Algebraic K-theory and trace invariants

TL;DR

This paper discusses the cyclotomic trace map from algebraic K-theory to topological cyclic homology, highlighting recent advances in understanding its properties and implications for algebraic K-theory.

Contribution

It reviews recent progress in understanding the cyclotomic trace and its relation to algebraic K-theory and cyclic homology, suggesting directions for future research.

Findings

Enhanced understanding of the cyclotomic trace map

Connections between K-theory and topological cyclic homology clarified

Potential for future developments in algebraic K-theory

Abstract

The cyclotomic trace of B\"okstedt-Hsiang-Madsen, the subject of B\"okstedt's lecture at the congress in Kyoto, is a map of pro-abelian groups K_*(A) -> TR_*^.(A;p) from Quillen's algebraic K-theory to a topological refinement of Connes' cyclic homology. Over the last decade, our understanding of the target and its relation to K-theory has been significantly advanced. This and possible future development is the topic of my lecture.