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Construction of the Global Parametrix for the Kissing Polynomials

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When trying to implement the Deift-Zhou method of nonlinear steepest descent to recover uniform asymptotics of orthogonal polynomials, one needs to construct solutions to a model Riemann-Hilbert problem (RHP). The solution to this model problem is known as the global parametrix. Typically, these model RHPs are of a standard form, and the global parametrix can be constructed with the use of theta functions on a certain Riemann surface. In the case when one is dealing with orthogonality in the complex plane and the limiting distribution of zeros is supported on multiple arcs, the associated model RHP is not of this standard form, and as such, new methods are needed to construct solutions to this problem. The goal of this talk is to outline the construction of the global parametrix which arises when one is trying to study asymptotics of a family of complex polynomials known as the Kissing polynomials. This is joint work with Guilherme Silva of the University of Michigan.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.

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