— 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.

— Spring 2019 —

4 Feb 2019

11 Feb 2019

Niles Johnson, Ohio State University

18 Feb 2019

25 Feb 2019

4 Mar 2019

Jonathan Campbell, Vanderbilt University