We discuss relations between superconformal field theory and noncommutative geometry. More specifically, we start with an irreducible, unitary positive energy representation of the Ramond algebra, which is one of the two N=1 super Virasoro algebras, and construct a net of spectral triples on the one-dimensional circle. (Here, a spectral triple is a noncommutaive version of a manifold.) This is a joint work with S. Carpi, R. Hillier and R. Longo.

Blob homology is a construction that takes an n-manifold, and roughly, an n-category with duals, and spits out a graded vector space. It's simultaneously a generalisation of: the skein module for a TQFT, the Hochschild homology of an algebra, and the usual singular homology of a space. We're not quite sure what it's good for yet, but it has such nice formal properties, generalises so much interesting mathematics, and is in principle sufficiently computable, that it's probably going to be fun finding out!

I'll introduce the notion of a "subfactor planar algebra", briefly making contact with its origin in von Neumann algebras, but mostly focusing on the 2-d dimensional combinatorial diagrams that encode everything we care about! The "principal graph" is the first interesting invariant, and a little graph theory gets us started on the classification problem. Next, I'll explain the "annular Temperley-Lieb category": this is important because every subfactor planar algebra gives a representation of this category, and the irreducible representations are easy to describe. Understanding how a planar algebra breaks up into irreducible representations gives us lots of information, including obstructions that rule out some potential principal graphs, as well as clues for constructing new planar algebras. I'll end with an overview of the current classification of small index subfactor planar algebras, and a brief account of a recent construction of the long mysterious "extended Haagerup planar algebra" (joint work with Stephen Bigelow, Emily Peters, and Noah Snyder).

The first examples of von Neumann algebras came from actions of countable groups on probability spaces. As it turns out, their study (up to isomorphism) is closely related to the study of countable equivalence relations (up to orbit equivalence). Recently, this connection proved to be very useful, leading to remarkable rigidity results in both orbit equivalence and von Neumann algebras theory. In this talk, I will survey some of these results including: existence of II_1 factors without symmetries, new cocycle superrigidity results, and existence of non-orbit equivalent actions for arbitrary non-amenable groups.