Some recent advances in topological Hochschild homology

TL;DR

This paper discusses recent developments in topological Hochschild homology, focusing on the motivic filtration and its connections to crystalline and de Rham--Witt theory in equal characteristic p.

Contribution

It provides an account of the construction of the Bhatt--Morrow--Scholze motivic filtration and explores its relations to crystalline and de Rham--Witt theories.

Findings

Construction of the motivic filtration on topological cyclic homology.

Connections established between topological cyclic homology and crystalline theory.

Insights into the case of equal characteristic p.

Abstract

We give an account of the construction of the Bhatt--Morrow--Scholze motivic filtration on topological cyclic homology and related invariants, focusing on the case of equal characteristic $p$ and the connections to crystalline and de Rham--Witt theory.