Topology seminar

 Hill-Hopkins-Ravanel norms give a lift of tensor-induction from the category of $H$-spectra to $G$-spectra for $H \subseteq G$ a subgroup of a finite group, and they were crucial to their resolution of the Kervaire Invariant one problem in all but one dimension; analogously, the influential Angelveit-Blumberg-Gerhardt-Hill-Lawson-Mandell perspective on (twisted) topological Hochschild homology expresses it as a norm from finite subgroups of the circle group. Following this, Blumberg-Hill-Mandell issued a series of conjectures concerning \emph{equivariant factorization homology} as a technique for constructing norms along closed subgroup inclusions between compact Lie groups (extending the above examples) and computing their geometric fixed points. Following work-in-progress with Juran, I will sketch a resolution of these conjectures.

VH 570

Green functors are an equivariant generalization of rings which underlie multiplicative equivariant cohomology theories. In this talk, I will introduce the notion of Green functors and discuss some recent work on understanding the categories of modules over Green functors through the lens of algebraic K-theory. I will also talk about some tools for computing algebraic K-groups of Green functors. This is joint work with Noah Wisdom.

VH 570

Transfer systems are combinatorial objects that encode information about equivariant operations. More precisely, a transfer system encodes the transfers (or wrong-way maps) carried by algebras over certain equivariant operads. Thus, transfer systems allow us to use combinatorial tools to study equivariant homotopy theory. This talk will be an overview of various properties, including some structural and combinatorial results. We will concentrate on multiplicative structures, using the idea of operad pairs of May. The results presented are part of various collaborations, including Scott Balchin, David Chan, Myungsin Cho, Evan Franchere, Usman Hafeez, Peter Marcus, Kristen Mazur, David Mehrle, Pablo S. Ocal, Kyle Ormbsy, Weihang Qin, Constanze Roitzheim, Rekha Santhanam, Ben Szczesny, Danika Van Niel, Paula Verdugo, Riley Waugh, and Valentina Zapata Castro.

VH 570

Topological Hochschild homology is a computable invariant for rings that approximates algebraic K-theory. If the rings have more symmetric structure, then both topological Hochschild homology and algebraic K-theory can be refined to capture the symmetry. For instance, given a ring with anti-involution, the notion of Real topological Hochschild homology was introduced, and approximates Real algebraic K-theory, whose induced symmetry captures Hermitian K-theory, L-theory and surgery of manifolds.

In this talk, I will present joint work with Angelini-Knoll and Merling, where we provide a unifying framework for variants of topological Hochschild homologies, and introduce new useful variants such as quaternionic topological Hochschild homology and topological symmetric homology. Each variant is based of a crossed simplicial group, generalizing Connes’ cyclic category, and introduced in the 90s by Fiedorowicz and Loday. They associated homologies for algebras over a field to each such crossed simplicial groups and our theories are the topological analogues.

VH 570

We'll review the infinity categorical perspective on Thom spectra, and proceed to talk about how this generalizes to the equivariant world. Along the way, we'll cover the appropriate toolbox required to deal with equivariant structures in the infinity categorical context.

VH570

 Double categories are a flexible 2-dimensional setting that allows us to encode two types of morphisms between objects, as well as a notion of higher cells. Surprisingly, unlike most categorical structures, there is no canonical notion of “equivalence of double categories”, as it seems that every possible definition requires us to make a choice. In this talk we will illustrate the issue that arises when defining double-categorical equivalences. Then, we will show how we can use homotopy theory to give a decisive answer as to who the “canonical double categorical equivalences” should be: the gregarious equivalences introduced by Campbell. In the process, we will show how to construct a plethora of model structures on double categories whose homotopy theories encode different 2-dimensional structures. Based on joint work with Lyne Moser and Paula Verdugo.

VH 570