Colloquium: Trace polynomials in analysis

Abstract: A {\it (pure) trace polynomial} is an expression like p(X,Y)=tr(XYXY)tr(X^2)-3tr(X^2Y), where the $X,Y$ are matrix variables and $tr$ denotes the usual trace. They first appeared in algebra (specifically, invariant theory) in a conjecture of Artin, which was proved by Procesi in 1976. (From this point of view they might be seen as generalizations of the classical symmetric functions.) However in recent years they have also attracted attention in analysis, especially in the emerging area of ``noncommutative'' analysis. I will survey some of these results, both over the real field (related to things like positivity, with applications to quantum theory) and over the complex field (including connections to random matrix theory and Hilbert function spaces).

There will be a tea reception in Room 200 (Lounge), Cupples I at 3:00pm.

Host: John McCarthy