# Past Talks

## Fall 2018

• Sep 172018

### Jeremy Hahn (MIT)

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#### The spectrum of units of a height 2 theory

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.

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• Sep 102018

### Akhil Mathew (University of Chicago)

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#### $p$-adic $K$-theory and topological cyclic homology

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.

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## Spring 2018

• May 142018

### Hiro Tanaka (Harvard University)

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#### Open and closed questions about cobordisms of Lagrangians

I'll talk about my attempts at applying modern homotopy-theoretic techniques to study (i) the theory of Lagrangian cobordisms, and (ii) its applications to Lagrangian Floer theory and Fukaya categories. It's a tantalizing world where Thom spaces aren't the obvious doorway to cobordism spectra, where there seem to be geometric explanations for really rich algebraic structures (like the paracyclic action on the s-dot construction), and where (frankly) not nearly enough is known.

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• May 072018

### Jeremiah Heller (University of Illinois Urbana-Champaign)

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#### A motivic homotopy limit problem

The seminar will meet at 3:00 PM in 2-236.

One formulation of Segal's conjecture is as a comparison between fixed points and homotopy fixed points of the sphere, for a finite group. I'll discuss a generalization for the motivic sphere, based on joint work with several collaborators.

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• May 072018

### Vesna Stojanska (University of Illinois Urbana-Champaign)

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#### Homotopical Motivic Galois Extensions

I will discuss a formal framework for homotopical Galois extensions, and consider a couple of different adaptations in the context of motivic homotopy theory. Each will be accompanied with interesting basic examples, including Eilenberg-MacLane and K-theory spectra. This is based on work with Beaudry, Heller, Hess, Kedziorek, and Merling.

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• Apr 302018

### Eva Belmont (MIT)

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#### Localization at $b_{10}$ in the stable category of comodules over the Steenrod reduced powers

Chromatic localization can be seen as a way to calculate a particular infinite piece of the homotopy of a spectrum. For example, the (finite) chromatic localization of a $p$-local sphere is its rationalization, and the corresponding chromatic localization of its Adams $E_2$ page recovers just the zero-stem. We study a different localization of Adams $E_2$ pages for spectra, which recovers more information than the chromatic localization. This approach can be seen as the analogue of chromatic localization in a category related to the derived category of comodules over the dual Steenrod algebra, a setting in which Palmieri has developed an analogue of chromatic homotopy theory. We work at $p=3$ and compute the $E_2$ page and first nontrivial differential of a spectral sequence converging to $b_{10}^{-1}Ext_P^*(\mathbb{F}_3,\mathbb{F}_3)$ (where $P$ is the Steenrod reduced powers), and give a complete calculation of other localized Ext groups, including $b_{10}^{-1}Ext_P^*(\mathbb{F}_3,\mathbb{F}_3[\xi_1^3])$.

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• Apr 232018

### David Gepner (Purdue)

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#### K-theoretic obstructions to bounded t-structures

No abstract

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• Apr 092018

### Gijs Heuts (University of Utrecht)

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#### Lie algebras and $v_n$-periodic spaces

I will discuss a homotopy theory obtained from that of pointed spaces by inverting the maps inducing isomorphisms in $v_n$-periodic homotopy groups. The case $n = 0$ corresponds to rational homotopy theory. In analogy with Quillen’s results in the rational case, I will outline how this $v_n$-periodic homotopy theory is equivalent to the homotopy theory of Lie algebras in $T(n)$-local spectra. One can also compare it to the homotopy theory of cocommutative coalgebras in $T(n)$-local spectra. For $n > 0$ these theories are no longer equivalent; the failure can be expressed in terms of the convergence of the Goodwillie tower of the identity in periodic homotopy.

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• Apr 022018

### Sam Nariman (Northwestern University)

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#### On the moduli space of flat symplectic surface bundles.

There are at least three different approaches to construct characteristic invariants of flat symplectic bundle. Reznikov generalized Chern-Weil theory for finite dimension Lie groups to the infinite dimensional group of symplectomorphisms. He constructed nontrivial invariants of symplectic bundles whose fibers are diffeomorphic to complex projective spaces. Kontsevich used formal symplectic geometry to build interesting classes that are not yet known to be nontrivial. For surface bundles whose holonomy groups preserve the symplectic form, Kotschick and Morita used the flux homomorphism to construct many nontrivial stable classes.

In this talk, we introduce infinite loop spaces whose cohomolgy groups describe the stable characteristic invariants of symplectic surface bundles. As an application, we give a homotopy theoretic description of Kotschick and Morita's classes and prove a result about codimension 2 foliations that implies the nontriviality of KM classes.

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• Mar 192018

### Maria Basterra (University of New Hampshire)

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#### Operads with Homological Stability detect Infinite Loop Spaces.

An earlier result of Tillmann shows that a carefully constructed surface operad detects infinite loop spaces in the sense that its algebras group complete to infinite loop spaces. An important ingredient in this result is Harer's homological stability theorem for mapping class groups of surfaces.

Motivated by this example, we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions (OHS). We prove that such operads detect infinite loop spaces in the sense above. In fact, we show that for these operads group completion provides a functor from their category of algebras to the category of infinite loop spaces.

I will sketch the proof of this result and, as time permits, present some examples and applications.

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• Mar 122018

### Mark Behrens (University of Notre Dame)

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#### A $C_2$-equivariant analog of Mahowald's Thom spectrum theorem

I will show that the $C_2$-equivariant Eilenberg-MacLane spectrum associated to the constant mod 2 Mackey functor can be realized as a Thom spectrum. This is joint work with Dylan Wilson.

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• Mar 052018

### Alex Suciu (Northeastern University)

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#### Algebraic models, duality, and resonance

I will discuss some of the interplay between duality and finiteness properties of spaces and groups, the structure of differential graded algebra models associated to them, and the geometry of the corresponding cohomology jump loci.

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• Feb 122018

### Bena Tshishiku (Harvard University)

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#### Cohomology of arithmetic groups and characteristic classes of manifold bundles

A basic problem in the study of fiber bundles is to compute the ring $H^*(BDiff(M))$ of characteristic classes of bundles with fiber a smooth manifold M. When M is a surface, this problem has ties to algebraic topology, geometric group theory, and algebraic geometry. Currently, we know only a very small percentage of the total cohomology. In this talk I will explain some of what is known and discuss some new characteristic classes (in the case dim M >>0) that come from the unstable cohomology of arithmetic groups.

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## Fall 2017

• Dec 112017

### Julie Bergner (University of Virginia)

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#### The Waldhausen $S$-construction as an equivalence of homotopy theories

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.

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• Dec 042017

### Michael Ching (Amherst College)

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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.

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• Nov 272017

### Kathryn Mann (Brown University)

$\begingroup \def\Homeo{\operatorname{Homeo}} \def\Diff{\operatorname{Diff}}$

#### New progress on spaces of foliated bundles and nonlinear character varieties

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.

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• Nov 202017

### Doug Ravenel (University of Rochester)

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#### Two equivariant approaches to the telescope conjecture

Reporting on joint work with Zhouli Xu and 4 others, I will talk about how equivariant techniques MIGHT be used to settle the telescope conjecture at the prime $2$.

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• Nov 132017

### Sander Kupers (Harvard University)

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#### Cellular $E_2$-algebras and homological stability

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.

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• Nov 062017

### Ben Knudsen (Harvard University)

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#### Subdivisional spaces and graph braid groups

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.

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• Oct 232017

### Dylan Wilson (University of Chicago)

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#### Slice spheres in equivariant and chromatic homotopy theory

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.

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• Oct 162017

### David Ayala (Montana State University)

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#### The Bruhat stratified orthogonal group acts on higher categories by adjoints

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.

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• Oct 022017

### Craig Westerland (University of Minnesota)

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#### Fox-Neuwirth/Fuks cells, quantum shuffle algebras, and Malle's conjecture for function fields

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.

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• Sep 282017

### Lars Hesselholt (Nagoya University and University of Copenhagen)

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#### Higher algebra and arithmetic

The seminar will meet at 4:30 PM in 2-190.

This talk is part of the Brandeis–Harvard–MIT–Northeastern joint Mathematics Colloquium

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.

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• Sep 252017

### Nora Ganter (University of Melbourne)

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#### Examples of categorical groups

The notion of categorical groups, also known as $2$-groups is increasingly recognised as the formalism governing symmetries in the context of string theory. The talk aims to fill this formalism with life by providing an interesting set of examples, together with a link to the stable three stem.

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• Sep 182017

### Nitu Kitchloo (Johns Hopkins University)

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#### Real Johnson-Wilson theories and Landweber flat exact pairs

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.

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• Sep 112017

### Zhouli Xu (MIT)

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#### Progress report on computing stable stems

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.

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## Spring 2017

• May 152017

### John Francis (Northwestern University)

$\begingroup \def\TQFT{\mathop{\mathrm{TQFT}}}\def\obj{\mathop{\mathrm{obj}}}$

#### Factorization homology and the cobordism hypothesis

The cobordism hypothesis—after Baez–Dolan, Costello, Hopkins–Lurie, and Lurie—asserts that for a symmetric monoidal $(\infty,n)$-category $C$ in which every object has a dual and every $k$-morphism has a left and right adjoint for $k<n$, there is an equivalence $\TQFT(C) = \obj(C)$ between $C$-valued framed topological field theories and objects of $C$. This is the formulation due to Lurie. I'll give a proof of the cobordism hypothesis based on factorization homology. Factorization homology is a multiplicative analogue of ordinary homology. Usual homology integrates an abelian group, chain complex, or spectrum over a manifold $M$, which one can think as the moduli space of points in $M$ itself. The result takes disjoint unions of manifolds to direct sums. The alpha version of factorization homology integrates an $E_n$-algebra over a moduli space of finite subsets of a manifold $M$. The beta version of factorization homology integrates an $(\infty, n)$-category over the moduli space of stratifications of $M$. The result takes disjoint unions to tensor products. I'll define this beta version of factorization homology. It satisfies a version of the Eilenberg–Steenrod axioms—this part is work in progress. These Eilenberg–Steenrod axioms together with an argument in the spirit of Pontryagin–Thom theory implies the cobordism hypothesis. This is joint work with David Ayala.

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• May 082017

### Hiro Tanaka (Harvard University)

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#### Morse theory and a stack of broken lines

In Morse theory, it's important to understand how a gradient trajectory can break into multiple gradient trajectories--for instance, this is how one proves the Morse chain complex is a chain complex. In this talk, I'll define a stack classifying families of domains of such broken and breaking trajectories. I'll then sketch a proof that the $\infty$-category of factorizable sheaves on this stack is equivalent to the $\infty$-category of non-unital $A_\infty$ algebras. Much of this is motivated by a strategy to formulate Morse theory and Floer theory as a deformation problem (similar to Cohen-Jones-Segal), but I may not talk too much about that because the details are not worked out yet. This is joint work with Jacob Lurie.

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• May 012017

### Jay Shah (MIT)

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#### Parametrized higher category theory

Parametrized or indexed $\infty$-category theory studies $\infty$-categories fibered over a given base $\infty$-category. This theory can be harnessed for the purposes of equivariant homotopy theory when one specializes to the case where is the base is the orbit category of a finite group. In this talk, we present a theory of parametrized homotopy limits and colimits that recovers and extends the Dotto-Moi theory of $G$-colimits. We apply this theory to prove that the $G$-$\infty$-category of $G$-spaces is freely generated under $G$-colimits by the contractible $G$-space, thereby affirming a conjecture of Mike Hill.

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• Apr 242017

### Angelica Osorno (Reed College)

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#### 2-Segal spaces and the Waldhausen construction

Due to the Simons lectures, the seminar will meet at 3:00 PM in 2-135.

The notion of 2-Segal spaces was introduced by Dyckerhoff and Kapranov as a higher dimensional version of Rezk's Segal spaces. In this talk we will explore the motivation for this notion, give examples, and show that it is related to a certain class of double categories via a version of Waldhausen's construction. This is joint work with J. Bergner, V. Ozornova, M. Rovelli, and C. Scheimbauer.

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• Apr 212017

### Dan Berwick-Evans (University of Illinois Urbana-Champaign)

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#### Field theories and elliptic cohomology from the vantage of characters

The seminar will meet at 3:00 PM in 2-131.

A basic way to probe a cohomology theory is through its complexification and the resulting Chern character map. Features of the cohomology theory can then be transferred to this complexification, e.g., pushforwards determine Riemann-Roch factors. Field theories also have a good notion of character theory via their partition functions, which are functions on certain moduli spaces. Constructions in field theories similarly endow partition functions with additional structure. I’ll apply these ideas to elliptic cohomology and 2-dimensional supersymmetric field theories, showing how the character theories of these objects line up with one another. Refining this comparison points toward a connection between field theories and topological modular forms.

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• Apr 102017

### Denis Nardin (MIT)

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#### Stability and distributivity over an orbital $\infty$-category

The category of $G$-spectra plays an important role in homotopy theory, even when a group action is not obviously involved. In this talk we will present a universal property for it providing an alternate proof of the Guillou-May theorem. If time permits, we will use it to provide a uniqueness theorem for the norm functors.

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• Apr 032017

• Mar 202017

### Bena Tshishiku (Harvard University)

$\begingroup \def\Mod{\mathop{\mathrm{Mod}}}\def\Diff{\mathop{\mathrm{Diff}}}$

#### Obstructions to Nielsen realization

Let $M$ be a manifold, and let $\Mod(M)$ be the group of diffeomorphisms of $M$ modulo isotopy (the mapping class group). The Nielsen realization problem for diffeomorphisms asks, "Can a given subgroup $G<\Mod(M)$ be lifted to the diffeomorphism group $\Diff(M)$?" This question about group actions is related to a question about flat connections on fiber bundles with fiber $M$. In the case $M$ is a closed surface, the answer is "yes" for finite $G$ (by work of Kerckhoff) and "no" for $G=\Mod(M)$ (by work of Morita). For most infinite $G<\Mod(M)$, we don't know. I will discuss some obstructions that can be used to show that certain groups don’t lift. Some of this work is joint with Nick Salter.

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• Mar 132017

### Mike Mandell (Indiana University)

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#### The strong Kunneth theorem for periodic topological cyclic homology

Hesselholt has recently been advertising "periodic topological cyclic homology" ($TP$) as potentially filling some of the same roles for finite primes as periodic cyclic homology plays rationally. It is constructed from topological Hochschild homology ($THH$) analogously to the way periodic cyclic homology is constructed from Hochschild homology. In joint work with Andrew Blumberg, we prove a strong Kunneth theorem for the periodic topological cyclic homology of smooth and proper dg categories over a finite field $k$, namely, the derived smash product $TP(X) \wedge_{TP(k)} TP(Y)$ is weakly equivalent to $TP(X \otimes_k Y)$.

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• Mar 062017

### Jack Morava (Johns Hopkins University)

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#### Some interesting perfectoid fields with possible topological applications

A Lubin-Tate group for a local field L defines a maximal totally ramified abelian extension of L, whose completion is perfectoid; Artin reciprocity identifies the Galois group of this extension with the group of units of L. Scholze and Nikolaus, following work of Hesselholt on the algebraic closure of $Q_p$, have calculated the topological Hochschild homology of such fields; these are complex orientable GEM spectra, whose associated formal groups seem to be rigid analytic versions of the Lubin-Tate group defining the extension, with the Galois group acting as generalized Adams operations. These spectra may have interesting connections with chromatic homotopy theory.

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• Feb 272017

### Ben Antieau (University of Illinois at Chicago)

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#### Negative and homotopy K-theory of ring spectra and extensions of the theorem of the heart

Barwick proved that the K-theory of a stable infinity-category with a bounded $t$-structure agrees with the K-theory of its heart in non-negative degrees. Joint work with David Gepner and Jeremiah Heller extends this to an equivalence of nonconnective K-theory spectra when the heart satisfies certain finiteness conditions such as noetherianity. Applications to negative K-theory and homotopy K-theory of ring spectra are provided, which were the original motivation for our work.

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• Feb 132017

### Sune Precht Reeh (MIT)

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#### Dimension functions, homotopy sphere actions, and fusion systems

Given a representation $V$ of a finite group $G$ we can associate a dimension function that to each subgroup $H$ of $G$ assigns the dimension of the fixed point space $V^H$. The dimension functions are "super class functions" that are constant on the conjugacy classes of subgroups in $G$. For a p-group the list of Borel-Smith conditions characterizes the super class functions that come from real representations.

In a joint paper with Ergün Yalcin we show that though we cannot lift Borel-Smith functions to real representations for a general group $G$, we can lift a multiple of any Borel-smith function to an action of $G$ on a finite homotopy sphere (which would be the unit sphere if we had a representation).

To solve the problem we localize at each prime p, and solve it in general for saturated fusion systems. That is, we give a list of Borel-Smith conditions for a fusion system that characterize the dimension functions of the fusion stable real representations. The proof for fusion systems involves biset functors and characteristic bisets for saturated fusion systems.

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## Fall 2016

• Dec 122016

### Ismar Volic (Wellesley College)

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#### Some new applications of manifold calculus of functors

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.

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• Dec 052016

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#### Understanding chromatic splitting

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.

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• Nov 212016

### Teena Gerhardt (Michigan State University)

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#### On the algebraic K-theory of $\Z[C_2]$

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.

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• Nov 142016

### Gabriel Drummond-Cole (Center for Geometry and Physics)

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#### Homotopy probability theory on a Riemannian manifold

Homotopy probability theory is a homological enrichment of algebraic probability theory, a toy model for the algebra of observables in a quantum field theory. I will introduce the basics of the theory and use it to describe a reformulation of fluid flow equations on a compact Riemannian manifold.

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• Nov 072016

### Ben Knudsen (Harvard University)

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#### Higher enveloping algebras and configuration spaces of manifolds

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.

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• Oct 242016

### Rade Zivaljevic (Mathematical Institute SANU)

$\begingroup$

#### Topology and Combinatorics of multiple chessboard complexes and a new Van Kampen-Flores theorem

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).

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• Oct 172016

### Chris Kottke (New College of Florida)

$\begingroup$

#### Transgression, fusion, and cohomology of free loop spaces

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.

$\endgroup$
• Oct 032016

### Kyle Ormsby (Reed College)

$\begingroup$

#### Vanishing in motivic stable stems

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.

$\endgroup$
• Sep 262016

### Emmanuel Dror Farjoun (Hebrew University)

$\begingroup$

#### Adjoint descent for infinity categories

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.)

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• Sep 192016

### Tyler Lawson (University of Minnesota)

$\begingroup$

#### Realizing the action of the Dyer-Lashof algebra

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 $A$.

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• Sep 122016

### Marc Hoyois (MIT)

$\begingroup$

#### Naive versus motivic $\Abb^1$-homotopy classes

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.

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## Spring 2016

• May 092016

### Nat Stapleton (Max Planck Institute)

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#### The infinite prime in chromatic homotopy theory

The seminar will meet at 3:30 PM in 2-131.

Nat will speak at 3:30 pm, followed by Saul at 4:30 pm

The behavior of the category of $E_n$-local spectra simplifies in various ways as $p \to \infty$. For a collection of categories indexed by the prime numbers we construct a category 'at the infinite prime' that captures behavior of all but finitely many of the input categories. We then apply this construction to the $E_n$-local and $K(n)$-local situations and analyze the resulting categories. This talk represents joint work with Barthel and Schlank.

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• May 092016

### Saul Glasman (Institute for Advanced Study)

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#### Categorical Tate constructions and cyclotomicity

I'll give an exposition of the stratification viewpoint on equivariant homotopy theory, and describe how this helps inspire new approaches to equivariant algebraic K-theory. I'll propose a categorification of the classical norm cofibration sequence for a cyclic group of prime order and the equivariant fracture square, with a version of the stable module category standing in for the Tate spectrum, leading to a K-theoretic approximation to the Hill-Hopkins-Ravenel norm. Cyclotomic spectra look enoticely perspicuous in the stratification picture, and if possible I'll begin to describe how it harmonizes with our theory of cyclonic objects to give a natural construction of the THH of a Waldhausen infinity-category. This is joint work in progress with Clark Barwick.

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• Apr 252016

### Tom Church (Stanford University)

$\begingroup$

#### Stable completed cohomology and excision in continuous K-theory

The seminar will meet at 3:30 PM in 2-131.

Time changed to 3:30 pm and room number changed to 2-136 due to Simons lectures

Two of the recent successes of representation stability are the description of the stable completed cohomology of arithmetic groups by Calegari-Emerton, and the proof of excision in continuous K-theory by Calegari. I'll explain these theorems, focusing on concrete cases such as H_1, K_1 and K_2 where we can work out explicitly exactly what is going on; I won't say anything fancy unless we can see it in at least one case. No knowledge of representation stability or K-theory needed; in fact, I'll give an introduction to many aspects of classical K-theory such as the congruence subgroup property and the Steinberg group.

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• Apr 112016

### Rune Haugseng (University of Copenhagen)

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#### Higher Lagrangian correspondences and topological field theories

The AKSZ construction, as implemented by Pantev-Toën-Vaquié-Vezzosi in the context of derived algebraic geometry, gives a symplectic structure on the derived stack of maps from an oriented compact manifold to a symplectic derived stack. I will discuss how this gives rise to a family of extended topological field theories valued in higher categories of symplectic derived stacks, with the higher morphisms given by a notion of higher Lagrangian correspondences. This is ongoing work with Damien Calaque and Claudia Scheimbauer.

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• Apr 042016

### Emanuele Dotto (MIT)

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#### On the classification of equivariant homogeneous functors

For every homotopy invariant endofunctor of the category of pointed G-spaces (G a finite group) equivariant calculus produces a 'genuine' tower, which under suitable circumstances converges to the original functor. The talk will focus on the description of the layers of this tower in terms of certain 'graph-equivariant' spectra, defined from Goodwillie's cross-effects.

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• Mar 282016

### Gijs Heuts (University of Copenhagen)

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#### Lie algebras and v_n-periodic spaces

We use the Goodwillie tower of the category of pointed spaces to relate the telescopic homotopy theory of spaces (in the sense of Bousfield) to the homotopy theories of Lie algebras and commutative coalgebras in T(n)-local spectra, in analogy with rational homotopy theory. As a consequence, one can determine the Goodwillie tower of the Bousfield-Kuhn functor in terms of topological Andre-Quillen homology, giving a different perspective on recent work of Behrens and Rezk.

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• Mar 142016

### Hiro Tanaka (Harvard University)

$\begingroup$

#### Lagrangian Cobordisms and Fukaya category

In symplectic geometry, the Fukaya category of a symplectic manifold is about as important as DbCoh is in algebraic geometry. As such, one wouldn't expect a totally topological way to characterize the Fukaya category of a symplectic manifold. However, it turns out that in a large class of non-compact symplectic manifolds, one expects to have a topological characterization; more surprisingly, it seems that the stable homotopy theory of Lagrangian cobordisms may recover the Fukaya category altogether. We'll begin with some basics of Fukaya categories, illustrate how there is a functor of oo-categories from a category of Lagrangian cobordisms of M to its Fukaya category, and explain structural theorems that move us closer to proving a conjecture that the Fukaya category can be recovered from the category of Lagrangian cobordisms.

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• Mar 072016

### Inna Zakharevich (University of Chicago)

$\begingroup$

#### The annihilator of the Lefschetz motive

The Grothendieck ring of varieties is defined to be the free abelian group generated by varieties, modulo the relation that for a closed subvariety $Y$ of $X$, $[X] = [Y] + [X \backslash Y]$; the ring structure is defined via the Cartesian product. For example, if $X$ and $Y$ are piecewise isomorphic, in the sense that there exist stratifications on $X$ and $Y$ with isomorphic strata, then $[X] = [Y]$ in the Grothendieck ring.

1. What does it mean when two varieties $X$ and $Y$ have equal classes in the Grothendieck ring? Must $X$ and $Y$ be piecewise isomorphic?

2. Is the class of the affine line a zero divisor?

Last December Borisov answered both of these questions with a single example, by constructing an element $[X] - [Y]$ in the kernel of multiplication by the affine line; in a beautiful coincidence, it turned out that $X \times \Abb^1$ and $Y \times \Abb^1$ were not piecewise isomorphic. In this talk we will describe an approach using algebraic K-theory to construct a topological version of the Grothendieck ring of varieties. We shall prove that $\pi_1$ of this space is generated by birational automorphisms of varieties which extend to piecewise automorphisms, which allows us to construct a group that surjects onto the kernel of multiplication by the affine line. By analyzing this group we will sketch a proof that Borisov's coincidence was not a coincidence at all: any element in the kernel of multiplication by the affine line can be represented as $[X]-[Y]$, where $X \times \Abb^1$ and $Y \times \Abb^1$ are not piecewise isomorphic.

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• Feb 082016

## Fall 2015

• Dec 072015

### Bhargav Bhatt (University of Michigan)

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#### Perfect rings and applications

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.

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• Nov 232015

### Ian Hambleton (McMaster University and Fields Institute)

$\begingroup$

#### Group actions on spheres with rank one isotropy

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.

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• Nov 162015

### David Treumann (Boston College)

$\begingroup$

#### The Maslov cycle and the J-homomorphism

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.

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• Nov 092015

### Eric Peterson (Harvard University)

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#### Determinantal K-theory and a few applications

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".

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• Nov 022015

### Kirsten Wickelgren (Georgia Tech)

$\begingroup$

#### The $\A^1$-Milnor number: counting zeros arithmetically

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.

$\endgroup$
• Oct 262015

### Marc Hoyois (MIT)

$\begingroup$

#### Secondary K-theory and the categorified Chern character

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.

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• Oct 192015

### John Terilla (CUNY)

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#### Homotopy probability theory with an application to fluids

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.

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• Oct 052015

### Cary Malkiewich (University of Illinois Urbana-Champaign)

$\begingroup$

#### Duality in topological Hochschild homology (THH)

Topological Hochschild homology ($THH$) is a beautiful and computable invariant of rings and ring spectra. In this talk, I will focus on the ring spectrum $DX$, and discuss a few different aspects of $THH(DX)$. For example, it splits when $X$ is a suspension, and we can use this for computations in topological cyclic homology. I will also recall the "Atiyah duality" between $THH(DX)$ and the free loop space $LX$, and prove that this duality preserves the genuine $S^1$-structure. This uses the new "norm" model of $THH$, and a surprising technical result about orthogonal $G$-spectra. If there is time, I will apply these tools once more and describe an enrichment of the character map from representation theory.

$\endgroup$
• Sep 282015

### Sune Reeh (MIT)

$\begingroup$

#### Saturated fusion systems as stable retracts of groups

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.

$\endgroup$
• Sep 212015

### Rob Thompson (CUNY)

$\begingroup$

#### An unstable change of rings for Morava $E$-theory

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.

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• Sep 142015

### Martin Gallauer Alves de Souza (University of California, Los Angeles)

$\begingroup$

#### Traces and homotopy colimits

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.

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## Spring 2015

• May 112015

### Emanuele Dotto (MIT)

$\begingroup$

#### Equivariant calculus of functors

Let $G$ be a finite group. There is a notion of "$J$-excision" of functors on pointed $G$-spaces, for every finite $G$-set $J$. When $J$ is the trivial $G$-set with $n$ elements it agrees with Goodwillie's definition of $n$-excision. When $J=G$ it recovers Blumberg's notion of equivariant excision.

The talk will focus on the $J$-excisive approximations of a homotopy functor, and how they fit together into a "Taylor tree". We will discuss the convergence of the tree, as well as possible classifications of $J$-homogeneous functors. Finally, we will relate the layers of the "genuine" tower of the identity functor on pointed $G$-spaces to partition complexes, and discuss possible applications of $\mathbb{Z}/2$-calculus to Real algebraic $K$-theory.

$\endgroup$
• May 042015

### Agnes Beaudry (University of Chicago)

$\begingroup$

#### The Chromatic Splitting Conjecture at $n=p=2$

In its strongest form, the chromatic splitting conjecture gives a precise description of the homotopy type of $L_{1}L_{K(2)}S$, which has been shown to hold for $p\geq 5$ by Hopkins and for $p=3$ by Goerss, Henn and Mahowald. In this talk, I will explain why this description cannot hold at the prime $p=2$. More precisely, let $V(0)$ be the mod $2$ Moore spectrum. I will give a summary of how one uses the duality resolution techniques to show that $\pi_{k}L_1L_{K(2)}V(0)$ is not zero when $k$ is congruent to $5$ modulo $8$. I will explain how this contradicts the decomposition of $L_1L_{K(2)}S$ predicted by the chromatic splitting conjecture.

$\endgroup$
• Apr 272015

$\begingroup$

#### Resolutions and duality in $K(2)$-local homotopy theory

The chromatic view of stable homotopy theory assembles a finite spectrum from its $K(n)$-localizations, focusing our attention on the $K(n)$-local category. This category has a number of interrelated dualities, which together go under the name of Gross-Hopkins duality. I'd like to explore this in the case $n=2$ using the topological resolutions developed with Henn, Mahowald, Rezk, and others. In particular, I'd like to explain how there is an elegant inevitability to calculations long regarded as impenetrable.

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• Apr 142015

### Guozhen Wang (MIT)

$\begingroup$

#### Unstable Chromatic Homotopy Theory

Using the self maps provided by the Hopkins-Smith periodicity theorem, we can decompose the unstable homotopy groups of a space into its periodic parts, except some lower stems. For fixed n, using the Bousfield-Kuhn functor we can associate to any space a spectrum, which captures the $v_n$-periodic part of its homotopy groups.

I will talk about the homotopy type of the Bousfield-Kuhn functor applied to spheres, which would tell us much about the $v_n$-periodic part of the homotopy groups of spheres provided we have a good understanding of the telescope conjecture. I will make use of the Goodwillie tower of the identity functor, which resolves the unstable spheres into spectra which are the Steinberg summands of the classifying spaces of the additive groups of vector spaces over finite fields.

By understanding the attaching maps of the Goodwillie tower after applying the Bousfield-Kuhn functor, we would be able to determine the homotopy type of its effect on spheres. As an example of how this works in concrete computations, I will compute the K(2)-local homotopy groups of the three sphere at primes p>3.

The computations show that the unstable homotopy groups not only have finite p-torsion, their K(2)-local parts also have finite $v_1$-torsion, which indicates there might be a more general finite $v_n$-torsion phenomena in the unstable world, conjectured by many people.

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• Apr 132015

### Michael Andrews (MIT)

$\begingroup$

#### The $v_1$-periodic part of the Adams spectral sequence at an odd prime - dancers to a discordant system

Algebraic topologists are interested in the class of spaces which can be built from spheres. For this reason, when one tries to understand the continuous maps between two spaces up to homotopy, it is natural to restrict attention to the maps between spheres first. The groups of interest are called the homotopy groups of spheres. Topologists soon realized that it is easier to work in a stable setting. Instead, one asks about the stable homotopy groups of spheres or, equivalently, the homotopy groups of the sphere spectrum. Calculating all of these groups is an impossible task but one can ask for partial information.

In particular, one can try to understand the global structure of these groups by proving the existence of recurring patterns; this is analogous to the fact that we cannot find all the prime numbers, but we can prove theorems about their distribution. These patterns are clearly visible in spectral sequence charts for calculating $\pi_*(S^0)$ and my thesis came about because of my desire to understand the mystery behind these powerful dots and lines, which others in the field appeared so in awe of. I will tell the story of the stable homotopy groups of spheres for odd primes at chromatic height 1, through the lens of the Adams spectral sequence.

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• Apr 062015

### Mike Shulman (University of San Diego)

$\begingroup$

#### Internal languages for $\infty$-toposes and sub-$\infty$-toposes

One of the most important insights of classical topos theory is that a topos (a category of sheaves) has an 'internal language', so that we can reason about its objects roughly 'as if they were sets'. The recent development of 'homotopy type theory' provides a similar internal language for $\infty$-toposes ($\infty$-categories of stacks), allowing us to reason about its objects 'as if they were spaces'. I will sketch this language and show how to apply it to study sub-$\infty$-toposes; these are represented internally as 'higher modalities' on a Martin-Löf-Voevodsky universe, generalizing Lawvere-Tierney operators from classical topos theory.

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• Mar 302015

### David Gepner (Purdue)

$\begingroup$

#### Localization sequences in the algebraic $K$-theory of ring spectra

Together with trace methods, the localization sequence comprises one of the only known methods for computing algebraic $K$-theory. If $R$ is a ring spectrum and $R[S^{-1}]$ is a localization of R, then there is a fiber sequence of $K$-theory spectra $K(\text{fiber}) \to K(R) \to K(R[S^{-1}])$. In this talk, we will show that (under mild conditions) the fiber term is compactly generated by a Koszul-type spectrum formed from $R$ and $S$, which when $R= BP \langle n \rangle$ and $S = \{v_n \}$ differs from $BP \langle n-1 \rangle = R/v_n$. We will then apply trace methods to show that their $K$-theories differ, answering a question of Rognes. Time permitting, we will sketch how this fits into a general program (primarily due to Waldhausen, Rognes, Ausoni, and others) to understand the $K$-theory of the sphere in terms of the chromatic filtration of the stable homotopy category. This is joint work with Benjamin Antieau and Tobias Barthel.

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• Mar 162015

### Emily Riehl (Harvard University)

$\begingroup$

#### The formal category theory of ($\infty$,n)-categories

Pioneering work of Andre Joyal and Jacob Lurie has shown that ordinary category theory can be extended to quasi-categories ("$\infty$-categories"), a type of ($\infty$,1)-category. In joint work with Verity, we show that the category theory of quasi-categories is 2-categorical: new definitions of the basic notions - (co)limits, adjunctions, fibrations - equivalent to the Joyal-Lurie definitions, can be encoded in the homotopy 2-category of quasi-categories. Our 2-categorical proofs in the homotopy 2-category restrict to the classical ones in the sub 2-category CAT. We give a short list of axioms, satisfied by (iterated) complete Segal spaces, that suffice for this homotopy 2-categorical development, so this "formal" approach to the category theory of quasi-categories immediately extends to other models of higher homotopical categories.

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• Mar 092015

$\begingroup$

#### Derived geometry as a tool for QFT, QFT as a tool in topology

In this talk I will highlight how formal moduli problems and derived stacks have a role in constructing quantum field theories. I will then focus on a specific example which allows one to give a new proof of the algebraic index theorem of Nest-Tsygan and Fedosov. If time permits, I will also discuss observables in QFT and their structure as a factorization algebra; giving a further reinterpretation of the algebraic index theorem and some insight into higher analogues of index theory.

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• Mar 022015

### Nicolo Sibilla (University of British Columbia)

$\begingroup$

#### The topological Fukaya category and mirror symmetry for toric Calabi-Yau threefolds

The Fukaya category of open symplectic manifolds is expected to have good local-to-global properties. Based on this idea several people have developed sheaf-theoretic models for the Fukaya category of punctured Riemann surfaces: the name topological Fukaya category appearing in the title refers to the (equivalent) constructions due to Dyckerhoff-Kapranov, Nadler and Sibilla-Treumann-Zaslow. The theory involved in setting up the topological Fukaya category has surprising connections with many different areas of geometry and topology, such as for instance the co-representability of the Waldhausen S-construction. In this talk I will focus on defining the topological Fukaya category and explain applications to Homological Mirror Symmetry for toric Calabi-Yau threefolds. This is work in progress joint with James Pascaleff.

$\endgroup$
• Feb 232015

### Mike Hill (University of Virginia)

$\begingroup$

#### Flavors of equivariant commutativity

Classically, there is essentially only one $E_\infty$ operad, and it parameterizes multiplications commutative up to all higher homotopies. In the $G$-equivariant context, the situation is muddied by the possible ways the group can interact with the powers of a space or spectrum. In this talk, I'll discuss the notion of an $N_\infty$ operad, an operad in $G$-space which just like the $E_\infty$ operad parameterizes multiplications commutative up to all higher homotopies but which also allows $G$ to permute factors. The use of these allows one to understand operadically the transfer map on equivariant infinite loop spaces, to see what structure is preserved by equivariant Bousfield localization, and to tease apart what sort of additional structure the category of modules over an equivariant commutative ring spectrum has.

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• Feb 172015

### John Harper (Ohio State University)

$\begingroup$

#### Derived Koszul duality and TQ-homology completion

Consider a flavor of structured ring spectra that can be described as algebras over an operad $O$ in spectra. A natural question to ask is when the fundamental adjunction comparing $O$-algebra spectra with coalgebra spectra over the associated Koszul dual comonad $K$ can be modified to turn it into an equivalence of homotopy theories. In a paper published in 2012, Francis and Gaitsgory conjecture that replacing $O$-algebras with the full subcategory of homotopy pro-nilpotent $O$-algebras will do the trick. In joint work with Kathryn Hess we show that every 0-connected $O$-algebra is homotopy pro-nilpotent; i.e. is the homotopy limit of a tower of nilpotent $O$-algebras.

This talk will describe recent work, joint with Michael Ching, that resolves in the affirmative the 0-connected case of the Francis-Gaitsgory conjecture; that replacing $O$-algebras with 0-connected $O$-algebras turns the fundamental adjunction into an equivalence of homotopy theories. This can be thought of as a spectral algebra analog of the fundamental work of Quillen and Sullivan on the rational homotopy theory of spaces, the subsequent work of Goerss and Mandell on the p-adic homotopy theory of spaces, and the work of Mandell on integral cochains and homotopy type. Corollaries include the following: (i) 0-connected $O$-algebra spectra are weakly equivalent if and only if their $TQ$-homology spectra are weakly equivalent as derived $K$-coalgebras, and (ii) if a $K$-coalgebra spectrum is 0-connected and cofibrant, then it comes from the $TQ$-homology spectrum of an $O$-algebra.

$\endgroup$
• Feb 132015

### Nathaniel Stapleton (Universität Bonn)

$\begingroup$

#### The character of the total power operation

We construct a total power operation on n-fold class functions compatible with the total power operation in Morava E-theory through the character map of Hopkins, Kuhn, and Ravenel. In essence, this gives a formula for the total power operation in Morava E-theory applied to a finite group. This is joint work with Barthel.

$\endgroup$
• Feb 092015

### Emily Riehl (Harvard University)

$\begingroup$

#### The formal category theory of ($\infty$,n)-categories

Pioneering work of Andre Joyal and Jacob Lurie has shown that ordinary category theory can be extended to quasi-categories ("$\infty$-categories"), a type of ($\infty$,1)-category. In joint work with Verity, we show that the category theory of quasi-categories is 2-categorical: new definitions of the basic notions - (co)limits, adjunctions, fibrations - equivalent to the Joyal-Lurie definitions, can be encoded in the homotopy 2-category of quasi-categories. Our 2-categorical proofs in the homotopy 2-category restrict to the classical ones in the sub 2-category CAT. We give a short list of axioms, satisfied by (iterated) complete Segal spaces, that suffice for this homotopy 2-categorical development, so this "formal" approach to the category theory of quasi-categories immediately extends to other models of higher homotopical categories.

$\endgroup$

## Fall 2014

• Dec 152014

### Andrew Ranicki (University of Edinburgh)

$\begingroup$

#### The total surgery obstruction

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.

$\endgroup$
• Dec 082014

### Mona Merling (Johns Hopkins University)

$\begingroup$

#### Equivariant algebraic K-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.

$\endgroup$
• Dec 012014

### Joseph Hirsh (MIT)

$\begingroup$

#### Model structures for coalgebras

Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.

$\endgroup$
• Nov 242014

### Sune Reeh (MIT)

$\begingroup$

#### Minimal characteristic bisets for fusion systems

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.

$\endgroup$
• Nov 172014

### Jesse Wolfson (University of Chicago)

$\begingroup$

#### The Index Map and Reciprocity Laws for Contou-Carrere Symbols

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.

$\endgroup$
• Nov 032014

### Dan Freed (UT Austin)

$\begingroup$

#### Invertible topological field theories

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.

$\endgroup$
• Oct 272014

### Bertrand Guillou (University of Kentucky)

$\begingroup$

#### The $\eta$-local motivic sphere

The motivic Hopf map $\eta$ is not nilpotent in the motivic stable homotopy groups of spheres, contrary to the situation in the homotopy of spaces. In the motivic Adams spectral sequence computing the motivic stable homotopy groups, there result a number of $h_1$-towers. The motivic Adams spectral sequence contains strictly more information than the classical case and is therefore quite complicated, but the $h_1$-local part is understood and computes the $\eta$-localization of the motivic sphere. I will discuss joint work with Dan Isaksen on the $\eta$-local motivic sphere and related topics.

$\endgroup$
• Oct 202014

### Joana Cirici (Freie Universität Berlin)

$\begingroup$

#### Topology of complex projective varieties with isolated singularities

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).

$\endgroup$
• Oct 062014

### Marc Hoyois (MIT)

$\begingroup$

#### The motivic Lefschetz fixed-point theorem

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.

$\endgroup$
• Sep 292014

### Marco Perez (MIT)

$\begingroup$

#### Computing Ext using $n$-projective modules

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}).

$\endgroup$
• Sep 252014

### Gunnar Carlsson (Stanford University)

$\begingroup$

#### The structure of spaces of persistence barcodes

[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.

$\endgroup$
• Sep 222014

### Ezra Getzler (Northwestern University)

$\begingroup$

#### Higher stacks and categories of fibrant objects

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.

$\endgroup$
• Sep 152014

### Ilya Grigoriev (University of Chicago)

$\begingroup$

#### Characteristic classes of manifold bundles

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.

$\endgroup$
• Sep 082014

### Emmanuel Farjoun (Hebrew University)

$\begingroup$

#### Group maps, Homotopically

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.

$\endgroup$

## Spring 2014

• May 122014

### Mark Behrens (MIT)

$\begingroup$

#### On the tmf-resolution

Mahowald used the bo-based Adams spectral sequence to compute the 2-primary v1-periodic stable homotopy groups of spheres, and from this he deduced the v1-periodic telescope conjecture for p = 2. I will discuss what I know about the tmf-resolution at p = 2, incorporating work of many collaborators over the years, most significantly Tyler Lawson, Kyle Ormsby, Vesna Stojanoska, and Nat Stapleton.

$\endgroup$
• May 052014

### Lars Hesselholt (Nagoya University and University of Copenhagen)

$\begingroup$

#### The unreasonable fibrancy of Waldhausen's construction

An old result of Gillet and Grayson shows that, to calculate the simplicial loop space of Waldhausen's S-construction of an exact category, it suffices to apply Kan's Ex-functor just once instead of infinitely many times. In this talk, I will explain that, in another instance, Waldhausen's construction turns out to be unreasonably fibrant, namely, that for every exact category with duality $(C,D,\eta)$, there is a canonical isomorphism $\Omega^{1,1}|NiS^{1,1}({C},D,\eta)| \simeq \Omega^{2,1}|NiS^{2,1}({C},D,\eta)|$ in the homotopy category of pointed real spaces. The proof uses the surprising fact, proved by Schlichting, that, on the set of components of the subspace of the left-hand side consisting of the points fixed by the canonical involution, the abelian monoid structure induced by orthogonal sum is an abelian group structure. It further uses that the real additivity theorem holds for both sides, as proved by Schlichting and by myself and Madsen, respectively, along with a new group-completion result due to Moi.

$\endgroup$
• Apr 282014

### Nat Stapleton (MIT)

$\begingroup$

#### The E-theory of the symmetric group

Strickland proved that the Morava E-theory of the symmetric group corepresents the scheme classifying subgroups of the formal group associated to E-theory after taking the quotient by a certain transfer ideal. In this talk I will discuss a new proof of this result using character maps from height n to height 1. I will emphasize different parts of the proof than were discussed in the Thursday seminar talk. This talk includes joint work with Tomer Schlank and Tobias Barthel.

$\endgroup$
• Apr 142014

### Matilde Marcolli (CalTech)

$\begingroup$

#### Feynman integrals, singular hypersurfaces, and motives

I will describe different recent approaches to the renormalization and computation of Feynman integrals in perturbative quantum field theory via differential forms on the complement of singular hypersurfaces and periods of motives.

$\endgroup$
• Apr 072014

### Ismar Volic (Wellesley College)

$\begingroup$

#### Homotopy-theoretic methods in the study of spaces of knots and links

I will survey the ways in which some homotopy-theoretic methods, manifold calculus of functors main among them, have in recent years been used for extracting information about the topology of spaces of knots and links. Cosimplicial and operadic models for these spaces will also be featured. I will also mention with some recent results about spaces of homotopy string links and in particular about how one can use functor calculus in combination with configuration space integrals to extract information about Milnor invariants as well as derive higher-order asymptotic invariants of vector fields.

$\endgroup$
• Mar 312014

### Jonathan Campbell (UT Austin)

$\begingroup$

#### Topological Hochschild Homology and Koszul Duality

In this talk I'll indicate how to prove a duality result relating the topological Hochschild homology (THH) of a ring spectrum with the THH of its Koszul dual. Both THH and Koszul duality will be defined, and a proof will be sketched.

$\endgroup$
• Mar 172014

### Andre Joyal (Université du Québec à Montréal)

$\begingroup$

#### Categorical Homotopy Type Theory

We show that Homotopy Type Theory can be formulated in the language of category theory. We will address the problem of finding an elementary notion of higher topos.

$\endgroup$
• Mar 102014

### Hakon Bergsaker (University of Bergen and MIT)

$\begingroup$

#### Towards TC(MU)

Given a ring spectrum R, there is an associated algebraic K-theory spectrum K(R). In general K(R) is very hard to compute; one method for approaching it is to use the cyclotomic trace map to topological cyclic homology, TC(R). This map turns out to be a good approximation in many cases, and TC(R) can be calculated provided one has a good grasp on the various cyclic fixed points of the topological Hochschild homology spectrum, THH(R).

In this talk I will focus on the case where R is the complex cobordism spectrum MU. In this case computing TC(MU) essentially reduces to computing the circle-Tate construction on THH(MU). I will describe and build on previous homological computations to study the Adams spectral sequence of the circle-Tate construction on THH(MU). This is work in progress.

$\endgroup$
• Mar 032014

### Tyler Lawson (University of Minnesota)

$\begingroup$

#### Comodules and connections

In this talk I'll discuss comodules in homotopy theory, specifically aimed at lifting $(MU_*, MU_*MU)$-comodules up to a homotopical notion. I'll then describe how Goodwillie calculus to give an iterative sequence of approximations to this structure.

$\endgroup$
• Feb 242014

### Emanuele Dotto (MIT)

$\begingroup$

#### Equivariant Diagrams and Equivariant excision

In a non-equivariant setting, a functor is excisive if it takes homotopy pushout squares to homotopy pullback squares. Given a finite group G and a functor from G-spaces to G-spaces (or G-spectra), this definition of excision does not 'capture enough equivariancy'. For example the category of endofunctors of G-spaces with this property does not model G-spectra. One solution is to replace squares by 'cubes with action', where the group is allowed to act on the whole diagram by permuting its vertices.

I will talk about the homotopy theory of these equivariant diagrams and relate the resulting notion of equivariant excision to previous work of Blumberg.

As an application of this theory, I will give a proof of the Wirthmuller isomorphism that uses only the equivariant suspension theorem and formal manipulations of limits and colimits.

$\endgroup$
• Feb 102014

### Tom Church (Stanford University)

$\begingroup$

#### A survey of representation stability

I will give a gentle survey of the theory of representation stability, viewed through the lens of its applications. These applications include: homological stability for configuration spaces of manifolds; understanding the stable (and unstable) homology of arithmetic lattices; uniform generators for congruence subgroups and congruence subgroups; and distributional stability for random squarefree polynomials over finite fields.

$\endgroup$

## Fall 2013

• Dec 092013

### Vesna Stojanoska (MIT)

$\begingroup$

#### Arithmetic duality for spectra

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.

$\endgroup$
• Dec 022013

### Kyle Ormsby (MIT)

$\begingroup$

#### Galois-equivariance and motivic homotopy

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 $k$.

$\endgroup$
• Nov 252013

### Emily Riehl (Harvard University)

$\begingroup$

#### Cell complex presentations for (generalized) Reedy categories

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.

$\endgroup$
• Nov 182013

### David Ayala (University of Southern California)

$\begingroup$

#### Poincaré Koszul duality and factorization homology

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.

$\endgroup$
• Nov 042013

### Dan Isaksen (Wayne State University)

$\begingroup$

#### A tour of the Adams spectral sequence

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.

$\endgroup$
• Oct 282013

### Lennart Meier (University of Virginia)

$\begingroup$

#### From affineness towards duality in chromatic homotopy theory

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.

$\endgroup$
• Oct 212013

### Craig Westerland (University of Minnesota)

$\begingroup$

#### The unit spectrum of Morava $E$-theory

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.

$\endgroup$
• Oct 072013

### Hiro Tanaka (Harvard University)

$\begingroup$

#### Bridgeland stability conditions, algebraic k theory, and factorization homology over the circle

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.

$\endgroup$
• Sep 302013

### David Carchedi (Max Planck Institute)

$\begingroup$

#### A differential graded approach to derived manifolds

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.

$\endgroup$
• Sep 232013

### Christian Haesemeyer (University of California, Los Angeles)

$\begingroup$

#### On the K-theory of toric varieties in characteristic p

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").

$\endgroup$
• Sep 162013

### S\o ren Galatius (Stanford University)

$\begingroup$

#### Homological stability for moduli spaces of manifolds

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.

$\endgroup$
• Sep 092013

### Anna Marie Bohmann (Northwestern University)

$\begingroup$

#### An equivariant infinite loop space machine

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.

$\endgroup$

## Spring 2013

• May 132013

### Marco Varisco (University at Albany, SUNY)

$\begingroup$

#### Algebraic K-Theory of Group Rings and Topological Cyclic Homology

The Farrell-Jones Conjecture predicts the structure of the algebraic K-theory K(ZG) of the integral group ring of an arbitrary discrete group G. It asserts that a so-called assembly map (whose target is the spectrum K(ZG) and whose source is the homotopy colimit of K(ZH) over all virtually cyclic subgroups H of G) is an equivalence. I will describe joint work with Wolfgang Lück, Holger Reich, and John Rognes, in which we prove partial injectivity results about the rationalized assembly map under finiteness assumptions on the group G, generalizing a theorem of Bökstedt-Hsiang-Madsen. The main tool is the cyclotomic trace map from algebraic K-theory to topological cyclic homology.

$\endgroup$
• Apr 302013

### Rune Haugseng (MIT)

$\begingroup$

#### Weakly Enriched Higher Categories

Categories enriched in symmetric monoidal categories such as spectra turn up in various places in algebraic topology. Unfortunately these can be difficult to work with in a homotopically meaningful way, which suggests that for many purposes it would be better to work with less rigid structures, where composition is only associative up to coherent homotopy. In this talk I will introduce a general theory of such weak or homotopy-coherent enrichment, built using a non-symmetric variant of Lurie's infinity-operads. I will then describe how the correct homotopy theory of these enriched infinity-categories can be constructed as a localization of a homotopy theory defined using infinity-operads; this is joint work with David Gepner. In addition, I will discuss some comparison results and, time permitting, mention analogues of natural transformations and correspondences in this setting.

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• Apr 292013

### Luis Pereira (MIT)

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#### Goodwillie Calculus and Algebras over a Spectral Operad

The overall goal of this talk is to apply the theory of Goodwillie calculus to the category $Alg_{\mathcal{O}}$ of algebras over a spectral operad. Its first part will deal with generalizing many of the original results of Goodwillie so that they apply to a larger class of model categories and hence be applicable to $Alg_{\mathcal{O}}$. The second part will apply that generalized theory to the $Alg_{\mathcal{O}}$ categories. The main results here are: an understanding of finitary homogeneous functors between such categories; identifying the Taylor tower of the identity in those categories; showing that finitary n-excisive functors can not distinguish between $Alg_{\mathcal{O}}$ and $Alg_{\mathcal{O_{\leq n}}}$, the category of algebras over the truncated $O_{\leq n}$; and a weak form of the chain rule between such algebra categories, analogous to the one studied by Arone and Ching in the case of Spaces and Spectra.

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• Apr 222013

### Geoffroy Horel (MIT)

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#### Operads, Modules and higher Hochschild cohomology

In this talk I will describe a general theory of modules over an algebra over an operad. Specializing to the operad Ed of little d-dimensional disks, I will show that each d-1 manifold gives rise to a theory of modules. I will then describe a geometric construction of the homomorphisms objects in these categories of modules inspired by factorization homology (also called chiral homology). A particular case of this construction is higher Hochschild cohomology (i.e. Hochschild cohomology for Ed-algebras). This construction enlightens the relationship between Hocshchild cohomology and geometric objects like the cobordism category and the spaces of long knots.

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• Apr 082013

### John Lind (Johns Hopkins University)

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#### Equivariantly Twisted Cohomology Theories

Twisted K-theory is a cohomology theory whose cocycles are like vector bundles but with locally twisted transition functions. If we instead consider twisted vector bundles with a symmetry encoded by the action of a compact Lie group, the resulting theory is equivariant twisted K-theory. This subject has garnered much attention for its connections to conformal field theory and representations of loop groups. While twisted K-theory can be defined entirely in terms of the geometry of vector bundles, there is a homotopy-theoretic formulation using the language of parametrized spectra. In fact, from this point of view we can define twists of any multiplicative generalized cohomology theory, not just K-theory. The aim of this talk is to explain how this works, and then to propose a definition of equivariant twisted cohomology theories using a similar framework. The main ingredient is a structured approach to multiplicative homotopy theory that allows for the notion of a G-torsor where G is a grouplike A_{\infty} space.

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• Apr 012013

### Jack Ullman (MIT)

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#### The regular slice spectral sequence

The equivariant slice spectral sequence was introduced by Hill, Hopkins and Ravenel in their solution of the Kervaire invariant problem, and is rapidly becoming an important computational tool in equivariant stable homotopy theory. In this talk, I will describe new results on a variant called the regular slice spectral sequence (or RSSS). I will explain how geometric fixed point and norm functors interact with the slice filtration, giving a Leibniz formula for the latter. I will then use Brown-Comenetz duality to relate the RSSS to the homotopy orbit and homotopy fixed point spectral sequences. Next, I will use model theory to obtain Toda bracket operations in the RSSS. Finally, I will use some of these tools to obtain a formula for the slice tower of a cofree spectrum, prove real Bott periodicity and prove a special case of the Atiyah-Segal completion theorem.

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• Mar 182013

### Nitu Kitchloo (Johns Hopkins University)

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#### The Stable Symplectic category and the Grothendieck-Teichmuller group

The Stable Symplectic category can the thought of as a category of Symplectic Motives. The objects in this topological category are symplectic manifolds, and the space of morphisms is an infinite loop space obtained by stabilizing the space of immersed totally-real correspondences between the source and target. A variant of this category can be traced back almost 30 years to early work of A. Weinstein on geometric quantization. In my talk, I will motivate the definition of the Stable Symplectic category. This will lead us to the construction of a canonical fiber functor F, on this category with values in the monoidal category of modules over a commutative ring spectrum Omega. The main aim of my talk is to explore the Motivic Galois group Aut(F) (i.e. the group of monoidal automorphisms of F). This group will be shown to be the abelian quotient of the Grothendieck-Teichmuller group as described by Kontsevich. Extending this observation along the lines of homotopy theory, we will motivate the topological hochschild homology of Omega:THH(Omega), as an integral candidate for Aut(F). If time permits, I would like to formulate some natural geometric questions in symplectic topology in terms of THH(Omega) and the Waldhausen K-theory K(Omega). This is joint work in part with Jack Morava.

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• Mar 112013

### Dustin Clausen (MIT)

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#### Arithmetic duality in algebraic K-theory

We will discuss an analog in algebraic K-theory of the Poitou-Tate global duality in Galois cohomology. A key point is that the use of algebraic K-theory instead of Galois cohomology allows to give a direct and pictorial construction of the fundamental class which is at the base of these dualities. It also allows to connect this arithmetic theory with some classical and modern work in homotopy theory, such as Quillen's on the J-homomorphism and Rezk's on logarithmic cohomology operations.

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• Mar 042013

### Ethan Devinatz (University of Washington)

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#### Higher homotopy commutativity of small ring spectra

Given a finite spectrum of type $n$, explicit $v_n$ self-maps are more easily constructed if that spectrum is a ring spectrum, by which I mean the spectrum is provided with a pairing which has a two-sided unit but is not necessarily homotopy commutative or homotopy associative. If in addition, the spectrum is homotopy associative and homotopy commutative, one can sometimes say more.

Twenty five years ago I proved that if $X$ is a finite ring spectrum of type n, then there exists a $v_n$ self-map f such that the cofiber $X(f^i)$ of the self-map $f^i$ is a ring spectrum for any $i$, and the pairing on $X(f^i)$ extends the pairing on $X$. In this talk, I will discuss my recent result that if $X$ is higher homotopy commutative up to some finite order, then $f$ may be chosen so that this higher homotopy commutative structure may be extended to such a structure on $X(f^i)$.

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• Feb 252013

### Matthew Gelvin (University of Copenhagen)

$\begingroup$

#### Parametrization of characteristic bisets of a saturated fusion system

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.

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• Feb 112013

### Michael Ching (Amherst College)

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#### Some examples of homotopic descent

I will describe a collection of theorems that exemplify homotopic descent. Each of these theorems says that a certain Quillen adjunction is 'comonadic' in a homotopical sense: that is, it identifies the homotopy theory on one side of the adjunction with the homotopy theory of coalgebras over a certain comonad that acts on the other side. I will say what I mean by the homotopy theory of such coalgebras and give a Barr-Beck comonadicity condition.

The examples concern operad theory and Goodwillie calculus. One result identifies the homotopy theory of 0-connected algebras over an operad of spectra with that of 0-connected divided power coalgebras over the Koszul dual operad. (This is joint work with John Harper.) Another describes the homotopy theory of n-excisive homotopy functors (between categories of spaces and/or spectra) in terms of appropriate comonads. (This is joint work with Greg Arone.) In the case of functors from spaces to spectra, and algebras over the commutative operad, there is a close connection between these two examples, which I shall describe.

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• Feb 052013

### André Henriques (University of Utrecht)

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#### Progress report on the construction of TMF

Roughly ten years ago, Stephan Stolz and Peter Teichner have set up a detailed plan for constructing TMF geometrically. Unfortunately, their idea of definition is still incomplete. A couple of months ago, I had an idea (which fits into the Stolz-Teichner program) about which I am quite excited: There should be a universal CFT, which I'll call $U$. The CFT $U$ should bear with respect to other CFTs a relationship that is analogous to the relationship that an infinite dimensional Hilbert space bears with respect to other finite dimensional vector spaces. Moreover, there should exist a property of quantum fields of $U$, which I'll call 'Fredholm' such that the space of Fredholm quantum fields of $U$ is a classifying space for the cohomology theory TMF. I'll explain what the theory $U$ is, and what it means for a quantum field to be 'Fredholm'. Disclaimer: this is all very speculative, and I don't think that, in its current form, this will yield TMF.

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## Fall 2012

• Dec 102012

### Steve Awodey (Carnegie Melon University)

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#### Homotopy Type Theory and Univalent Foundations

I will give a survey of the recently discovered connection between constructive logic and homotopy theory. This forms the basis of Voevodsky's Univalent Foundations program, a new approach to foundations with intrinsic geometric content and a computational implementation. Time permitting, I will explain the Univalence axiom.

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• Dec 032012

### Angélica Osorno (University of Chicago)

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#### Equivariant infinite loop space machines

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.

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• Nov 262012

### Kirsten Wickelgren (Harvard University)

$\begingroup$

#### $H_1$ of the Abel-Jacobi Map to the Compactified Jacobian gives Poincaré Duality

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.

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• Nov 192012

### Emily Riehl (Harvard University)

$\begingroup$

#### The Formal Category Theory of Quasi-Categories

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.

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• Nov 052012

### Tomer Schlank (MIT)

$\begingroup$

#### Obstruction Theory for Topoi and Sections for (Pro)finite Group

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.

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• Oct 292012

### Matthew Gelvin (University of Copenhagen)

$\begingroup$

#### Parametrization of Characteristic Bisets of a Saturated Fusion System

For $G$ a finite group with Sylow subgroup $S$, the conjugation action of $G$ on the subgroups of $S$ gives rise to the data of a \emph{saturated fusion system} $\mathcal{F_S(G)}$ on $S$. On the other hand, $S$ acts on $G$ by left and right multiplication. The resulting $(S,S)$-biset $_SG_S$ turns out to contain much of the same information as $\mathcal{F_S(G)}$, in that the biset determines the fusion system, but not conversely. These notions can be abstracted to make no reference to the ambient group $G$, resulting in an \emph{abstract saturated fusion system} $\mathcal{F}$ on $S$ and a \emph{characteristic biset} $\Omega$ for $\mathcal{F}$. Again, $\Omega$ determines $\mathcal{F}$, but each $\mathcal{F}$ has many associated characteristic bisets. This talk will focus on the failure of a saturated fusion system to uniquely determine a characteristic biset. We will show that there is a parametrization of all characteristic bisets for a fixed $\mathcal{F}$, which will have as a consequence the surprising result that each saturated fusion system has a unique \emph{minimal} associated characteristic biset.

$\endgroup$
• Oct 222012

### Vesna Stojanoska (MIT)

$\begingroup$

#### A Study of tmf Cooperations

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.

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• Oct 152012

### Nathaniel Stapleton (MIT)

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#### An Application of Transchromatic Generalized Character Theory

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.

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• Oct 012012

### Kyle Ormsby (MIT)

$\begingroup$

#### Motivic Brown-Peterson invariants of the rationals

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.

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• Sep 242012

### Larry Guth (MIT)

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#### Contraction of areas and homotopy-type of mappings

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 $c(m) > 0$.

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• Sep 172012

### Paul Arne Østvær (University of Oslo)

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#### Motivic slices and the graded Witt ring

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.

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• Sep 102012

### Søren Galatius (Stanford University)

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#### Homological stability for moduli spaces of high dimensional manifolds

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 $n=1$.

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## Spring 2012

• May 142012

### Man Chueng Cheng (MIT)

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#### Duality of differentiable stacks

It was shown by Greenlees-Sadofsky that classifying spaces of finite groups satisfy a Morava K-theory version of Poincare duality. This duality map can be viewed as coming from a Spanier-Whitehead type construction for differentiable stacks. In this talk I will define differentiable stacks and explain this construction. I will also discuss the generalization of the above result to a more general class of stacks and give some examples.

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• May 072012

### Marcy Robertson (University of Western Ontario)

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A multicategory, also known as a colored operad, is simply a generalized non-commutative algebra. In this talk we focus on studying maps between multicategories enriched in simplicial sets. We show that the homotopy function complex of maps between any two multicategories can be computed as the moduli space of a certain small category of (operatic) bimodules. As an application, we show how this description leads to several important decompositions which allow one to compute various geometric invariants.

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• Apr 302012

### Robin Koytcheff (Brown University)

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Budney recently constructed an operad that encodes splicing of knots and extends his little 2-cubes operad action on the space of (long) knots. He further decomposed the space of knots as the space freely generated over the splicing operad by the subspace of torus and hyperbolic knots. Infection of knots (or links) by string links is a generalization of splicing from knots to links and is useful for studying concordance of knots. In joint work with John Burke, we have constructed a colored operad that encodes this infection operation. This suggests looking for other ways to decomposes spaces of knots and links, which is a main direction of our work in progress.

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• Apr 232012

• Apr 182012

### Bertrand Toën (Université de Montpellier)

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#### Quantization of derived moduli spaces

I will report on a work in progress in collaboration with Pantev-Vaquié-Vezzosi. The central notion will be of a shifted symplectic structure on a derived scheme or derived stack, which I will explain in details. The main result is an existence statement of shifted symplectic structures on derived mapping spaces toward a symplectic target, that will be used to construct many examples (moduli of sheaves on $CY$ varieties, moduli of representation of $\pi_1$ of a compact oriented manifold, Lagrangian intersections ...). Finally, I will explain what quantization means in this context as well the general strategy to prove existence of canonical quantizations.

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• Apr 092012

### Burt Totaro (University of Cambridge)

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#### New bounds for the cohomology and Chow ring of finite groups

Symonds (2010) showed that the cohomology ring of a finite group $G$ with a faithful complex representation of dimension $n$ is generated by elements of degree at most $n^2$. This was a remarkable advance, since no bound was known before. Symonds's proof combined equivariant cohomology with commutative algebra (Castelnuovo-Mumford regularity). We give better bounds for the cohomology ring of a p-group. The methods also apply to the Chow ring of algebraic cycles on $BG$.

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• Apr 022012

### David Ayala (Harvard University)

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#### Higher categories are sheaves on manifolds

Many proposed higher categories come from geometric situations. This talk will demonstrate a constructive connection between a homotopy theory of local invariants of n-manifolds and that of weak n-categories in the sense of Rezk. Connections to specific topological field theories will be discussed. This is a report on joint work with Nick Rozenblyum.

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• Mar 192012

### Inna Zakharevich (MIT)

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#### Scissors congruence and K-theory

Hilbert's third problem asks the following question: given two polyhedra with the same volume, is it possible to dissect one into finitely many polyhedra and rearrange it into the other one? The answer (due to Dehn in 1901) is no: there is another invariant that must also be the same. Further work in the 60s and 70s generalized this to other geometries by constructing groups which encode scissors congruence data. Though most of the computational techniques used with these groups related to group homology, the algebraic K-theory of various fields appears in some very unexpected places in the computations. We will give a different perspective on this problem by examining it from the perspective of algebraic K-theory: we construct the K-theory spectrum of a scissors congruence problem and relate some of the classical structures on scissors congruence groups to structures on this spectrum.

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• Mar 132012

### Po Hu (Wayne State University)

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#### Topological hermitian cobordism

I will discuss topological Hermitian cobordism, an RO(G)-graded Z/2 x Z/2- spectrum constructed in a joint paper with Igor Kriz. I will also talk about the method for calculating its RO(G)-graded coefficients completely.

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• Mar 122012

### Igor Kriz (University of Michigan)

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#### Khovanov homology and k-theory

I plan to discuss my recent joint work with Po Hu and Daniel Kriz on a lifting of Khovanov homology to connective k-theory. I will also talk about its relation to recent work by Lipshitz and Sarkar. As a technical tool, our approach exhibits a curious link between modular functors and the Elmendorf-Mandell approach to multiplicative infinite loop space theory.

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• Mar 052012

### David Gepner (Universität Regensburg and MIT)

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#### The $A^1$- localization of noncommutative motives and homotopy K-theory

The motivic perspective on algebraic K-theory is a useful means of constructing trace maps as well as other additive or localizing invariants which are often easier to compute. For instance, a homotopy invariant form of K-theory is obtained by forming the $A^1$-localization of noncommutative motives before passing to homotopy groups. In order to relate this to the usual notion of homotopy K-theory it is useful to have a direct construction of this localization, and this involves understanding the additive group $A^1$ in the noncommutative context. This is joint work with Andrew Blumberg and Goncalo Tabuada.

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• Feb 272012

### Dominic Verity (Center of Australian Category Theory)

$\begingroup$

The aim of this talk is to discuss the homotopy coherence properties of adjunctions between quasi-categories.

Taking as our lead the theory of the "walking adjunction" A of 2-category theory, we generalise to categories enriched in quasicategories and show that this same 2-category plays a similar role in this new context. Specifically, using insights drawn from the calculus of string diagrams we give an explicit presentation of A as a simplicially enriched category. We then use this to show that if C is any quasicategory enriched category and u is a right adjont 0-arrow in C, in some suitable sense to be discussed, then this data may be completed to give a simplicially enriched functor A->C. Furthermore, we show that the space of all such exensions is contractible.

That adjunctions of quasicategories may be completed up to enriched functors on A in this way contains, in its very essence, the adjunction data discussed by Jacob Lurie. Such enriched functors encapsulate both the coherent monad and the coherent comonad generated by such an adjunction and provide the building blocks upon which to found a formal theory of such things along the lines established by Street in the 2-categorical context.

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• Feb 212012

### Stefan Schwede (Universität Bonn)

$\begingroup$

#### Equivariant properties of symmetric products

The filtration on the infinite symmetric product of spheres by number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. In this talk I will discuss the equivariant stable homotopy types, for finite groups, obtained from this filtration for the infinite symmetric product of representation spheres. The filtration is more complicated than in the non-equivariant case, and already on the zeroth homotopy groups an interesting filtration of the augmentation ideal of the Burnside ring functor arises. Our method is by 'global' homotopy theory, i.e., we study the simultaneous behaviour for all finite groups at once. The equivariant subquotients are no longer rationally trivial, nor even concentrated in dimension 0.

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• Feb 132012

### Roman Mikhailov (Steklov Mathematical Institute and Institute for Advanced Study)

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#### A combinatorial description of homotopy groups of spheres

The talk is based on recent results obtained jointly with Jie Wu. For all $n>k$, we construct a finitely generated group by explicit generators and relations such that its center is the $n$-th homotopy group of the $k$-th sphere.

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## Fall 2011

• Dec 122011

### Emily Riehl (Harvard University)

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#### Algebraic Model Structures and Cellularity

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.

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• Dec 052011

### Andrew Blumberg (UT Austin)

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#### Progress towards TC(MU)

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.

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• Nov 282011

### John Francis (Northwestern University)

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#### Factorization Homology

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.

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• Nov 212011

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#### The Fundamental Theorem via Derived Morita Invariance, Localization, and $\mathbb{A}^{1}$-Homotopy Invariance

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.

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• Nov 142011

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#### Filtered spaces and homology of $E_n$-algebras

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.

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• Nov 072011

### Kirsten Wickelgren (Harvard University)

$\begingroup$

#### Massey Products in Galois Cohomology via Rational Points

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 $\pi_{1}^{\text{et}}(\mathbb{P}^1 - \{0,1,\infty\})$.

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• Oct 312011

### Kyle Ormsby (MIT)

$\begingroup$

#### Tate Normal Form in Level Resolutions of the $K(2)$-Local Sphere

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.

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• Oct 242011

### Nathaniel Stapleton (MIT)

$\begingroup$

#### Towards Transchromatic Twisted Character Maps

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 $n$.

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• Oct 172011

### Thomas Kragh (MIT)

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.

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• Oct 032011

### Chris Schommer-Pries (MIT)

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#### On the Uniqueness of the Homotopy Theory of Higher Categories

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.

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• Sep 262011

### Oscar Randal-Williams (University of Copenhagen)

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#### Stable Moduli Spaces of High-Dimensional Manifolds

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 $n = 3, 4, \ldots$

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• Sep 122011

### Vesna Stojanoska (MIT)

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#### Duality for Topological Modular Forms

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.

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## Summer 2011

• Aug 092011

### Matthew Gelvin (University of Copenhagen)

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#### What is a $p$-local finite group?

Fusion systems are an algebraic description of the $p$-local structure of a finite group. A centric linking system is the algebraic data needed to construct a topological classifying space for a fusion system; recent work of Andy Chermak shows that in fact, every saturated fusion system has a unique associated centric linking system. The collection of these data is what is known as a $p$-local finite group.

In this talk, I will try to give some insight about the structure of $p$-local finite groups. I will also give a brief discussion of the more topological notion of retractive transfer triple, which appears to be equivalent to the data of a $p$-local finite group.

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• Aug 022011

### Gabriel Katz (MIT)

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#### Morse Theory, gradient flows, concavity, and complexity on manifolds with boundary

Let $f: X \rightarrow \mathbb{R}$ be a Morse function on a manifold $X$ and $v$ its gradient-like vector field. Classically, the topology of a closed $X$ can be described in terms of the spaces of v-trajectories that link the singular points of $f$. On manifolds with boundary, the situation is somewhat different: there, a massive set of nonsingular functions is available. For such Morse data $(f, v)$, the interactions of the gradient flow with the boundary dX take central stage. We will introduce and measure the convexity and concavity of a $v$-flow relative to $dX$. "Some manifolds are intrinsically more concave than others with respect to any gradient flow" is the main slogan of the talk. Stated differently, the intrinsic concavity of $X$ is a reflection of its complexity. We will explain how this approach leads to new topological invariants, both of the flow $v$ and of the manifold $X$. In 3D, we have a good grasp of these invariants and their connection to the classification of 3-folds.

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• Jul 192011

### John Harper (University of Western Ontario)

$\begingroup$

#### Quillen homology completion and strong convergence of the

Quillen's derived functor notion of homology provides interesting and useful invariants in a variety of homotopical contexts, and includes as special cases (i) singular homology of spaces, (ii) homology of groups, and (iii) Andre-Quillen homology of commutative rings. Working in the topological context of symmetric spectra, we study topological Quillen homology of commutative ring spectra, $E_n$ ring spectra, and more generally, algebras over any operad $O$ in spectra. Using a QH-completion construction---analogous to the Bousfield-Kan R-completion of spaces---we prove under appropriate conditions (a) strong convergence of the associated homotopy spectral sequence, and (b) that connected O-algebras are QH-complete---thus recovering the O-algebra from its topological Quillen homology plus extra structure. A key problem in usefully describing this extra structure was solved recently using homotopical ideas in joint work with Kathryn Hess that describes a rigidification of the derived comonad that coacts on the object underlying topological Quillen homology, and plays the analogous role (in symmetric spectra) of the Koszul cooperad associated to a Koszul operad in chain complexes. This talk is an introduction to these results with an emphasis on proving (a) and (b) which is joint work with Michael Ching.

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• Jul 122011

### Michael Ching (Amherst College)

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#### Functors from spaces to spectra and the little disc operads

Greg Arone and I have been trying to understand the structure that exists on the derivatives of a functor in the sense of Goodwillie calculus. Previously we have shown that these derivatives possess the structure of a coalgebra over a certain comonad on the category of symmetric sequences. In this talk I'll try to describe in more depth what this structure amounts to for functors from based spaces to spectra. Specifically I'll relate these coalgebras to right modules over the (Koszul duals of) the little disc operads. This is all joint work with Greg, with substantial input also from Bill Dwyer.

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• Jul 052011

• Jun 062011

## Spring 2011

• May 032011

### Aravind Asok (University of Southern California)

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#### Some low-dimensional examples in $A^1$-homotopy theory

We will discuss connectedness in unstable $A^1$-homotopy theory, focusing on some examples. We will also discuss the classification of low dimensional $A^1$-connected smooth proper varieties and its some similarities to and differences from the corresponding topological classifications.

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• May 022011

• Apr 292011

### Nick Rozenblyum (MIT)

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#### Connections on Conformal Blocks

For an algebraic group $G$ and a projective curve $X$, we study the category of $D$-modules on the moduli space $Bun_G$ of principal $G$-bundles on $X$ using ideas from conformal field theory. We describe this category in terms of the action of infinitesimal Hecke functors on the category of quasi-coherent sheaves on $Bun_G$. This family of functors, parametrized by the Ran space of $X$, act by averaging a quasi-coherent sheaf over infinitesimal modifications of $G$-bundles at prescribed points of $X$. We show that sheaves which are, in a certain sense, equivariant with respect to infinitesimal Hecke functors are exactly $D$-modules, i.e. quasi-coherent sheaves with a flat connection. This gives a description of flat connections on a quasi-coherent sheaf on $Bun_G$ which is local on the Ran space.

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• Apr 112011

### Aravind Asok (University of Southern California)

$\begingroup$

#### Some low-dimensional examples in $A^1$-homotopy theory

We will discuss connectedness in unstable $A^1$-homotopy theory, focusing on some examples. We will also discuss the classification of low dimensional $A^1$-connected smooth proper varieties and its some similarities to and differences from the corresponding topological classifications.

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• Apr 042011

### Don Davis (Lehigh University)

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#### Nonimmersions of real projective spaces and tmf

We use the spectrum tmf to obtain new nonimmersion results for many real projective spaces $RP^n$ for n as small as 113. The only new ingredient is some new calculations of tmf-cohomology groups. We present an expanded table of nonimmersion results. We also present several questions about tmf.

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• Mar 282011

### Justin Thomas (University of Notre Dame)

$\begingroup$

#### Kontsevich's swiss cheese conjecture

We prove a conjecture of Kontsevich which states that if $A$ is an $E_{d-1}$ algebra then the Hochschild cohomology object of $A$ is the universal $E_d$ algebra acting on $A$. The notion of an $E_d$ algebra acting on an $E_{d-1}$ algebra was defined by Kontsevich using the swiss cheese operad of Voronov. We prove a homotopical property of the swiss cheese operad from which the conjecture follows.

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• Mar 072011

### Teena Gerhardt (Michigan State University)

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#### Cyclotomic spectra and computations in algebraic K-theory

In this talk I will describe joint work with Vigleik Angeltveit, Mike Hill, and Ayelet Lindenstrauss, yielding new computations of algebraic K-theory groups. In particular, we consider the K-theory of truncated polynomial algebras in several variables. Techniques from equivariant stable homotopy theory are often key to algebraic K-theory computations. In this case we use n-cubes of cyclotomic spectra to compute the topological cyclic homology, and hence K-theory, of the rings in question.

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• Feb 282011

### Tyler Lawson (University of Minnesota)

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#### Relative Picard and Brauer groups

Picard and Brauer groupoids, which capture information about invertible modules and central simple algebras, are objects of classical interest in algebra and number theory. The calculation of these objects is often aided by the use of Galois cohomology. We will discuss the generalizations to ring spectra, due to Hopkins-Mahowald-Sadofsky and Baker-Richter respectively. We then discuss how to compute these using Galois cohomology in higher category theory. (This talk based on joint work with David Gepner.)

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• Feb 142011

### Michael Ching (University of Georgia)

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#### A classification of Taylor towers for functors of spaces and spectra

I'll talk about joint work with Greg Arone to describe the data needed to reconstruct the Taylor tower of a functor from its layers. That data consists of a bimodule over the derivatives of the relevant identity functors, together with a coaction of a particular cotriple on the category of bimodules. I'll describe what we know about this cotriple for functors of based spaces and/or spectra.

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• Feb 072011

### John Harper (University of Western Ontario)

$\begingroup$

#### On a finiteness theorem and Quillen homology completion for

We prove a finiteness theorem relating finiteness properties of topological Quillen homology groups and homotopy groups --- this result should be thought of as an algebras over operads in spectra analog of Serre's finiteness theorem for the homotopy groups of spheres. We describe a rigidification of the derived cosimplicial resolution with respect to topological Quillen homology, and use this to define Quillen homology completion --- in the sense of Bousfield-Kan --- for algebras over operads in symmetric spectra. We prove that under appropriate connectivity conditions, the coaugmentation into Quillen homology completion is a weak equivalence --- in particular, such algebras over operads can be recovered from their topological Quillen homology. Many of the results described are joint with K. Hess.

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## Fall 2010

• Dec 072010

• Dec 062010

### Mark Walker (University of Nebraska)

$\begingroup$

#### What is the algebraic K-theory of a monoid?

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.

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• Nov 292010

### Robin Koytcheff (Brown University)

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#### A homotopy-theoretic view of Bott-Taubes integrals and knot spaces

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.

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• Nov 222010

### David Ayala (Harvard University)

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#### A combinatorial $E_n$ operad

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.

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• Nov 152010

### Srinikanth Iyengar (University of Nebraska)

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#### Colocalizing subcategories of the stable module category of (possibly infinite dimensional) modules over a finite group

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.

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• Nov 082010

### Kirsten Wickelgren (Harvard University)

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#### 2 nilpotent real section conjecture

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.

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• Nov 012010

### Kate Ponto (University of Kentucky)

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#### Fixed points and traces

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.

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• Oct 182010

### Kyle Ormsby (MIT)

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#### Equivariant motivic homotopy and the completion problem for Hermitian K-theory

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.

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• Oct 042010

### Ieke Moerdijk (University of Utrecht)

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#### To what extent are Lie groupoids like Lie groups?

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.

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• Sep 272010

### Chris Schommer-Pries (MIT)

$\begingroup$

#### Dualizability and Locality in 3D Topological Field Theory

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.

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• Sep 202010

### David Spivak (MIT)

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#### Categorical informatics

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.

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• Sep 132010

### Thomas Kragh (University of Oslo and MIT)

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#### Nearby Lagrangians

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).

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## Spring 2010

• May 102010

### John Palmieri (University of Washington)

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#### Algebraic topology vs. Sage

The computer software package Sage (\texttt{www.sagemath.org}) understands some basic constructions from algebraic topology: it understands simplicial complexes, cubical complexes, and it seems to be the only major mathematical software package with an implementation of $\Delta$-complexes. It can perform basic operations like joins, products, and connected sums. It can compute homology and cohomology over the integers or over a field. In this talk, we will discuss and demonstrate some of these capabilities, present some related unsolved problems, and discuss future directions.

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• May 032010

• Apr 272010

### Martin Frankland (MIT)

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#### Quillen cohomology of $\Pi$-algebras and application to their realization

A $\Pi$-algebra is a graded group with additional structure that makes it look like the homotopy groups of a space. Given one such object $A$, one may ask if it can be realized topologically: Is there a space $X$ such that $\pi_*X$ is isomorphic to $A$ as a $\Pi$-algebra, and if so, can we classify them?

Work of Blanc-Dwyer-Goerss provided an obstruction theory to realizing a $\Pi$-algebra $A$, where the obstructions (to existence and uniqueness) live in certain Quillen cohomology groups of $A$. What do these groups look like, and can we compute them?

We will tackle this question from the algebraic side, focusing on Quillen cohomology of truncated $\Pi$-algebras. We will then use the obstruction theory to obtain results on the classification of certain 2-stage homotopy types, and compare them to what is known from other approaches.

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• Apr 262010

### Martin Frankland (MIT)

$\begingroup$

#### Quillen cohomology of $\Pi$-algebras and application to their realization

A $\Pi$-algebra is a graded group with additional structure that makes it look like the homotopy groups of a space. Given one such object $A$, one may ask if it can be realized topologically: Is there a space $X$ such that $\pi_*X$ is isomorphic to $A$ as a $\Pi$-algebra, and if so, can we classify them?

Work of Blanc-Dwyer-Goerss provided an obstruction theory to realizing a $\Pi$-algebra $A$, where the obstructions (to existence and uniqueness) live in certain Quillen cohomology groups of $A$. What do these groups look like, and can we compute them?

We will tackle this question from the algebraic side, focusing on Quillen cohomology of truncated $\Pi$-algebras. We will then use the obstruction theory to obtain results on the classification of certain 2-stage homotopy types, and compare them to what is known from other approaches.

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• Apr 132010

### Angélica Osorno (MIT)

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#### An Infinite Loop Space Structure for $K$-theory of Bimonoidal Categories

In recent work of Baas-Dundas-Richter-Rognes, the authors introduce the notion of the $K$-theory of a bimonoidal category $R$, and show that it is equivalent to the algebraic $K$-theory space of the ring spectrum $HR$. In this thesis we show that $K(R)$ is the group completion of the classifying space of the 2-category $Mod_R$ of modules over $R$, and show that $Mod_R$ is a symmetric monoidal 2-category. We explain how to use this symmetric monoidal structure to produce a $\Gamma$-(2-category), which gives an infinite loop space structure on $K(R)$. We show that the equivalence mentioned above is an equivalence of infinite loop spaces.

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• Apr 052010

### Fabien Morel (Ludwig-Maximilians-Universität)

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#### An $A^1$-homological criterium for the rigidity property

This is one of the main technical step in the proof of the Friedlander-Milnor conjecture, and uses a lot of "classical" topological arguments (lower central series, $A^1$-derived functors of the free Lie algebra functors, devissage of the Lie algebra functor in terms of other polynomial functors.

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• Mar 292010

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#### Non-commutative motives

In this talk I will describe the construction of the category of non-commutative motives $[1,2,3]$ in Drinfeld-Kontsevich's non-commutative algebraic geometry program. In the process, I will present the first conceptual characterization of Quillen's higher K-theory since Quillen's foundational work in the 70's. As an application, I will show how these results allow us to obtain for free the higher Chern character from K-theory to cyclic homology.

References:

[1] D.-C. Cisinski and G. Tabuada, Symmetric monoidal structure on Non-commutative motives. Available at arXiv:1001.0228.

[2] D.-C. Cisinski and G. Tabuada, Non-connective K-theory via universal invariants. Available at arXiv:0903.3717.

[3] G. Tabuada, Higher K-theory via universal invariants. Duke Math. Journal, 145 (2008), no.1, 121-206. 2010/03/30,,,,Matthew Gelvin"

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• Mar 152010

### Rob Thompson (CUNY)

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#### $v_n$-periodic $\Pi$ algebras

A $\Pi$-algebra is a sequence of groups, along with an action of the primary homotopy operations, which are indexed by homotopy groups of wedges of spheres. It is natural to view the homotopy groups of a space as an object in the category of $\Pi$-algebras. The $v_n$-periodic homotopy groups of a space are obtained by inverting non-nilpotent self maps of finite complexes. In this talk I will investigate $v_n$-periodic $\Pi$-algebras.

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• Mar 082010

### Andrew Salch (Johns Hopkins University)

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#### $v_2$ and $v_4$

We discuss some new methods for computing the cohomology of Morava stabilizer groups at large heights. We apply these methods to compute a good approximation to the cohomology of the 5-primary height 4 stabilizer group, giving us some interesting information about $v_4$-periodic stable homotopy groups of spheres, and we discuss what is involved in computing the entire cohomology of the Morava stabilizer group for heights 4 and up.

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• Mar 012010

### Mark Behrens (MIT)

$\begingroup$

#### The homotopy groups of the $E(2)$-local sphere, revisited

Shimomura and Yabe computed the homotopy groups of the $E(2)$-local sphere at primes greater than or equal to $5$. However, their computation is difficult to understand. I will describe a conceptual way to understand the answer, and the computation. The goal is to make the answer as concrete as the image of the $J$ homomorphism. We'll see of this goal is achieved...

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• Feb 082010

### Simona Paoli (Penn State Altoona)

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#### Homotopical properties of a weakly globular model of homotopy types

Homotopy $n$-types are an important class of topological spaces: they amount to CW complexes whose homotopy groups vanish in dimension higher than $n$. The problem of modeling homotopy types is relevant both in higher category theory and homotopy theory and received contributions from both areas. There is a particularly simple model of homotopy types in the path connected case, consisting of n-fold categories internal to groups, also called $cat^n$-groups. This model, however, has the disadvantage that is it does not have an algebraic description of the Postnikov decomposition nor it is easy to establish algebraically when a map of $cat^n$-groups is a weak equivalence.

In this talk we introduce a new model of connected $n$-types through a subcategory of $cat^n$-groups, which we call weakly globular, for which the above issues are resolved in transparent way. We also describe other homotopical properties of this model, and discuss the relevance of these structures for higher category theory.

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## Fall 2009

• Dec 072009

• Nov 302009

### Barry Walker (Northwestern University)

$\begingroup$

#### Orientations and $p$-adic analysis

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.

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• Nov 242009

### Charles Rezk (University of Illinois Urbana-Champaign)

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#### Koszul resolutions for algebras of power operations

This will be a Tuesday talk, at 4pm in 2-132.

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.

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• Nov 232009

### Rekha Santhanam (Johns Hopkins University)

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#### Equivariant $\Gamma$-spaces

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 $G$.

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• Nov 162009

### Martin Bendersky (CUNY)

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#### The unstable chromatic spectral sequence

The (stable) chromatic spectral sequence has had a significant impact on our understanding of the stable homotopy groups of the spheres. I will talk about preliminary attempts to construct an unstable version. I will try to describe a filtration of the stable chromatic spectral sequence induced by the Hopf rings for the odd spheres. There are natural questions that arise in the unstable world (e.g. an unstable version of the Morava stabilizer algebra) and a chromatic interpretation of the Hopf invariant.

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• Nov 092009

### Tyler Lawson (University of Minnesota)

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#### $p$-divisible groups, automorphic forms, and displays

Lurie's theorem allows the functorial construction of $E_\infty$ ring spectra associated to certain $p$-divisible groups. In this talk I will discuss three situations in which we can apply this and attempts to understand the computational results. The first is joint work with Behrens on the relationship between the moduli of elliptic curves and certain moduli of abelian surfaces with complex multiplication. The second is joint work with Hill on Shimura curves that parametrize "false elliptic curves", and in particular trying to obtain computations of the homotopy of the associated spectra without niceties such as $q$-expansions and Weierstrass equations. The third is on using Zink's work on displays to produce $E_\infty$ ring spectra from purely algebraic input data, in the form of invertible matrices over Witt rings.

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• Nov 022009

• Oct 262009

### Kirsten Wickelgren (Harvard University)

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#### Étale $\pi_1$ obstructions to homotopy sections of curves

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

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• Oct 192009

• Oct 052009

### Gereon Quick (WWU Muenster)

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#### Some applications of profinite homotopy theory

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.

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• Sep 292009

### Søren Galatius (Stanford University)

#### Monoids of moduli spaces of manifolds.

This will be a Tuesday talk, at 4pm in 2-132.

• Sep 282009

• Sep 212009

### David Ayala (University of Copenhagen)

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#### Singular field theories

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.

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• Sep 142009

### Chris Schommer-Pries (Harvard University)

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#### Extensions of $2$-Groups and a Finite Dimensional Model of the String Group

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.

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## Spring 2009

• May 182009

### Mike Hopkins (Harvard University)

#### The Kervaire Invariant

This talk will take place at the normal time (4:30) in 4-270.

• May 122009

### José Gómez (University of British Columbia)

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#### Stable splittings and almost commuting elements

This will be a Tuesday talk, at 4:00 in 2-151

In this talk we will show that the space of almost commuting elements in a compact Lie group splits after one suspension.

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• May 112009

• May 042009

### David Spivak (University of Oregon)

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#### Derived smooth manifolds

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

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• Apr 272009

• Apr 152009

### Fred Cohen (University of Rochester)

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#### Generalized moment-angle complexes

This will be a Wednesday talk, at 4:00 in 2-143.

A second subspace of a product is the generalized moment-angle complex first defined in generality by Neil Strickland. Definitions, examples, as well as connections will be addressed.

One notable case is given by subspaces of products of infinite dimensional complex projective space 'indexed by a finite simplicial complex'. These spaces appearing in work of Goresky-MacPherson, Davis-Januskiewicz, Buchstaber-Panov-Ray, Denham-Suciu, Franz as well as many others encode information ranging from the structure of toric varieties in one guise, Stanley-Reisner rings, as well as 'motions of certain types of robotic legs' in another guise.

What do these spaces have to do with the motions of legs of a cockroach? This feature will be illustrated with slides.

Features of these spaces are developed within the context of classical homotopy theory based on joint work with A. Bahri, M. Bendersky, and S. Gitler.

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• Apr 142009

### Fred Cohen (University of Rochester)

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#### On natural subspaces of products, and their applications

This will be a Tuesday talk, at 4:00 in 2-151

The first basic example here is the configuration space of unordered $k$-tuples of distinct points in a space $M$. When specialized to the case of $M$ given by the complex numbers, these spaces can be identified as the space of classical complex, monic polynomials of degree $k$ which have exactly k distinct roots.

Elementary features of these spaces as well as their connections to spaces of knots, links, and homotopy groups of spheres will be addressed. These topics are based on joint work with R. Budney as well as J. Berrick, Y. Wong and J. Wu.

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• Apr 132009

### Fred Cohen (University of Rochester)

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#### On spaces of homomorphisms and spaces of representations

The subject of this talk is the structure of the space of homomorphisms from a free abelian group to a Lie group $G$ as well as quotients spaces given by the associated space of representations. These spaces as well as further spaces of representations admit the structure of a simplicial space at the heart of the work here. Features of geometric realizations will be developed.

What is the fundamental group or the first homology group of the associated space in case $G$ is a finite, discrete group?

This deceptively elementary question as well as more global information given in this talk is based on joint work with A. Adem, E. Torres, and J. Gomez.

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• Apr 062009

### Nitu Kitchloo (University of California, San Diego)

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#### The two dimensional cobordism category with flat connections

I will describe the stable (in genus) structure of the universal moduli space of flat connections on riemann surfaces. I will also introduce the category of $1$-manifolds and $2$-cobordisms endowed with flat connections. Using classical techniques of Atiyah-Bott, and more recent techniques introduced by Madsen-Weiss and coauthors, we will give a complete description of the classifying space of this category. This is joint work with R. Cohen and S. Galatius.

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• Mar 302009

### Sam Isaacson (Harvard University)

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#### Cubical homotopy theory and monoidal model categories

Let $C$ be a model category. In a 2001 paper, Dan Dugger showed that if $C$ is combinatorial, it can be realized as a left Bousfield localization of simplicial presheaves on some small site. I'll describe a variation of this theorem: by replacing simplicial sets with a cubical model for the homotopy category, we can produce a presentation for $C$ when $C$ is symmetric monoidal that retains the monoidal structure of $C$ as the Day convolution product.

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• Mar 162009

• Mar 092009

• Mar 032009

### Pascal Lambrechts (University of Louvain)

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#### Rational homology of spaces of smooth embeddings

This will be a Tuesday talk, at 4:00 in 2-151.

.For a given compact smooth manifold $M$ we consider the space $\mathrm{Emb}(M,R^k)$ of smooth embeddings of M into some large Euclidean space $R^k$, or rather some geometric variant of it, which is a homotopy invariant of $M$.

I will explain how Goodwillie's cutting method enables us to understand the homotopy type of this space of emeddings. I will then prove that the rational homology of that space is actually an invariant of the rational homotopy type of $M$. The proof is based on Kontsevich's theorem on the formality of the little cube operad and Arone's description of the layers of Weiss' orthogonal tower for the space of embeddings. This is a joint work with Greg Arone and Ismar Volic.

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• Mar 022009

### John Harper (École Polytechnique Fédérale de Lausanne)

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#### Bar constructions and Quillen homology of modules over operads

In Haynes Miller's proof of the Sullivan conjecture on maps from classifying spaces, Quillen's derived functor notion of homology (in the case of commutative algebras) is a critical ingredient. This suggests that homology for the larger class of algebraic structures parametrized by an operad will also provide interesting and useful invariants. Working in the two contexts of symmetric spectra and unbounded chain complexes, we establish a homotopy theory for studying Quillen homology of modules and algebras over operads, and we show that this homology can be calculated using simplicial bar constructions. A key part of the argument is proving that the forgetful functor commutes with certain homotopy colimits. A larger goal is to determine the extra structure that appears on the derived homology and the extent to which the original object can be recovered from its homology when this extra structure is taken into account. This talk is an introduction to these results with an emphasis on several of the motivating ideas.

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• Feb 232009

### Samik Basu (Harvard University)

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#### $A_\infty$ structures on Thom spectra

Let $R$ be an $E_\infty$ ring spectrum. Given a map $f:X \rightarrow \mathrm{BGL}_1(R)$, we can construct a Thom spectrum $Xf$. If $f$ is a loop map, then there is an $A_\infty$ $R$ module structure on the Thom spectrum. I will consider various examples of these Thom spectra and construct $A_\infty$ structures on them. I will then use this identification to calculate Topological Hochschild Homology.

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• Feb 092009

### Brian Munson (Wellesley College)

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#### A stable range description of the space of link maps

For smooth manifolds $P$, $Q$, and $N$, let $\mathrm{Link}(P,Q;N)$ denote the space of smooth maps of $P$ in $N$ and $Q$ in $N$ such that their images are disjoint. I will discuss the connectivity of a "generalized linking number" from the homotopy fiber of the inclusion of $\mathrm{Link}(P,Q;N)$ into $\mathrm{Map}(P,N)\times\mathrm{Map}(Q,N)$ to a certain cobordism space of manifolds over a space which is a homotopy theoretic model for the intersections of $P$ and $Q$. The proof of the connectivity uses some easy statements about connectivities in the world of smooth manifolds as a guide for obtaining similar estimates in a setting where the tools of differential topolgy do not apply. This is joint work with Tom Goodwillie.

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## Fall 2008

• Dec 152008

### Boris Botvinnik (University of Oregon)

#### Cobordism category of manifolds with positive scalar curvature

• Dec 092008

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#### Fusion in the Burnside ring

This talk is to be held at 4pm in room 2-151.

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.

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• Dec 082008

### Matthias Kreck (Universität Bonn)

#### Codes, arithmetic and $3$-manifolds.

This talk will begin at 5:00. Please note the time change.

• Dec 012008

• Nov 242008

### Kathryn Lesh (Union College)

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#### An interesting filtration of bu and an analogue of the Whitehead Conjecture

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.

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• Nov 172008

### Bertrand Guillou (University of Illinois Urbana-Champaign)

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#### Enriched and equivariant homotopy 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.

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• Nov 122008

### Dev Sinha (University of Oregon)

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#### Configuration spaces and homotopy theory.

This talk is to be held at 4pm in room 2-142.

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.

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• Nov 032008

### Carl-Friedrich Bödigheimer (Universität Bonn)

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#### Symmetric groups and moduli spaces of surfaces

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 $h =3D 2g + m$).

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• Oct 272008

### Matthew Ando (University of Illinois Urbana-Champaign)

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#### Parametrized spectra, Thom spectra, and twisted Umkehr maps

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.

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• Oct 202008

### Mark Hovey (Wesleyan University)

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#### Ring spectra of finite dimension

In joint work with Keir Lockridge, we have been developing theories of global and weak dimensions for ring spectra. We have good results for ring spectra of dimension zero, and partial results but good conjectures for the finite dimensional case.

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• Oct 152008

### Larry Smith (Georg-August-Universität Göttingen)

#### Local cohomology, Poincare duality algebras, and Macaulay dual systems.

This is held in room 2-142!

• Oct 062008

### Alex Suciu (Northeastern University)

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#### Cohomology jumping loci

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 $X$.

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• Sep 292008

### Ismar Volic (Wellesley College)

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#### Link invariants through multivariable manifold calculus

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.

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• Sep 152008

### Søren Galatius (Stanford University)

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#### hocolim decomposition of compactified moduli space

The moduli space of Riemann surfaces $M$ is a classifying space for families of Riemann surfaces. It has a compactification $\bar M$, which is a classfying space for families of modal Riemann surfaces. A nodal Riemann surface is allowed to have singularities which look like the solutions to $zw=0$ in complex $2$-space. I will describe how to decompose $\bar M$ as a homotopy colimit of spaces which look more like $M$. Then I will use this to study part of the homology of $\bar M$, using what is known about the homology of $M$.

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• Sep 082008

### Birgit Richter (Universität Hamburg)

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#### An involution on the K-theory of (some) bimonoidal categories

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.

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