Recent developments on noncommutative motives (1611.05439v4)

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.