Mathematics Graduate Students Organization

Seminar:

Date: September 9

Group von Neumann algebras have been a key class of examples since Francis Murray and John von Neumann’s original papers dating back to the 1930s. However, little progress was made for the next several decades regarding the Free Group Factor Isomorphism Problem: is the group von Neumann algebra for the free group of N generators *-isomorphic to that of M generators if N does not equal M? In the 1980s, in the hopes of better understanding these vNAs, Dan Virgil Voiculescu invented free probability theory, which essentially treats elements of a vector algebra as ``random variables’’ and emphasizes their ``moments’’ with respect to some positive linear functional (``expected value’’). In this talk, we’ll discuss the connections between free groups and the notion of free independence, as well as the compression techniques of Voiculescu, Florin Radulescu, and Ken Dykema which led to the closest we’ve come to an answer: either the free group factors are all isomorphic, or no two are isomorphic. (No prior experience with von Neumann algebras will be necessary, as I will begin with a light introduction to the subject.)