Functional Analysis Seminar

A group is said to be matricially stable if every function from the group to unitary matrices that is "almost multiplicative" in the point-operator norm topology is "close," in the same topology, to a genuine representation. A result of Dadarlat shows that even cohomology obstructs matricial stability. The obstruction in his proof can be realized as follows. To each almost-representation,  a vector bundle may be associated. This vector bundle has topological invariants, called Chern characters, which lie in the even cohomology of the group. If any of these invariants are nonzero, the almost-representation is far from a genuine representation. The first Chern character can be computed with the "winding number argument" of Kazhdan, Exel, and Loring, but the other invariants are harder to compute explicitly. In this talk, Professor Forrest Glebe will discuss results that allow the computation of higher invariants in specific cases: when the failure to be multiplicative is scalar (joint work with Marius Dadarlat) and when the failure to be multiplicative is small in a Schatten p-norm.

An epsilon-representation of a discrete group G is a map from G to the unitary group U(n) that is epsilon-multiplicative in norm uniformly across the group. In the 1980s, Kazhdan showed that surface groups of genus at least 2 are not uniform-to-local stable in the sense that they admit epsilon-representations that cannot be perturbed, even locally (on the generators), to genuine representations.
 

In this talk, Marius Dadarlat of Purdue University will discuss the role of bounded 2-cohomology in Kazhdan's construction and explain why many rank-one lattices in semisimple Lie groups are not uniform-to-local stable, using certain K-theory properties reminiscent of bounded cohomology.

to follow