Let V be a suvariety of a manifold M. We say that V has extension property, if any bounded holomorphic function on V extends to a holomorphic function on M with the same sup-norm. In the talk we shall explain connections between this problem and operator theory (von Neumann inequality, interpolation problem) as well as with the theory of invariant functions and metrics

The essential dimension of an algebraic object E over a field L is heuristically the number of parameters it takes to define it. This notion was formalized and developed by Buhler and Reichstein in the late 90s, who noticed at the time, that several classical results could be interpreted as theorems about essential dimension. Since the paper of Buhler and Reichstein, most of the progress on essential dimension has had to do with essential dimension of versal G-torsors for an algebraic group G. But recently Farb, Kisin and Wolfson showed that interesting theorems can be proved for certain (usually) non-versal torsors arising from congruence covers of Shimura varieties. I'll explain this work, some extensions of it proved by me and Fakhruddin, and a conjecture on period maps which generalizes the picture.