Spring 2022 Talks

Day Speaker Title
02/11 Brian An Introduction to Exodromy

One of the first major topics we learn about in algebraic topology is the classification of locally constant sheaves of sets (i.e. covering spaces) of a sufficiently nice topological space in terms of its fundamental group. This classification is mediated by an equivalence of categories known as the monodromy equivalence. An insight of Kan was that, in order to classify locally constant sheaves of more interesting objects, one must pass from fundamental groups to fundamental infinity-groupoid. In this expository talk, I’d like to talk about work of Barwick-Glasman-Haine pushing this circle ideas further into the realm of stratified spaces. The main result is the exodromy equivalence, which classifies constructible sheaves on a stratified space in terms of its profinite stratified shape.
02/18 Timmy Formalism of six operations and derived algebraic stacks

The formalism of six operations was originally introduced by A. Grothendieck and his collaborators in the study of étale cohomology. It naturally leads to many well-known results in cohomology theory like duality and Lefschetz trace formula. This partially justifies the slogan that the formalism of six operations are enhanced cohomology theories. In this talk, I will introduce the formalism of six operations. I will explain the relation between it and some cohomology theories (topological, coherent, ℓ-adic). Moreover, I will talk about the application of it to a nice class of derived algebraic stacks. And show that this leads to some nontrivial results of algebraic (homotopy) K-theory for stacks.
02/25 Zach Introduction to Equivariant Homotopy Theory and RO(G)-graded Cohomology

In its simplest form, equivariant homotopy theory is the study of homotopy theory with the addition of actions by a group G. By mixing representation theory into homotopy theory, we create additional structure and complexities to consider. To every representation of our group, we may take the one point compactification and the group will then act on the resulting sphere. In so called “genuine” G-spectra, these representation spheres are invertible, so in particular we may grade cohomology theories on (virtual) G-representations. In this introductory talk, I will go over some of the additional properties and structures of equivariant homotopy theory and if time permits illustrate these structures with a cohomology computation.
03/04 CANCELLED
03/11 Likun Applications of Dualizing Complex in Commutative Algebra

Starting with a finitely generated module of finite projective dimension over the completion of a Noetherian local ring A, a natural question is when does this module descend, i.e. when is this module the completion of a finite A-module of finite projective dimension? We will need the theorem of local duality to show it happens over some “good” rings. We will introduce dualizing complex both in the language of derived categories, and in an explicit form for commutative Noetherian ring. We will apply them to study some descent problems for module of finite projective dimension. The techniques employed also allow one to recover a theorem of Horrocks about vector bundles over a punctured spectrum of a local ring. We follow Section I.5 in Dimension projective finie et cohomologie locale by Peskine and Szpiro.
03/25 Yigal An introduction to 2-categories

This talk will be a survey of some of the basic notions and facts about 2-categories. After introducing 2-categories and various types of functors between them, we will consider constructions that map 2-categories to more familiar objects. On one hand, viewing a 2-category as a “poor” higher category, we can “reduce” it to an ordinary category, via a homotopy construction. On the other hand, viewing a 2-category as a “rich” ordinary category, we can “uplift” it to a simplicial set, via the duskin nerve. I will talk about these and related ideas, indicate some quirks of the theory, and provide examples along the way.
04/01 Liz Splitting BP<2>⋀BP<2> at primes p≥5

In the 1980s, Mahowald and Kane used Brown−Gitler spectra to construct splittings of bo⋀bo and ℓ⋀ℓ. These splittings helped make it feasible to do computations using the bo− and ℓ−based Adams spectral sequences. In this talk, we will construct an analogous splitting for BP<2>⋀BP<2> at primes p≥5.
04/08 Sam Towards a universal property of the ∞-equipment of enriched (∞,1)-categories

One way or another, enriched 1-category theory has held an important spot in the study of homological and homotopical phenomena practically since the very start of ordinary category theory. For many purposes, enriched 1-categories or their model 1-categorical counterparts are simply too rigid, or they might not even exist at all. In recent years various models of enriched (∞,1)-categories have been introduced, and some comparisons at differing levels have been made e.g. the underlying parameterizing ∞-operads or their ∞-categories (with a closed left action over Cat_∞). We are interested in a universal property that can compare these theories at a level which can detect pointwise Kan extensions for example. Part of one approach to this involves the representability of a certain class of double ∞-categorical fibrations. This talk will be heavily focused on examples and justifying why we would want such theories anyway. The only prerequisite is some knowledge of enriched 1-category theory and an appetite for homotopy theory. Time permitting, we may discuss the situation with enriched (∞,1)-operads and (∞,1)-properads, or other possible uses of intermediate results.
04/15 Doron You Already Care About ∞-Topoi

One of the most important roles played by topological spaces is being a base for geometry, i.e. “something to have sheaves on”. As is often the case, however, this classical notion falls short when it comes to describing homotopical geometry. The correct generalization is that of an ∞-topos. In this talk, I will describe the theory of ∞-topoi, how they generalize classical objects from topology and geometry, and several applications. No prior knowledge of 1-topoi or presentable ∞-categories will be assumed.
04/22 Langwen Rational Homotopy Theory

Rational homotopy theory is homotopy theory modulo torsion. This simplification reduces topology to algebra. More precisely, Quillen proved that the rational homotopy theory of 2-connected spaces is equivalent to that of (1) 1-connected dg Lie algebras (2) 2-connected dg cocommutative coalgebras. This is subsequently augmented by Sullivan, who provides a dg commutative algebra model of rational homotopy theory with computational strength. Time permitting, I will also discuss interesting applications to geometry and local algebra.
04/29 CANCELLED
05/06 Johnson Calculus for Algebraic Topologists

Early on in our mathematical studies, we learn that instead of studying a problem directly it is useful to study a linearization of the problem. For example, to understand a smooth map between manifolds we can look at the resulting linear map between the tangent spaces at a point. In this talk we will be looking at a categorification of this idea through the lens of Goodwillie Calculus. In particular, given a map between sufficiently nice infinity categories we will define what it means for such a map to be “linear” and furthermore how one can approximate by such maps. Time permitting, we will explore further generalizations of this idea.