Title: Topological obstructions to stability in operator norm and larger norms

Abstract: A group is said to be stable in the operator norm if every function from the group to unitary matrices that is "almost multiplicative" in the point-operator norm topology is "close" to an honest representation of the group. A result of Dadarlat shows that even rational cohomology obstructs stability in the operator norm for a very broad class of groups. The obstruction in his proof can be realized as follows. To each almost-representation, we may associate a vector bundle. 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, I will explore the extent to which these techniques can be generalized to stability in unnormalized Schatten p-norms. The obstruction coming from 2k cohomology cannot obstruct stability in the p-norm for p ≤ k , and examples show that p ≤ k is sharp.