## Riemann-Hilbert techniques and Painlevé equations

### Dr. Andrej Kapaev

Number of cycles: 1

First semester: Mon 11-12:30, Tue 9-10:30 (starting from November 7)

Content:

- Linear differential systems.
- Monodromy data.
- Isomonodromic deformations.
- Transformations of the linear systems.
- Schlesinger transformations.
- Middle convolution.
- Harnad's duality.
- Direct and inverse monodromy probem.
- Painlevé equations.
- Riemann-Hilbert (RH) boundary value problem.
- Uniqueness/nonuniqueness of the solution to the RH problem.
- Existence of the RH problem solution.
- Fredholm theory of singular integral operators approach.
- Birkhoff-Grothendieck approach.
- Asymptotic solvability of the RH problem.
- Steepest descent analysis.
- Boutroux equations.
- Asymptotic analysis of the second Painlevé equation (PII) via the RH problem.
- Regular generic asymptotics on the real line.
- Regular degenerate asymptotics and quasi-linear Stokes phenomenon.
- Singular asymptotics on the real line.
- Elliptic asymptotics in a complex sector and the nonlinear Stokes phenomenon.
- Coefficient asymptotics in a degenerate solution of PII.
- Linear equations with Painlevé potential and the induced Stokes phenomenon.
- Scaling limits in isomonodromy systems.
- n-large asymptotics of the polynomials orthogonal on the line w.r.t. the exponential weight with a quartic potential.
- Integrable integral operators and TBA.