### Noncommutative numerical motives, Tannakian structures, and motivic Galois groups, by Matilde Marcolli and Goncalo Tabuada

In this article we further the study of noncommutative numerical motives. By
exploring the change-of-coefficients mechanism, we start by improving some of
our main previous results. Then, making use of the notion of Schur-finiteness,
we prove that the category NNum(k)_F of noncommutative numerical motives is
(neutral) super-Tannakian. As in the commutative world, NNum(k)_F is not
Tannakian. In order to solve this problem we promote periodic cyclic homology
to a well-defined symmetric monoidal functor HP on the category of
noncommutative Chow motives. This allows us to introduce the correct
noncommutative analogues C_NC and D_NC of Grothendieck's standard conjectures C
and D. Assuming C_NC, we prove that NNum(k)_F can be made into a Tannakian
category NNum'(k)_F by modifying its symmetry isomorphism constraints. By
further assuming D_NC, we neutralize the Tannakian category NNum'(k)_F using
HP. Via the (super-)Tannakian formalism, we then obtain well-defined
noncommutative motivic (super-)Galois groups. Finally, making use of
Deligne-Milne's theory of Tate triples, we construct explicit homomorphisms
relating these new noncommutative motivic (super-)Galois groups with the
classical ones.

Matilde Marcolli <matilde@caltech.edu>

Goncalo Tabuada <tabuada@math.mit.edu>