Veranstaltungen am Mathematischen Institut

Abstract: In the higher-categorical setting, Galois theory extends beyond finite groups: one can study Galois extensions for an arbitrary E_1-group. We develop such a theory for higher semiadditive categories, where finite groups are naturally replaced by pi-finite groups. Under suitable height assumptions, we construct an n-truncated pro-pi-finite "absolute Galois group" classifying all Galois extensions. We also relate abelian extensions to the Picard spectrum through a higher version of Kummer theory. Time permitting, we will compute the absolute Galois group in some chromatic and categorified settings.

Given an action of a countable group on the Cantor space, clopen subsets can be compared by cutting and moving through the action or by evaluating them on invariant measures. Dynamical comparison asserts that the measure-theoretic notion controls the dynamical one. Since its first appearance, dynamical comparison has been verified for a wide variety of actions. In this talk, I will survey some recent positive results in this direction and present the construction of the first examples of topologically free minimal actions that fail dynamical comparison, obtained in a joint work with Akshara Prasad.