Elias Klakken Angelsen (UiO): Differential cohomology (and Hodge-filtered K-theory)

Abstract: Topological invariants are vital to algebraic topologists. They are also useful if we restrict ourselves to the study of smooth manifolds, but in that context, they seem to not care about geometric information such as curvature, as they allow for deformations changing the geometry. On the other hand, we know of many ways to construct such topological invariants from geometric data. Both Chern-Weil theory and de Rham cohomology are good examples of such constructions. We may therefore ask if and how we can "refine" topological invariants such as cohomology theories so that they see more of the geometry. This question leads us directly into the field of differential cohomology, and the first part of this talk will be on the main motivations, definitions, results and examples in the field. In the second part, we also ask what happens when we work with complex manifolds instead of smooth manifolds. This gives us Hodge-filtered cohomology theories as defined by Hopkins-Quick, and if time permits, we hint towards explicit constructions and open problems regarding Hodge-filtered K-theory.