## Midwest Topology Seminar

**The 2018 Midwest Topology Seminar was hosted by Northwestern University at Swift Hall on March 3rd**.

This seminar has run continuously since the middle of the 1960s, meeting Fall, Winter and Spring of most years. Originating in the desire of topologists in the Chicago area to meet to discuss recent developments, it grew into a meeting of algebraic and geometric topologists from the entire upper Midwest and Great Lakes regions of the United States and Canada. It typically features speakers drawn from both the local and international communities and speakers at all stages of their careers.

**2018 Midwest Topology Seminar ****Scheduled Speakers and Abstracts:**

**Akhil Mathew: Rigidity in Algebraic K-theory and Topological Cyclic Homology**

We show that for a henselian pair (R, I), the relative K-theory and relative topological cyclic homology with mod p coefficients agree. This yields a generalization of the Gabber-Gillet-Suslin-Thomason rigidity theorem (when p is invertible on the ring) and the Dundas-McCarthy theorem (when I is nilpotent). This recovers several computations in p-adic algebraic K-theory and leads to some new structural results. Our methods are based on the new description of the homotopy theory of cyclotomic spectra given by Nikolaus and Scholze. This is joint work with Dustin Clausen and Matthew Morrow.

**Mona Merling**:

**Toward the Equivariant Stable Parametrized H-cobordism Theorem**Waldhausen's introduction of A-theory of spaces revolutionized the early study of pseudo-isotopy theory. Waldhausen proved that the A-theory of a manifold splits as its suspension spectrum and a factor Wh(M) whose first delooping is the space of stable h-cobordisms, and its second delooping is the space of stable pseudo-isotopies. I will describe a joint project with C. Malkiewich aimed at telling the equivariant story if one starts with a manifold M with group action by a finite group G.

**Bhargav Bhatt:**

*Topological Hochschild homology and p-adic cohomology*I will give an overview of my joint work with Matthew Morrow and Peter Scholze giving precise relations between topological Hochschild homology and integral *p*-adic cohomology theories arising in arithmetic geometry (such as crystalline cohomology or the recently constructed *A*_{inf}-cohomology). Instead of giving the most general possible results, I will try to emphasize certain features of *p*-adic cohomology theories that are most easily seen through the topological perspective**Nat Stapleton: The character of the total power operation**

In the 90's Goerss, Hopkins, and Miller proved that the Morava E-theories are E_{∞}-ring spectra in a unique way. Since then several people including Ando, Hopkins, Strickland, and Rezk have worked on explaining the effect of this structure on the homotopy groups of the spectrum. In this talk, I will present joint work with Barthel that shows how a form of character theory due to Hopkins, Kuhn, and Ravenel can be used to reduce this problem to a combination of combinatorics and the GL_{n}(Q_{p})-action on the Drinfeld ring of full level structures which shows up in the local Langlands correspondence.

Previous MTS Seminars Hosted by Northwestern University