Spring 2024 Algebraic topology seminar
Where:
507 Mathematics
When:
Fridays 10-11am
Date | Speaker (affiliation) | Title & Abstract |
January 26 | Elden Elmanto (University of Toronto) | Atiyah-Segal completion theorems in algebraic geometry.Abstract: I will speak on joint work in progress with Kubrak and Sosnilo on the K-theory of algebraic stacks. Inspired by a theorem of Atiyah-Segal in topology, we prove a similar result for quotient stacks in characteristic zero, generalizing results of Thomason, Krishna, Tabuada-van den berg and others. This leads to a definition of motivic cohomology of stacks in characteristic zero. |
February 2 | Toni Annala (IAS) | Atiyah duality and applicationsAbstract: In topology, Atiyah duality provides a geometric model for the dual of the suspension spectrum of a smooth manifold. In this talk, we export this into algebraic geometry by proving an analogous claim in the non-$\mathbb{A}^1$-invariant stable motivic homotopy theory of Annala-Hoyois-Iwasa. Besides recovering many Poincaré duality type results, it has quite interesting consequences for the behavior of the $\mathbb{A}^1$-colocalization functor R. Namely, R is a way of turning a cohomology theory into an $\mathbb{A}^1$-invariant one without changing the on smooth projective varieties. Using this observation, we can prove the independence of logarithmic cohomology groups from the choice of good compactification, and that certain cohomology groups are birational invariants. |
February 9 | Morgan Opie (UCLA) | Enumerating stably trivial topological vector bundles with higher real K-theoriesAbstract: The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking questions about topological vector bundles. But, in general, there are many non-equivalent vector bundles with the same K-theory class. Bridging the gap between K-theory and actual bundle theory is challenging, even for the simplest CW complexes. Building on work of Hu, we use Weiss-theoretic techniques in tandem with a little chromatic homotopy theory to translate vector bundle enumeration questions to tractable stable homotopy theory computations. Our main result is to compute lower bounds for the number of stably trivial rank complex rank r topological vector bundles on complex projective n-space, for infinitely many n and r. The talk will include a gentle discussion of the tools involved. This is joint work with Hood Chatham and Yang Hu. |
February 16 | Tasos Moulinos (CNRS/IAS) | Twists of stable homotopy theoryAbstract: Twisted stable homotopy theory was introduced by C. Douglas in his 2005 PhD thesis, to accomodate a need in Floer homotopy theory, of dealing with infinite-dimensional manifolds that are "non-trivially polarised". Roughly one can think of a twisted spectrum over a fixed topological space B as a global section of a bundle of stable infinity-categories over B, which has fiber the category of spectra. I will talk about recent work developing the theory of twisted spectra from an infinity-categorical perspective. I will describe several ways of thinking about such objects, as well how their ensuing functoriality is determined by being fibered over the Brauer space of the sphere spectrum. I will also mention some examples, both of an elementary nature and some arising from Seiberg-Witten Floer theory. This is joint work with Alice Hedenlund. |
February 23 | Shubhodip Mondal (IAS) | $p$-typical curves on Tate twistsAbstract: In this talk, I will discuss joint work with Sanath Devalapurkar in which we show that de Rham–Witt forms are naturally isomorphic to $p$-typical curves on $p$-adic Tate twists. I will attempt to discuss the proof of this result which is obtained by more generally equipping a related result of Hesselholt on topological cyclic homology with the motivic filtrations introduced by Bhatt–Morrow–Scholze. |
March 1 | Achim Krause (IAS/Münster) | On the K-theory of $\mathbb{Z}/p^n$Abstract: Algebraic K-theory of $\mathbb{Z}/p$ was computed by Quillen shortly after defining higher K-groups. On the contrary, $K(\mathbb{Z}/p^n)$ for $n>1$ has so far eluded computation in all but the smallest degrees. Based on the recently discovered prismatic cohomology, we compute algebraic K-theory of those rings. We do not quite obtain closed-form descriptions, but our calculation is completely effective and we have turned it into a program computing K-theory through an arbitrary range of degrees. In this talk, I want to give an overview over our methods. |
April 12 | Abigail Hickok (Columbia) | TitleAbstract: |