Geometry Colloquium: Discrete Riemann-Hilbert problem, d-connections on P^1 and discrete Painlevé equations (Anton Selemenchuk)

Time： 16:30-17:30

Place：Faculty of Science Building  #3, Room 204

Speaker：Anton Selemenchuk（Saint Petersburg State University）

Title：Discrete Riemann-Hilbert problem, d-connections on P^1 and discrete Painlevé equations

Abstract：The formalism of the Discrete Riemann–Hilbert Problem (DRHP), developed by A. Borodin and collaborators, provides a unified framework for describing analytic properties of orthogonal polynomials on discrete sets, the Christoffel–Darboux kernel, Fredholm determinants and the linear difference (or q-difference) equations they satisfy. Under the usual hypotheses on the discrete data (e.g. a locally finite affine lattice and a weight with rational log-derivative), the DRHP can be realised as a discrete Lax pair for an auxiliary matrix problem. After a brief motivation from asymptotic representation theory, I will outline the main steps of this construction and discuss the geometric interpretation: namely, the identification of these isomonodromic transformations with (rational) isomorphisms of moduli spaces of (rational) d-connections on $\mathbb{P}^1$ with prescribed singularity structure, and the resulting connection to discrete Painlevé equations.