L6

This talk will provide an overview of the landscape of bicommutant categories, these are tensor categories with a strong functional-analytic flavour. I will discuss the evolution of the definition (and give the current version of the definition) and explain precisely how they categorify von Neumann algebras, in the same way a tensor category can be viewed as a categorification of an algebra. We will also introduce the string-calculus that renders the coherences in the definition transparent and workable. 

The necessary background from functional analysis (in particular, operator theory) will be reviewed, and I will conclude with open questions (if waiting for the end of talk is not your style, there are 75 Open problems on André’s website). 

A 3-manifold is a space which locally looks like R^3. A major theme in 3-manifold Topology is to understand and classify 3-manifolds. Given two compact 3-manifolds M_1,M_2 we can form another 3-manifold by taking what’s called the “connect sum” of M_1 and M_2. Under this operation, 3-manifolds can be decomposed uniquely into prime pieces just like the integers can be decomposed uniquely as a product of primes. We will discuss this prime decomposition theorem for 3-manifolds while also giving a wide variety of examples.