Matthew Kennedy, Carleton University
“Operator algebraic geometry”
In this talk I will give an overview of an area of mathematics that combines the theory of operator algebras with ideas and methods from commutative algebra and classical algebraic geometry. The basic idea is to study operators (or matrices) satisfying a system of polynomial equations. For example, given the polynomial x3 + y3 − z3, we consider triples (X, Y, Z) of commuting operators satisfying X3 + Y 3 − Z3 = 0. The self-adjoint algebra generated by these operators, which is typically highly noncommutative, can be seen as a non-classical counterpart of the (classical) variety determined by the polynomials. The big problem in this area is to determine how the structure of this algebra relates to the geometric structure of the variety.
At the centre of much of this work is a conjecture of Arveson and Douglas, made over a decade ago, about the behaviour of certain “almost” commuting families of operators. What makes this conjecture particularly interesting is that it can be reformulated as a quantitative statement in algebraic geometry. To date, however, it has been proved only in certain special cases.
In this talk, I will give high level introduction to this area of research, and explain some of the interest in these problems.
Refreshments will be served in MC 5046 at 3:30 p.m. Everyone is welcome to attend.