Speaker:  Alexander Efimov (Steklov Mathematical Institute of Russian Academy of Sciences and National Research University Higher School of Economics)

Title:  Localizing motives and corepresentability of TR and TC

Abstract:  I will explain some of my recent results on the category of localizing
motives – the target of the universal localizing invariant commuting with
filtered colimits. The main surprising result about this category is that it
is rigid as a symmetric monoidal category (in the sense of Gaitsgory and
Rozenblyum).

As an application of the proof of rigidity, we will deduce that the func-
tors TR (topological restriction) and TC (topological cyclic homology) are
corepresentable in this category, if we restrict to connective E₁ - rings. If time permits, I will explain how rigidity of M ot^l oc allows to con-
struct refined versions of (topological) Hochschild homology and its vari-
ants, which contain much more information about the E₁ − algebra than
the usual variants of (T)HH.