Categorification is the process of replacing sets by categories, and functions by functors, while trying to keep the rest "as unchanged as possible". The categorification of a vector space (over a field k) is a category (linear over k), and the categorification of a ring is a tensor category. The word "categorification" is very flexible and has been used in a variety of contexts: algebra, representations theory, algebraic geometry, topology...

In my talk, I will add "analysis" to the above list, by explaining how to categorify von Neumann algebras. Thes are certain subalgebras of B(H), the algebra of all bounded operators on a Hilbert space (they're the ones which are closed in the weak operator topology). In this talk, I will tell you which tensor category categorifies B(H), and which sub-tensor categories are the "categorified von Neumann algebras".

Examples of categorified von Neumann algebras include unitary fusion categories, and the category of representations of $\Omega G$, the based loop group of a compact Lie group G.

Later in the talk, I will speculate about the relationship between categorified von Neumann algebras and elliptic cohomology. At last, I will present a conjecture about elliptic curves over number fields which comes out of the above speculations (the conjecture mentions neither elliptic cohomology nor Neumann algebras -- categorified or not).