The Adelic Grassmannian, Calogero-Moser Matrices and Exceptional Hermite Polynomials

Explore the connections between exceptional Hermite polynomials, the Adelic Grassmannian, and Calogero-Moser systems in this 58-minute lecture by Alex Kasman. Delve into the world of integrable systems, bispectrality, and orthogonal polynomials as part of the Workshop on the Role of Integrable Systems dedicated to John Harnad. Discover how exceptional orthogonal polynomials, first defined in 2009, were actually hidden within George Wilson's Adelic Grassmannian, waiting to be uncovered. Learn about the importance of the KP Hierarchy's second time variable in revealing these connections. Examine how Wilson's bispectral algebras and involution help answer open questions about exceptional Hermites. Investigate the unexpected link between exceptional Hermites and Calogero-Moser systems, and gain insights into ongoing research exploring this connection. Cover topics including bispectral differential operators, classical and generalized orthogonal polynomials, and the historical development of Calogero-Moser particles.

Intro
Bispectral Differential Operators
The KP Hierarchy
Classical Orthogonal Polynomials
Generalizations: Orthogonal Polynomials
Exceptional Hermites
Brainstorming in Halifax
First Corollary: Producing "Recurrence Relations"
Calogero-Moser Particles in the 1970s
Concluding Remarks