Abstract: The idea of making spacetime into a noncommutative space goes back to the late 60's, and deformation quantization was first introduced in 1978 by Bayen, Flato, Fronsdal, Lichnerowicz, and Sternheimer. In my talk, I will define a star product on a spacetime $M$, which is a noncommutative deformation of $C^\infty(M)$ and (essentially uniquely) quantize the star product. My goal is to use this to construct a noncommutative field theory on the principal $G$-bundle $P$, considered as a finitely generated projective module in the quantized star product over the star product algebra $C^\infty(M)$.

Abstract: Classical Tannaka duality is a duality between groups and their categories of representations. The two basic questions it answers are the reconstruction problem (when can a group be reconstructed from its category of representations?) and the recognition problem (can we characterize categories of representations abstractly?).

I will outline how the notion of a Tannakian category can be weakend in order to solve the recognition problem for categories of coherent sheaves of algebraic stacks (if you are an algebraic geometer), respectively categories of comodules of Hopf algebroids (if you are an algebraic topologist). I will end with an application that illustrates one of the differences between these two perspectives.