Recent developments on noncommutative motives
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 semisimplicity conjecture; prove a farreaching 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 Ktheory; construct a noncommutative motivic Gysin triangle; compute the localizing A1homotopy invariants of corner skew Laurent polynomial algebras and of noncommutative projective schemes; relate Kontsevich's category of noncommutative mixed motives to MorelVoevodsky's stable A1homotopy category, to Voevodsky's triangulated category of mixed motives, and to Levine's triangulated category of mixed motives; prove the Schurfiniteness conjecture for quadric fibrations over lowdimensional bases; and finally extend Grothendieck's theory of periods to the setting of dg categories.
 Publication:

arXiv eprints
 Pub Date:
 November 2016
 arXiv:
 arXiv:1611.05439
 Bibcode:
 2016arXiv161105439T
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  KTheory and Homology;
 Mathematics  Representation Theory
 EPrint:
 Survey article