— Spring 2019 —

4 Feb 2019

Mona Merling, University of Pennsylvania

The equivariant stable parametrized h-cobordism theorem

The stable parametrized h-cobordism theorem provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. For a compact smooth manifold M it gives a decomposition of Waldhausen's A(M) into QM_+ and a delooping of the stable h-cobordism space of M. I will talk about joint work with Malkiewich on this story when M is a smooth compact G-manifold.

11 Feb 2019

Niles Johnson, Ohio State University

The algebra of stable 2-types

This talk will describe recent work using symmetric monoidal 2-categories to build algebraic models of stable homotopy 2-types (equivalently, 3-connected 6-types). We will describe a dictionary between homotopy-theoretic constructions among stable 2-types and algebraic constructions among symmetric monoidal 2-categories. As applications, we obtain a model for the 2-type of the sphere and the 2-type of algebraic K-theory spectra. This work is joint with Nick Gurski and Angélica Osorno.

18 Feb 2019

25 Feb 2019

Jonathan Campbell, Vanderbilt University

Iterated Traces in Bicategories and Applications

Kate Ponto and Mike Shulman have developed a powerful categorical framework for defining traces in bicategories. This framework of "shadows" has wide application, in particular to algebraic K-theory and fixed point theory. In this talk I'll discuss another application: proving a very general Lefschetz fixed point theorem that recovers ones of Lunts, Shklyarov, and Cisinski-Tabuada. Time permitting, I'll discuss further applications to Topological Hochschild Homology (THH) and Hopkins-Kuhn-Ravenel theory. This is work joint with Kate Ponto.

4 Mar 2019

11 Mar 2019

Michael Ching, Amherst College

Tangent ∞-categories and Goodwillie calculus

Goodwillie calculus is a set of tools in homotopy theory developed, to some extent, by analogies with ordinary differential calculus. The goal of this talk is to make that analogy precise by describing a common higher-category-theoretic framework that includes both the calculus of smooth maps between manifolds, and the calculus of functors, as examples. This framework is based on the notion of "tangent category" introduced first by Rosicky and recently developed by Cockett and Cruttwell in connection with models of differential calculus in logic. In joint work with Kristine Bauer and Matthew Burke (both at Calgary) we generalize to tangent structures on an (∞,2)-category and show that the (∞,2)-category of presentable ∞-categories possesses such a structure. This allows us to make precise, for example, the intuition that the ∞-category of spectra plays the role of the real line in Goodwillie calculus. As an application we show that Goodwillie's definition of n-excisive functor can be recovered purely from the tangent structure in the same way that n-jets of smooth maps are in ordinary calculus. If time permits, I will suggest how other concepts from differential geometry, such as connections, may play out into the context of functor calculus.

18 Mar 2019

25 Mar 2019

Bernardo Villarreal, UNAM

Classifying spaces for commutativity

In this talk I’ll give a brief summary of the most important results on the theory of classifying spaces for commutativity for a topological group G, denoted BcomG. The second part of the talk will treat TC structures: There is a natural inclusion of BcomG into the classifying space BG, and a natural question is when does the classifying map of a G-bundle lifts to BcomG. A homotopy class of such a lift is called “Transitionally Commutative” (TC) structure. In recent work with O. Antolín-Camarena, S. Gritschacher and D. Ramras we have constructed characteristic classes for the example cases of the orthogonal groups O(n) (but mostly O(2)), that give obstructions to such structures.

1 Apr 2019

David White, Denison University

The homotopy theory of homotopy presheaves

I will present a model category structure that encodes the homotopy theory of (small) homotopy functors, from a combinatorial model category to simplicial sets. The fibrant objects are the homotopy functors, i.e. functors that preserve weak equivalences. Next, I will explain how the homotopy theory of homotopy functors is homotopy invariant, i.e. a Quillen equivalence on domain categories induces a Quillen equivalence on homotopy functor categories. I will demonstrate the importance of this result with examples drawn from numerous fields, including spaces, spectra, chain complexes, simplicial presheaves, motivic spectra, infinity categories, and infinity operads. This is joint work with Boris Chorny.

8 Apr 2019

Hiro Lee Tanaka, Texas State University

Morse theory on a point: Broken lines and associativity

I'll introduce a stack of Morse trajectories on a point. It turns out this stack classifies associative algebras in a large class of categories, and this is a first step toward constructing stable homotopy enrichments of invariants that people in mirror symmetry care about (Lagrangian Floer theory and, more generally, Fukaya categories). I'll begin with a basic review of Morse theory and give some feel for what this stack is doing. This is joint work with Jacob Lurie.

15 Apr 2019

22 Apr 2019

Maximilien Peroux, UIC

Coalgebras and comodules in stable homotopy theory

We investigate how to use homotopy-theorical methods in order to study coalgebras and comodules, using model categories and ∞-categories. We explain how model categories fail to represent the correct homotopy theory of coalgebras in spectra, from joint work with Brooke Shipley. We also present current progress on rectification results for coalgebras and comodules in spectra over the Eilenberg-Mac Lane spectrum of a field.

29 Apr 2019

— Fall 2018 —

6 Sept 2018

K-theory of endomorphisms, Witt vectors, and cyclotomic spectra

There is an endofunctor of the category of categories which associates to a category C the category End(C) of endomorphisms of objects of C. If C is a stable infinity category then End(C) is as well, and the associated K-theory spectrum KEnd(C):=K(End(C)) is called the K-theory of endomorphisms of C. Using calculations of Almkvist together with the theory of noncommutative motives, we classify equivalence classes of endofunctors of KEnd in terms of a noncompeleted version of the Witt vectors of the polynomial ring Z[t], answering a question posed by Almkvist in the 70s. As applications, we obtain various lifts of Witt rings to the sphere spectrum as well as a more structured version of the cyclotomic trace via cyclic K-theory, as studied in recent work of Kaledin and Nikolaus-Scholze.

10 Sept 2018

Kevin Carlson, UCLA

The shadow of ∞-category theory in category theory

Any ∞-category has an underlying 2-category, in which the 2-morphisms are all invertible, which associates to any object of an ∞-category a presheaf of groupoids on a 2-category. Under appropriate conditions we can get a Whitehead-type theorem for the original ∞-category, in which homotopy groups are replaced by homotopy groupoids, and even a Brown representability theorem, constructing objects of the original ∞-category from this purely categorical data. These conditions hold notably for the ∞-categories of spaces and of small ∞-categories. If time allows, I'll describe joint work with Christensen proving that the 2-dimensional aspect is unavoidable for these theorems, even in the case of spaces.

17 Sept 2018

24 Sept 2018

1 Oct 2018

8 Oct 2018

Liang Ze Wong, University of Washington

Enriched fibrations and the relative nerve

The Grothendieck construction relates (pseudo)functors $B^{op} \to Cat$ with fibrations over $B$. In this talk, I will present an enriched version of this correspondence, which holds when the enriching category $V$ satisfies certain conditions. Applied to $V = sSet$, (the dual of) this result provides an alternative construction of Lurie's nerve of $B$ relative to a functor $B \to sCat \to sSet$, as well as a factorization of the operadic nerve. If time permits, I will discuss applications to coalgebras of an operad. This is joint work with Jonathan Beardsley.

15 Oct 2018

Chris Kapulkin, University of Western Ontario

Cubical sets and higher category theory

I will report on the recent work joint with Voevodsky on using cubical
sets to gain a better understanding of a number of constructions in
higher category theory. This work is inspired by the use of cubical
sets in Homotopy Type Theory by Coquand and his group.

22 Oct 2018

29 Oct 2018

5 Nov 2018

Aaron Mazel-Gee, USC

The geometry of the cyclotomic trace

K-theory is a means of probing geometric objects by studying their vector bundles, i.e. parametrized families of vector spaces. Algebraic K-theory, the version applying to varieties and schemes, is a particularly deep and far-reaching invariant, but it is notoriously difficult to compute. The primary means of computing it is through its "cyclotomic trace" map K→TC to another theory called topological cyclic homology. However, despite the enormous computational success of these so-called "trace methods" in algebraic K-theory computations, the algebro-geometric nature of the cyclotomic trace has remained mysterious.

In this talk, I will describe a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry. This rests of a reinterpretation of genuine-equivariant homotopy theory in terms of stratified geometry (following Glasman and others). This represents joint work with David Ayala and Nick Rozenblyum. By the end of the talk, you will be able to take home with you a very nice and down-to-earth fact about traces of matrices. If there is time, I will also indicate joint work with Reuben Stern on a 2-dimensional version of this story that applies to "2-traces" for the 2-vector bundles of Baas--Dundas--Rognes.

In this talk, I will describe a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry. This rests of a reinterpretation of genuine-equivariant homotopy theory in terms of stratified geometry (following Glasman and others). This represents joint work with David Ayala and Nick Rozenblyum. By the end of the talk, you will be able to take home with you a very nice and down-to-earth fact about traces of matrices. If there is time, I will also indicate joint work with Reuben Stern on a 2-dimensional version of this story that applies to "2-traces" for the 2-vector bundles of Baas--Dundas--Rognes.

12 Nov 2018

Morgan Opie, Harvard

Localization in homotopy type theory

I will discuss a formulation of localization at a prime in homotopy type theory. The main goal of my talk is prove type-theoretic analogues of classical results on the effect of localization of spaces on algebraic invariants. The main theorem is that for a pointed, simply connected type, the natural $p$-localization map induces algebraic localization on all homotopy groups. I'll preface these results by summarizing key ideas of homotopy type theory and the theory of localization of spaces, and throughout my talk I will emphasize ways in which the type-theorietic story differs from the classical one. This is joint work with J. D. Christensen, E. Rijke, and L. Scoccola.

19 Nov 2018

26 Nov 2018

Aaron Royer (CANCELLED)

3 Dec 2018

Agnès Beaudry, University of Colorado

The Linearization Conjecture

For G a nice profinite group (such as the Morava stabilizer groups), I will discuss the construction of a p-adic sphere which comes equipped with a natural action of G. The linearization conjecture predicts that this sphere is a kind of one point compactification of the p-adic Lie algebra of G. I will explain how to show that this holds when the action is restricted to certain finite subgroups of G and discuss an application to chromatic homotopy theory.