Noncommutative numerical motives and the Tannakian formalism

I will describe recent work with Goncalo Tabuada, where we consider analogs
of Grothendieck's standard conjectures C and D for a suitable category of
noncommutative numerical motives and we show that, assuming these conjectures,
one can make this category into a Tannakian category. The motivic Galois group
of this category surjects onto the kernel of the homomorphism from the motivic Galois
group of the category of (commutative) numerical motives to the multiplicative group,
determined by the inclusion of the subcategory of Tate motives.