On a geometric realization of the center of the small quantum group

In Lie theory, the center of many important representation categories admits geometrical interpretations as singular cohomology of certain algebraic varieties. We will explain a new realization of this type, relating the center of the small quantum group to the cohomology of certain affine Springer fibres. As an application, we obtain a conjectural formula for the dimension of the center of the small quantum group. This is based on a joint work with R. Bezrukavnikov, P. Boixeda-Alvarez, and E. Vasserot.