## Geometry Seminar - Abstracts

### Talk

Thursday 31 August 2017, 16:00-17:00 in HG03.085

**Pieter Roffelsen** (University of Sydney)

*Singularities of Painlevé IV Rationals and their Distribution*

### Abstract

The fourth Painlevé equation can be realised as the nonlinear differential
equation governing the isomonodromic deformation of certain rank two linear
systems.
Given a solution and point z in the complex plane, corresponding inverse
monodromy problem furnishes the value of the solution at z, except when z is a
zero or pole, in which case the inverse monodromy problem does not have any
solution. In this talk I will discuss how the zeros and poles can in fact be
characterised as the solutions of a simpler inverse monodromy problem
concerning an anharmonic oscillator of degree two. Upon specialising to
rational solutions of the fourth Painlevé equation, we find that their zeros
and poles are classified by the monodromy representation of a class of
meromorphic functions with a finite number of branch points introduced by
Nevanlinna. In particular this allows us to compute the asymptotic distribution
of zeros and poles for Hermite type rationals, as their degree becomes large.

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