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Title: Riemann-Hilbert problem for a type of planar orthogonal polynomials, Part III Speaker: Abril Arenas Time: 4:00pm–5:00pm Place: CMC 130
Title: Riemann-Hilbert problem for a type of planar orthogonal polynomials, Part II Speaker: Abril Arenas Time: 4:00pm–5:00pm Place: CMC 130
Title: Riemann-Hilbert problem for a type of planar orthogonal polynomials Speaker: Abril Arenas Time: 4:00pm–5:00pm Place: CMC 130
Planar orthogonal polynomials are fundamental objects of interest in the random matrix theory, the statistics of a particular model are encoded in the asymptotics of planar orthogonal polynomials. The family of planar orthogonal polynomials, and the matrix model, depends on a weight of the form \(\exp[-Q(z)]\). When \(Q(z)=|z|^2\) the associated planar orthogonal polynomials are just monomials \(z^n\). Starting with P. Di Francesco et al, taking the simplest harmonic deformation, \(Q(Z)=|z|^2+A\left(z^2 + \bar(z)^2\right)\), leads to the study of Hermite polynomials. Beginning with a paper by P. Bleher and A. Kuijlaars; they considered polynomials associated to the potential \(Q(z)=|z|^2+A\left(z^3+\bar(z)^3\right)\) which are no longer classical yet they managed to find asymptotics using the Riemann-Hilbert apparatus, although in this setting one has to introduce a cutoff to handle convergence issues.
F. Balogh, M. Bertola, S. Lee, and K. McLaughlin studied planar orthogonal polynomials associated to the potential \(Q(z)=|z|^2-2c\log|z-a|\). The polynomials associated to this potential are again no longer classical orthogonal polynomials, yet their idea was to translate planar orthogonality into type II Hermite-Padé multiple orthogonality and then use the Riemann Hilbert technology to study asymptotics of the polynomials.
Following their idea I will present a family of planar orthogonal polynomials associated to the potential \(Q(z)=|z|^2p+2\Re H(z)\); \(H(z)\) a polynomial of degree at most \(2p\). In this talk I will present how to pass from planar orthogonality to a kind of contour orthogonality. This contour orthogonality then allows us to pose a novel Riemann-Hilbert problem characterizing this family of polynomials. The talk should be accessible to any student having taken a semester of complex analysis. As always we will start from the beginning.
No seminar this week.
Title: Algebra, PDE, Analytic Continuation of the Solutions, Part III Speaker: Dima Khavinson Time: 4:00pm–5:00pm Place: CMC 130
Title: Algebra, PDE, Analytic Continuation of the Solutions, Part II Speaker: Dima Khavinson Time: 4:00pm–5:00pm Place: CMC 130
Title: Algebra, PDE, Analytic Continuation of the Solutions, Part I Speaker: Dima Khavinson Time: 4:00pm–5:00pm Place: CMC 130
We shall discuss Hesse’s conjecture for homogeneous polynomials and Korenblum’s conjecture on algebras of harmonic functions from the standpoint of nonlinear first-order PDE. Also, we show how to extend a recent theorem of McKinley and Shekhtman for homogeneous polynomial partial differential operators to a much wider class of linear PDE with entire coefficients by putting it into a general framework of analytic continuation of solutions of linear holomorphic PDE.
Most of the talk should be accessible to students.
Title: The Joint Spectral Radius problem in a random setting, Part II Speaker: Razvan Teodorescu Time: 4:00pm–5:00pm Place: CMC 130
Title: The Joint Spectral Radius problem in a random setting, Part I Speaker: Razvan Teodorescu Time: 4:00pm–5:00pm Place: CMC 130
Computing the Joint Spectral Radius (JSR) for the switching problem from a finite set, \(S\), of positive-definite square matrices of size \(k\), is a classical example of an NP-hard case of analytic extraction. Furthermore, for matrices over the rational field, sub-unitary boundness of JSR is known to be undecidable. To make matters worse, we consider a random switching protocol from a matrix ensemble, in the infinite \(k\) limit. Surprisingly, this allows for a complex-analytic approach and points to possible new conjectures.