In the 1960s, Ulam proposed the following question: Suppose $G$ is a group and $(H,d)$ is a group endowed with a metric. If there a map $\varphi: G \to H$ has the property that $d(\varphi(st) , \varphi(s) \varphi(t)) < \epsilon$ for all $s,t \in G$, then is $\varphi$ ''close to'' an actual homomorphism? In the case which $H$ is $B(\mathcal H)$ for some Hilbert space $\mathcal H$, this question can be reformulated to ask when an approximate representation implies the existence of a nearby actual representation. This talk will introduce some results for particular classical groups and sketch how they might be extended to quantum groups.

This is joint work with M. Brannan.