Sept. 27, 2010: Dualizability and Locality in 3D Topological Field Theory. MIT Algebraic Topology Seminar
In this talk I will report on recent work, joint with Christopher Douglas and Noah Snyder, on understanding the nature of fully extended (a.k.a. local) 3-dimensional topological quantum field theories. Specifically, we show that fusion categories are fully-dualizable objects in the 3-category of tensor categories, a natural categorification of the bicategory of algebras, bimodules, and bimodule maps. Fusion categories themselves are well-known are arise in several areas of mathematics and physics - conformal field theory, operator algebras, representation theory of quantum groups, and others. In light of Hopkins and Lurie's work on the cobordism hypothesis, this provides a fully local TQFT for arbitrary fusion categories. Moreover, we will discuss how many familiar structures from the theory of fusion categories are given a natural explanation from this point of view.
May, 4, 2010: 2-Groups and 2-Group Cohomology. MIT Baby Topology Seminar
2-Groups are a categorification of the notion of group, which appear in many guises in various parts of mathematics. They also provide the garden for a romantic tryst between homotopy theory and higher category theory. In this talk we will discuss 2-groups, what they are and what we can do with them, focusing on aspects relevant to algebraic topology. We hope to explain "2-group cohomology" and how to categorify derived functors.