**Title**: Results and Problems on Perturbations of $\textrm{II}_1$ Factors

**Speaker**: Alan Wiggins (University of Michigan-Dearborn)

**Abstract**: Let $A$ and $B$ be C*-algebras represented on the same Hilbert $H$ space and let $$d(A,B) = \max \{ \sup_{x \in A} \inf_{y \in B} \|x-y\|, \sup_{x \in B} \inf_{y \in A} \|x-y\| \}.$$

Kadison and Kastler asked whether there is a universal constant $c>0$ such that if $d(A,B) < c$, then $A$ is *-isomorphic to $B$. We will discuss fairly recent progress on this problem in the case where $A$ (and hence, $B$) is a $\textrm{II}_1$ factor. We will describe a technique that allows one to represent both $A$ and $B$ in standard form while still maintaining control over $d(A,B)$ and the resulting consequences when $A$ is a crossed product of a maximal abelian *-subalgebra by a discrete group $G$. We will conclude with a discussion on invariants for finite index subfactors that is work in progress with Dave Penneys and Stuart White.