LisMath Seminar 

13/07/2023, 11:00 — 11:30 — Online
Nino Scalbi, Instituto Superior Técnico, Universidade de Lisboa

Skeletal Diffeologies and Differential Cohomology

Differential cohomology combining both homotopy-theoretic and geometric invariants is interpreted to be a refinement of ordinary cohomology. More precisely, a differential cocycle refines an ordinary cocycle just as a bundle with connection refines its underlying bundle. Deligne cohomology and Cheeger-Simons differential characters are the most common models of differential cohomology.

From a different perspective, parallel transport systems have been used successfully to model connections on higher bundles, relying on the concept of thin homotopy. Using the language of diffeological spaces and their homotopy theory, we will examine the relationship between Cheeger-Simons differential characters and the smooth cohomology of skeletal diffeological spaces. Here the notion of thinness is directly built into the skeletal diffeology. We will finish by showing how the skeletal diffeology naturally appears in the recently conjectured geometric cobordism hypothesis by Grady-Pavlov, suggesting a link to geometric field theories.