**Talk Announcement:**

**Title:**Special polynomials associated with the Painlevé equations

**Speaker**: Peter A. Clarkson (University of Kent, UK)

**When:**Thursday, Aug 5, 2021 - 3:55 PM - 5:00 PM (IST)

**Where:**Zoom

**(Please write at sfandnt@gmail.com for a link)**

**Live Link:**https://youtu.be/uLOGBnOh0v0

**Tea or Coffee**: Please bring your own.

**Abstract:**

The
six Painlevé equations, whose solutions are called the Painlevé
transcendents, were derived by Painlevé and his colleagues in the
late 19th and early 20th centuries in a classification of second order
ordinary differential equations whose solutions have no movable critical
points.

In the 18th and 19th centuries, the classical special
functions such as Bessel, Airy, Legendre and hypergeometric functions,
were recognized and developed in response to the problems of the day in
electromagnetism, acoustics, hydrodynamics, elasticity and many other
areas.

Around the middle of the 20th century, as science and
engineering continued to expand in new directions, a new class of
functions, the Painlevé functions, started to appear in
applications. The list of problems now known to be described by the
Painlevé equations is large, varied and expanding rapidly. The list
includes, at one end, the scattering of neutrons off heavy nuclei, and
at the other, the distribution of the zeros of the Riemann-zeta function
on the critical line $\mbox{Re}(z) =\tfrac12$. Amongst many others,
there is random matrix theory, the asymptotic theory of orthogonal
polynomials, self-similar solutions of integrable equations,
combinatorial problems such as the longest increasing subsequence
problem, tiling problems, multivariate statistics in the important
asymptotic regime where the number of variables and the number of
samples are comparable and large, and also random growth problems.

The
Painlevé equations possess a plethora of interesting properties
including a Hamiltonian structure and associated isomonodromy problems,
which express the Painlevé equations as the compatibility condition
of two linear systems. Solutions of the Painlevé equations have some
interesting asymptotics which are useful in applications. They possess
hierarchies of rational solutions and one-parameter families of
solutions expressible in terms of the classical special functions, for
special values of the parameters. Further the Painlevé equations
admit symmetries under affine Weyl groups which are related to the
associated Bäcklund transformations.

In this talk I shall
discuss special polynomials associated with rational solutions of
Painlevé equations. Although the general solutions of the six
Painlevé equations are transcendental, all except the first
Painlevé equation possess rational solutions for certain values of
the parameters. These solutions are expressed in terms of special
polynomials. The roots of these special polynomials are highly symmetric
in the complex plane and speculated to be of interest to number
theorists. The polynomials arise in applications such as random matrix
theory, vortex dynamics, in supersymmetric quantum mechanics, as
coefficients of recurrence relations for semi-classical orthogonal
polynomials and are examples of exceptional orthogonal polynomials.

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