Noncommutative motives and their applications

This survey is based on lectures given by the authors during the program “Noncommutative algebraic geometry and representation theory” at the MSRI in the Spring 2013. It covers the recent work [44, 45, 46, 47, 48, 49, 50] on noncommutative motives and their applications, and is intended for a broad mathematical audience. In Section 1 we recall the main features of Grothendieck’s theory of motives. In Sections 2 and 3 we introduce several categories of noncommutative motives and describe their relation with the classical commutative counterparts. In Section 4 we formulate the noncommutative analogues of Grothendieck’s standard conjectures of type C and D, of Voevodsky’s smash-nilpotence conjecture, and of Kimura-O’Sullivan finite-dimensionality conjecture. Section 5 is devoted to recollections of the (super-)Tannakian formalism. In Section 6 we introduce the noncommutative motivic Galois (super-)groups and their unconditional versions. In Section 7 we explain how the classical theory of (intermediate) Jacobians can be extended to the noncommutative world. Finally, in Section 8 we present some applications to motivic decompositions and to Dubrovin’s conjecture