Historically, bundles in geometry arose from several sources: covering spaces, tangent bundles and homogeneous bundles. Only with this rich supply of examples could the theory of bundles (vector bundles, principal bundles and fibre bundles more generally) arise. It was a marvellous convergence when it was discovered that physical theories such as electromagnetism and electroweak theory, and even general relativity, were also described by the data of connections on such geometric structures. Contemporary physical(-ish) field theories that arise from, say, string theory require a much richer setting, and the geometry behind this setting has become known as "higher geometry". Not surprisingly the geometry has its roots dating back to none other than the differential geometer and category theorist Charles Ehresmann. It is the marriage of geometry and (higher) category theory as exemplified in Lie groupoids that provides this richer setting that physics requires. However, the kind of examples that drove the theory of bundles in the first half of the 20th century have been lacking until recently. By going back to those old sources it is in hindsight obvious how to generate many examples in higher geometry, and this talk will give a flavour of how one does this. No prior knowledge of Lie groupoids or higher category theory will be necessary.
Normal supercharacter theory is a mechanism to substitute conjugacy classes by normal subgroups in such a way that this coarsening simulates some useful features of irreducible characters. We construct a normal supercharacter theory for the group of unipotent upper-triangular matrices. The supercharacters in this supercharacter theory are indexed by Dyck paths. We show that this construction is identical with Scott Andrews' construction after gluing the superclasses and supercharacters by the action of torus group. In the end, we build up some Hopf structures base on these supercharacters.
Normal supercharacter theory, Dyck paths, and Hopf Structures Sponsored by the Meyer Fund
Low-dimensional higher geometry by examples