# Equivariant Stable Homotopy Theory and p-adic Hodge Theory (20w5217)

## Organizers

(University of Texas, Austin)

(Michigan State University)

(University of California, Los Angeles)

## Description

The Banff International Research Station will host the "Equivariant Stable Homotopy Theory and p-adic Hodge Theory" workshop in Banff from March 1 to March 06, 2020.

Algebraic topology has had a long and fruitful collaboration with algebraic geometry, with each providing techniques and problems to the other. This workshop is aimed at an exciting, evolving incarnation of this story: applications of equivariant stable homotopy to number theory. Recent work on the foundations of equivariant stable homotopy theory (starting with the Hill--Hopkins--Ravenel work on the Kervaire invariant one problem) and Lurie's development of the foundations of derived algebraic geometry'' now allows systematic exploration and organization of equivariant derived algebraic geometry''. This allows us to do ordinary algebraic geometry in commutative ring spectra.

New foundations in this area have been spectacularly applied to phenomena seen in the trace methods approach to computing algebraic $K$-theory.
For instance, although the theory of equivariant commutative ring spectra was described decades ago, few of the subtleties in the theory were understood or explored.
The modern approaches to computing algebraic $K$-groups step through equivariant commutative ring spectra via the natural $S^1$-action on topological Hochschild homology. Ongoing and transformative work by Bhatt--Morrow-Scholze in $p$-adic Hodge theory uses cyclotomic spectra and therefore subtle equivariant information. This workshop, at the vanguard of work in this area, seeks to bring together experts in algebraic topology, (derived) algebraic geometry, and number theory to explore these exciting new connections.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).