MIT Combinatorics Seminar

Haagerup's Inequality and Free Cumulants

Todd Kemp
(Massachusetts Institute of Technology)

Wednesday November 1, 2006   4:15 pm    Room 2-136


In 1978, Uffe Haagerup proved an important inequality relating two functional analytic norms (the $\ell^2$-norm and the convolution norm) on the group algebra of a free group. While invented to be used in operator algebras, the inequality also found important uses in geometric group theory, probability theory, and other fields.

From a combinatorial standpoint, one can analyze the above-mentioned norms in terms of {\em free cumulants} associated to elements in the free group. Free cumulants are multilinear functions defined using M\"obius inversion on the lattice of non-crossing partitions $NC$.

In this talk, I will discuss recent work on a surprisingly strong improvement of Haagerup's inequality, which extends far beyond the context of the free group. The results rely fundamentally on the structure of $NC$.

A working knowledge of partitions and basic combinatorial constructions such as M\"obius inversion will be an asset, but {\em no knowledge of functional analysis} will be assumed.