Department of Mathematics

The families of polynomials going with the names of Hermite, Laguerre and Jacobi are the only ones that are orthogonal with respect to a measure supported on the real line and are joint eigenfunctions of some second order differential operator. They are the workhorse of classical mathematical physics starting with Laplace and all the way to quantum mechanics.

I will talk about matrix valued orthogonal polynomials, a notion introduced by M.G. Krein. The determination of those ones going along with differential operators with matrix coefficients has received some attention very recently.

All these examples and their appropriate deformations play an important role in connection with subjects such as the Korteweg-deVries equation, the Toda flows, commutative algebras of differential operators, algebraic curves, etc.