Cohomology in algebra, geometry, physics and statistics

Bornological coarse spaces are "large scale" generalizations of metric spaces (up to quasi-isometry). Homological invariants of such spaces are given by coarse homology theories, which are functors from the category of bornological coarse spaces to a stable cocomplete ∞-category, satisfying additional axioms. Among the main examples of coarse homology theories, there are coarse versions of ordinary homology, of topological
and algebraic K-theory. In the talk we define G-equivariant coarse versions of the classical Hochschild and cyclic homologies (of algebras). If k is a field, the evaluation at the one point space induces equivalences with the classical Hochschild and cyclic homology of k. In the equivariant setting, the G-equivariant coarse Hochschild (cyclic) homology of the discrete group G agrees with the classical Hochschild (cyclic) homology of the associated group algebra k[G].