This semester we're organizing a reading seminar on various topics pertaining to appearances of higher categories in representation theory, to some recent work on the positive Grassmannian, and to extension problems in algebraic geometry.
The first three talks will be about tensor categories and their Morita theory - centers, module and bi-module equivalences, classification, using ideas from higher categories and homotopy theory.
Some relevant sources are:
1) Lectures on tensor categories and modular functor
2) On fusion categories
3) Module categories, weak Hopf algebras and modular invariants
4) Fusion categories and homotopy theory
5) Morita equivalence for convolution categories
6) Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry
The next three talks will be about the positive Grassmannian. Some relevant sources are:
1) Total positivity, Grassmannians, and networks
2) Enumeration of totally positive Grassmann cells
3) Matching polytopes, toric geometry, and the non-negative part of the Grassmannian
4) Scattering amplitudes and the positive Grassmannian
5) A Jewel at the heart of quantum physics
6) A talk by Arkani-Hamed on the "Amplituhedron", relative of positive Grassmannian
The next two talks will be about Khovanov and Lauda's graphical approach to categorifying quantum groups.
Finally, we will have a lecture about the classical problem in algebraic geometry of extending sheaves over closed subvarieties.
Here's the calendar of speakers we agreed to at the organizational meeting. I'll add titles to talks as they become known.