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Belavin-Polyakov-Zamolodchikov equation and rational functions


Term:

IAP

Department:

18: Mathematics

Faculty Supervisor:

Yilin Wang

Faculty email:

yilwang@mit.edu

Apply by:

Dec. 1

Contact:

Yilin Wang: yilwang@mit.edu

Project Description

Shapiro's conjecture states that if a rational function has only real critical points has to be a real rational function up to post-composition by a Mobius function. This was proved first by Eremenko-Gabrielov. Recently, it is proved using a different method, via a link to the Loewner potential of multichords introduced by Peltola-Wang [arXiv: 2006.08574]. This potential is motivated by the conformal field theory and large deviations of Schramm-Loewner evolutions. In particular, each such rational function is associated with the potential of its real locus and satisfies a semiclassical limit of the Belavin-Polyakov-Zamolodchikov equation. The goal of the project is to understand better the link between the potential and rational functions and to simulate the corresponding multichords. Even if the outcome is not satisfactory, you will learn about Loewner potential, quasiconformal mappings, rational functions, Riemann surfaces, conformal field theory, Schramm-Loewner evolution, etc.

Pre-requisites

Advanced knowledge and familiarity with analysis, differential equations, complex analysis. Simulation skills, e.g. Matlab or Mathematica. Please take a peek at [arXiv: 2006.08574] before applying.