Joint IAS/PU Arithmetic Geometry

Pretannakian categories, that is k-linear abelian categories with finite dimensional Hom groups, given with a commutative and associative tensor product, with a unit object (such that End(1)=k) and duals, can be viewed as generalizations of linear algebraic groups over k : to G corresponds the category Rep(G) of representations of G, and for k algebraically closed, G can be recovered (up to inner automorphisms) from the category. In characteristic 0, for a long time, the only known pretannakian categories were obtained by "interpolation" from categories Rep(G). This changed a few years ago, with a new point of view leading to unexpected beautiful such categories.