behrens, Nat Stapleton, and Vesna Stojanoska.

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 $n$.

In this talk I will start by describing what Hamiltonian Floer homology is and how it relates to 1-periodic 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 spectrum-module over these coefficient rings with a set of generators in 1-1 correspondence with periodic orbits.

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.

Madsen and Weiss' theorem --- identifying the stable homology of moduli spaces of Riemann surfaces --- and the theorem of Barratt-Priddy, 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 $(n-1)$-connected $2n$-manifolds. I will discuss recent joint work with Søren Galatius which as a special case proves these statements for all $n = 3, 4, \ldots$

It has been observed that certain localizations of the spectrum of topological modular forms tmf are self-dual (Mahowald-Rezk, Gross-Hopkins). 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 well-behaved 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 Grothendieck-Serre duality, tmf($p$) is self dual. The vanishing of the associated Tate spectra then makes tmf itself Anderson self-dual.

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 K-theory of the monoid-ring $k[A]$ 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 K-theory of toric varieties in arbitrary characteristic.

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 Pontrjagin-Thom 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 connect-sum. If time permits, I will mention some progress towards further understanding these classes using a cosimplicial model for knot spaces coming from the Goodwillie-Weiss embedding calculus.

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 quasi-categories (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 well-developed methods for recognizing $n$-fold loop spaces: as certain algebras over the little $n$-disk operad, as certain algebras over the Barratt-Eccles $E_n$ 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.

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.

We show a 2-nilpotent section conjecture over $R$: for a smooth curve $X$ over $R$ with negative Euler characteristic, $\pi_0(X(R))$ is determined by the maximal 2-nilpotent 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 2-nilpotent quotient of $Gal(C(X))$ with its $Gal(R)$ action, showing a 2-nilpotent birational real section conjecture.

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.

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 K-theory 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 Grothendieck-Witt 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.

Lie groupoids (and Lie algebroids) play an increasingly important role in foliation theory, symplectic and Poisson geometry, and non-commutative geometry. In this lecture, I will explain how some basic properties of Lie groups extend to groupoids, and how some other properties don't.

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) 3-dimensional topological quantum field theories. Specifically, we show that fusion categories are fully-dualizable objects in the 3-category of tensor categories, a natural categorification of the bicategory of algebras, bimodules, and bimodule maps. Fusion categories themselves are well-known 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.

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 category-theoretic 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 real-world store-houses 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.

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 (non-vanishing 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 Chas-Sullivan product).

Ando constructed power operations for the Lubin-Tate 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 Lubin-Tate spectra of Morava are very rigid objects. Using ideas of Ando, Hopkins and Rezk, we can classify those orientations of complex K-Theory 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 p-adic K-Theory. 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.

Morava E-theory (the complex oriented cohomology theories associated to deformations of formal groups) are structured commutative ring spectra, and so support a well-behaved 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.

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 $G$.

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.

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.

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.

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 Lubin-Tate spectra. If time permits we will discuss this idea in progress as well.

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 Gromov-Witten theory.

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 non-linear 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.

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 p-group 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 long-standing questions on the stable splitting of classifying space and generalizing a variant of the Adams-Wilderson theorem, as well as the obvious implications for $p$-local finite groups.

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 K-theory. It turns out that key constructions in Kuhn and Priddy's proof have bu-analogues, and there is a surprising connection to the stable rank filtration of algebraic K-theory.

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.

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.

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 $S_p \rightarrow S_{p-1}$ 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 $h =3D 2g + m$).

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 K-theory, 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.

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.

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, quasi-projectivity, and homological finiteness propoerties of the fundamental group of $X$.

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 homotopy-theoretic framework. Higher-dimensional analogs will also be discussed. This is joint work with Brian Munson.

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 $M$.

On every bimonoidal category with anti-involution, R, there is an involution on the associated K-theory. This K-theory is the algebraic K-theory of the spectrum associated to R. In the talk I will construct this involution, discuss examples and indicate why the involution is non-trivial in several examples.