Some explicit solutions to the Riemann-Hilbert problem

TL;DR

This paper presents explicit solutions to the Riemann-Hilbert problem by realizing certain irreducible non-rigid local systems, exploring their connections to isomonodromy, Painleve equations, and algebraic solutions.

Contribution

It provides new explicit solutions to the Riemann-Hilbert problem related to non-rigid local systems and their links to Painleve equations and algebraic functions.

Findings

Explicit solutions to the Riemann-Hilbert problem are constructed.

Connections between solutions, isomonodromy, and Painleve VI are established.

The work relates algebraic solutions to special groups and Belyi maps.

Abstract

Explicit solutions to the Riemann-Hilbert problem will be found realising some irreducible non-rigid local systems. The relation to isomonodromy and the sixth Painleve equation will be described. Keywords: Riemann-Hilbert problem, Painleve equations, algebraic solutions, Heun equations, tetrahedral/octahedral group, triangle groups, Belyi maps.