Recent developments on noncommutative motives

TL;DR

This survey reviews recent advances in noncommutative motives, including proofs of major conjectures, computations of invariants, and connections to classical motivic theories, highlighting significant progress in the field.

Contribution

It provides new proofs of key conjectures, computes invariants for complex structures, and establishes links between noncommutative and classical motivic categories.

Findings

Proved Kontsevich's semi-simplicity conjecture.

Established noncommutative analogs of the Weil conjectures.

Connected noncommutative motives with classical motivic categories.

Abstract

This survey covers some of the recent developments on noncommutative motives and their applications. Among other topics, we compute the additive invariants of relative cellular spaces and orbifolds; prove Kontsevich's semi-simplicity conjecture; prove a far-reaching noncommutative generalization of the Weil conjectures; prove Grothendieck's standard conjectures of type C+ and D, Voevodsky's nilpotence conjecture, and Tate's conjecture, in several new cases; embed the (cohomological) Brauer group into secondary K-theory; construct a noncommutative motivic Gysin triangle; compute the localizing A1-homotopy invariants of corner skew Laurent polynomial algebras and of noncommutative projective schemes; relate Kontsevich's category of noncommutative mixed motives to Morel-Voevodsky's stable A1-homotopy category, to Voevodsky's triangulated category of mixed motives, and to Levine's triangulated category of mixed motives; prove the Schur-finiteness conjecture for quadric fibrations over low-dimensional bases; and finally extend Grothendieck's theory of periods to the setting of dg categories.