Special Colloquium
Speaker: James Eldred Pascoe (Drexel University)
Host: Sourav Pal
Title: Matrix convex functions
Time, day and date: 2:30:00 PM - 3:30:00 PM, Thursday, August 28
Venue: Ramanujan Hall
Abstract: Given a function on some subset of the real numbers, there is a natural way to evaluate it on self-adjoint matrices with eigenvalues in the domain known as the functional calculus. For a simple function like a polynomial, power series or a rational function, such agrees with substituting the matrix into the expression and evaluating. Kraus analyzed the so-called matrix convex functions which are the functions such that $f_v(X) = \langle f(X)v, v\rangle$ is convex as a function of matrix inputs $X.$ For classical convex functions, we have the simple condition that some weak version of the second derivative must be nonnegative, but for matrix convex functions it turns out they must automatically be analytic and have a very particular form which can be codified in terms of an integral representation formula. (That is, matrix convex functions are essentially moment generating functions.) Helton, McCullough and Vinnikov generalized the Kraus representation to rational expressions of several noncommuting variables as their so-called Butterfly realization. With Ryan Tully-Doyle, we generalized their result to arbitrary functions of noncommuting matrix inputs leveraging a powerful technique known as the ``royal road" to lift the automatic analyticity from the one variable Kraus theorem to several variables for free and thus reduce to an algebraic problem.
In this talk, we will give a gentle introduction to the functional calculus, its rapidly maturing multivariate generalization known as free noncommutative function theory, and the theory of matrix convex functions in both contexts.