|Location:||MSRI: Simons Auditorium, Online/Virtual|
Recent Progress On Planar Orthogonal And Skew-Orthogonal Polynomials
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Coulomb gases in 2D are an exciting subject with many open problems. At least at $\beta=2$ we can use the technique of planar orthogonal polynomials. In this talk I will address the inverse question, when orthogonal polynomials on the real line also are orthogonal in the plane, with respect to some a priori unknown weight and domain. While for Hermite, Laguerre and Chebyshev polynomials this is known, we have recently added planar Gegenbauer polynomials to the list, a subset of the Jacobi polynomials. Being a one parameter family, like the other examples they allow to interpolate to the classical ensembles of Hermitian random matrices. In the second part I will present a general construction of planar skew-orthogonal polynomials from orthogonal ones, for general weight functions under certain conditions. These are useful e.g. in the symmetry class of the symplectic Ginibre ensemble, which is yet another 2D Coulomb gas.
This is based on joint works with Markus Ebke, Taro Nagao, Ivan Parra and Graziano Vernizzi.No Notes/Supplements Uploaded