Talk

Seminar webpage:  https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html

Webpage:  https://people.mpim-bonn.mpg.de/scholze/veranstaltungen.html

A crucial problem in geometric quantization is to understand the relationship among quantum spaces associated to different polarizations. Two types of polarizations on toric varieties, Kähler and real, have been studied extensively. In this talk, I will introduce the quantum spaces associated to mixed toric polarizations and explore their relationships with those associated with Kähler polarizations.

Seminar webpage:  https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html

The cobordism hypothesis classifies extended topological quantum
field theories (TQFTs) in terms of algebraic information in the target
category. One of the core principles in quantum field theory - unitarity -
says that state spaces are not just vector spaces, but Hilbert spaces.
Recently in joint work with many others, we have defined unitarity for
extended TQFTs, inspired by Freed and Hopkins. Our main technical tool is a
higher-categorical version of dagger categories; categories $C$ equipped
with a strict anti-involution $\dagger: C \to C^{op}$ which is the identity
on objects. I explain joint work in progress with Theo Johnson-Freyd,
Cameron Krulewski and Lukas Müller in which we prove a version of the
cobordism hypothesis for unitary TQFTs. The main observation is that the
\emph{stably} framed bordism n-category is freely generated as a symmetric
monoidal dagger n-category with unitary duals by a single object: the point.

Given the data of a variety, an algebra, or more generally a triangulated category, Bridgeland stability produces a complex manifold (the space of stability conditions). What does the geometry of this manifold tell us about the starting data? In this talk we'll investigate this question by looking for so-called "geometric" stability conditions on surfaces that arise as free quotients by finite groups.