## Math 259B, L^{2}-invariants: groups, equivalence
realtions, von Neumann algebras.

Instructor: D. Shlyakhtenko

MWF 1–2pm in MS6229

This course will be about L^{2}-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 L^{2} homology of the graph
Γ.

Specifically, we will discuss:

- L
^{2} homology and cohomology for discrete groups and manifolds;
- L
^{2} Betti numbers for groups; computations, amenable groups, residually finite groups;
- L
^{2} Betti numbers and measure equivalence (work of D. Gaboriau);
- L
^{2} Betti numbers for tracial algebras and connections with free probability theory;
- deformations of von Neumann algebras and L
^{2} cocycles (work of Popa, Peterson, ...)
and some applications;
- if time permits, we’ll discuss L
^{2} torsion.

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

Office hours (I am either in MS6356 or MS7901):
by appointment