Topology Seminar
Past Talks
Winter 2021

 Jan 052021
Chris Lloyd (UVA)
Calculating the nth Morava Ktheory of the real Grassmannians using C4equivariance
In this talk we will demonstrate how letting the cyclic group of order four act on the real Grassmannians can show the AtiyahHirzebruch spectral sequence calculating their 2local nth Morava Ktheory collapses. This uses chromatic fixed point theory coming from the classification of the equivariant Balmer spectrum of the cyclic groups. This work is joint with Nicholas Kuhn.
Fall 2020

 Dec 012020
Arpon Raksit (Stanford)
q–de Rham cohomology and complex Ktheory
Work of Bhatt–Morrow–Scholze uncovered a "qdeformation" of de Rham cohomology in padic algebraic geometry, and moreover a striking relation of this invariant to stable homotopy theory via topological Hochschild homology, i.e. Hochschild homology relative to the sphere spectrum. I will discuss ongoing work concerning a related phenomenon involving Hochschild homology relative to the complex Ktheory spectrum.

 Nov 242020
Morgan Opie (Harvard)
Vector bundles on projective spaces
Given the ubiquity of vector bundles, it is perhaps surprising that there are so many open questions about them  even on projective spaces. In this talk, I will outline some results about vector bundles on projective spaces, including my ongoing work on complex rank 3 topological vector bundles on CP^5. In particular, I will describe a classification of such bundles which involves a surprising connection to topological modular forms; a concrete, rankpreserving additive structure which allows for the construction of new rank 3 bundles on CP^5 from simple" ones; and future directions related to this project."

 Nov 172020
Calista Bernard (Stanford)
Twisted homology operations
In the 70s, Fred Cohen and Peter May gave a description of the mod p homology of a free E_{n}algebra in terms of certain homology operations, known as DyerLashof operations, and the Browder bracket. These operations capture the failure of the E_{n} multiplication to be strictly commutative, and they prove useful for computations. After reviewing the main ideas from May and Cohen's work, I will discuss a framework to generalize these operations to homology with certain twisted coefficient systems and give a complete classification of twisted operations for E_{∞}algebras. I will also explain computational results that show the existence of new operations for E_{2}algebras.

 Nov 102020
Hana Jia Kong (Uchicago)
The motivic Chow tstructure
In this talk, I will introduce the Chow tstructure on the motivic stable homotopy category over a general base field. This tstructure is a generalization of the ChowNovikov tstructure defined on a pcompleted cellular motivic module category in work of GheorgheWangXu. Moreover, we identify the heart of this tstructure with a purely algebraic category, and expand the results of GheorgheWangXu to integral results on the entire motivic category over general base fields. This leads to computational applications on determining the Adams spectral sequences in the classical stable homotopy category, as well as that in the motivic stable homotopy category over C, R, and F_p. This is joint work with Tom Bachmann, Guozhen Wang and Zhouli Xu.

 Nov 032020
Peter Haine (MIT)
Stratified étale homotopy theory
Étale homotopy theory was invented by Artin and Mazur in the 1960s as a way to associate to a scheme X, a homotopy type with fundamental group the étale fundamental group of X and whose cohomology captures the étale cohomology of X with locally constant constructible coefficients. In this talk we’ll explain how to construct a stratified refinement of the étale homotopy type that classifies constructible étale sheaves of spaces. We’ll also explain how this refinement gives rise to a new, concrete definition of the étale homotopy type. This is joint work with Clark Barwick and Saul Glasman.

 Oct 272020
Ang Li (U. Kentucky)
A real motivic v_{1} selfmap
We consider a nontrivial action of C_{2} on the type 1 spectrum Y:=S/2 S/η. This can also be viewed as the complex points of a finite realmotivic spectrum. One of the v_{1}−selfmaps of Y can be lifted to a C_{2} equivariant selfmap as well as a realmotivic selfmap. Further, the cofiber of the selfmap of the Rmotivic lift of Y is a realization of the realmotivic Steenrod subalgebra A(1). This is joint work with Prasit Bhattacharya and Bertrand Guillou.

 Oct 202020
Weinan Lin (Uchicago)
On the May Spectral Sequence at the Prime 2
The May spectral sequence is one of the first effective methods to compute the cohomology of the Steenrod algebra. In addition to a conjecture of May about all the indecomposables of the E2 page of the May spectral sequence, we are going to state a conjecture about all the relations in the E2 page. We also conjecture that this E2 page is nilpotent free. We will show that these conjectures are all true in a big subalgebra of E2 which covers a large range of dimensions.

 Oct 132020
Richard Wong (UT Austin)
The Picard Group of the Stable Module Category for Quaternion Groups
One problem of interest in modular representation theory of finite groups is in computing the group of endotrivial modules. In homotopy theory, this group is known as the Picard group of the stable module category. This group was originally computed by CarlsonThévenaz using the theory of support varieties. However, I provide new, homotopical proofs of their results for the quaternion group of order 8, and for generalized quaternion groups, using the descent ideas and techniques of Mathew and MathewStojanoska. Notably, these computations provide conceptual insight into the classical work of CarlsonThévenaz.

 Oct 062020
Andy Smith (UCLA)
The structure of incomplete equivariant stable homotopy categories
Gspectra indexed on Ninfinity operads, introduced by Blumberg and Hill, refine the theory of Guniverses to give a finer interpolation between "naive’’ and "genuine’’ equivariant stable homotopy theory. I will explore some of the structural properties of these categories of Gspectra, in particular their description as incomplete Mackey functors and the appropriate version of the Segal conjecture for this setting. Then I will use this structure to describe the categories’ Picard groups.

 Sep 292020
William Balderrama (UIUC)
Approximating higher algebra by derived algebra
A general heuristic in homotopy theory tells us that by understanding the operations which act naturally on the homotopy groups of a class of objects, one can build obstruction theories and so forth for working with these objects. For instance, in the setting of highly structured ring spectra, this heuristic leads one to obstruction theories built on top of power operations. In this talk I'll describe a general framework that makes it easy to set up these kinds of obstruction theories, with a focus on the particular example of K(n)local Einfinity algebras over a LubinTate spectrum. I'll explain how the picture one obtains is very pleasant at heights 1 and 2, and in particular can be applied to produce new Einfinity complex orientations.