Spring 2024 Algebraic topology seminar

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.

Abstract: 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.

Abstract: 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.

Abstract: 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.

Abstract: 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.

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.

Abstract: I will introduce the general approach to duality in a six-functor formalism known as "Lu-Zheng magic" and explain how it implies and clarifies classical Poincaré duality in the case of the six-functor formalism on anima.

Abstract: The theory of dualizable presentable $\infty$-categories has received a lot of attention in recent years, in large parts due to Efimov's recent work on continuous K-theory.
In this talk, I will go over some motivations for this theory, as well as its basic inner workings; up to a sketch of the proof that the $\infty$-category of dualizable presentable $\infty$-categories is itself presentable.

Abstract: Topological data analysis (TDA) is a way to understand the “shape” of a data set (e.g., a collection of points in R^n) by using algebraic topology. The primary tool of TDA is persistent homology, which tracks the connected components, holes, and higher-dimensional homology classes as they emerge and disappear at increasing scale. I’ll start with an overview of TDA, and then we’ll talk about methods for analyzing how the topology of a data set changes as multiple parameters vary. This is a very active area of research. In particular, I’ll discuss a construction called “persistence diagram bundles” that I introduced, as well as its relation to other objects in TDA such as “vineyards”, “multiparameter persistent homology”, and the “persistent homology transform”.

Abstract: The talk is about cohomological invariants of topological spaces with an action of a compact Lie group $G$. If we insist on these taking values in rational vector spaces, one may hope to find a small and calculable algebraic model. This is now known for many examples, and there is evidence in general. The talk will describe the structural features of the category and the general character of the model, illustrating it with examples. The methods are fairly general and should apply in other similar contexts. (The talk will feature joint work with S.Balchin and T.Barthel).