Fall 2020

• • Dec 012020

  Arpon Raksit (Stanford)

  q–de Rham cohomology and complex K-theory

  Work of Bhatt–Morrow–Scholze uncovered a "q-deformation" of de Rham cohomology in p-adic 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 K-theory 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, rank-preserving 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 Dyer-Lashof 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 E₂-algebras.

• • Nov 102020

  Hana Jia Kong (Uchicago)

  The motivic Chow t-structure

  In this talk, I will introduce the Chow t-structure on the motivic stable homotopy category over a general base field. This t-structure is a generalization of the Chow-Novikov t-structure defined on a p-completed cellular motivic module category in work of Gheorghe-Wang-Xu. Moreover, we identify the heart of this t-structure with a purely algebraic category, and expand the results of Gheorghe-Wang-Xu 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 v₁ selfmap

  We consider a nontrivial action of C₂ on the type 1 spectrum Y:=S/2 S/η. This can also be viewed as the complex points of a finite real-motivic spectrum. One of the v₁−self-maps of Y can be lifted to a C₂ equivariant self-map as well as a real-motivic self-map. Further, the cofiber of the self-map of the R-motivic lift of Y is a realization of the real-motivic 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 endo-trivial modules. In homotopy theory, this group is known as the Picard group of the stable module category. This group was originally computed by Carlson-Thévenaz 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 Mathew-Stojanoska. Notably, these computations provide conceptual insight into the classical work of Carlson-Thévenaz.

• • Oct 062020

  Andy Smith (UCLA)

  The structure of incomplete equivariant stable homotopy categories

  G-spectra indexed on N-infinity operads, introduced by Blumberg and Hill, refine the theory of G-universes 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 G-spectra, 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 E-infinity algebras over a Lubin-Tate 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 E-infinity complex orientations.