This course will be about L²-invariants. To give you a brief example, let Γ be a graph (e.g. the infinite binary tree, or the graph A_∞ which forms an infinite line). Let φ be a function on the edges of Γ (e.g., you can think of it as the flow of some liquid along the edges of Γ), so that the sum of values of φ at edges emanating from every vertex is zero (in terms of flows, this is the conservation law insisting that no liquid gets lost where edges are joined together). The questions are: (i) Can you choose such φ to have compact support? What about φ square summable? (It’s NO for both in the case of A_∞ and NO-YES for the tree). (ii) How “many” such φ can you find? The difference between the two answers in (i) and (ii) is, roughly, the L² homology of the graph Γ.

Specifically, we will discuss:

• L² homology and cohomology for discrete groups and manifolds;
• L² Betti numbers for groups; computations, amenable groups, residually finite groups;
• L² Betti numbers and measure equivalence (work of D. Gaboriau);
• L² Betti numbers for tracial algebras and connections with free probability theory;
• deformations of von Neumann algebras and L² cocycles (work of Popa, Peterson, ...) and some applications;
• if time permits, we’ll discuss L² torsion.

What you need to know: you should know what a von Neumann algebra is, or be willing to learn quickly.