Abstract:

Lecture 1. An introduction to the Stokes phenomenon and isomonodromy deformation. The first lecture gives an introduction to the Stokes matrices of a linear system of meromorphic ordinary differential equations, and the associated nonlinear isomonodromy deformation equation. In the case of Poncare rank 1, the nonlinear equation naturally arises from the theory of Frobenius manifolds, stability conditions, Poisson geometry and so on, and can be seen as a higher rank generalizations of the sixth Painlevé equation.

Lecture 2. Stokes phenomenon and Yang-Baxter equation. This talk introduces the quantum Stokes matrices of the universal quantum linear ordinary differential equations at a second order pole. It then proves that the quantum Stokes matrices satify the RLL relation of the Drinfeld-Jimbo quantum group. Further study of the connection gives a dictionary between the Stokes phenomenon at 2nd order pole and the representation theory of quantum groups.

Lecture 3. WKB approximation、spectral curves and crystal basis. Abstract: This talk studies the WKB approximation of the linear meromorphic systems of Poncaré rank 1, via the isomonodromy approach. In the classical setting, it unveils a relation between the WKB approximation of the Stokes matrices, the Cauchy interlacing inequality and cluster algebras, based on a joint work with Anton Alekseev, Andrew Neitzke and Yan Zhou. In the quantum setting, it gives a transcendental realization of the crystal structures via the WKB approximation in the Stokes phenomenon.

Lecture 4. Quantum Riemann-Hilbert-Birkhoff maps. The Riemann-Hilbert-Birkhoff map is a highly transcendental Poisson map between the de Rham and Betti moduli spaces of meromorphic connections at a k-th order pole. This talk studies its quantization. It first introduces the universal quantum linear ordinary differential equations at an arbitrary order pole. It then proves that the quantum Stokes matrices, of the differential equation at a k-th order pole, give rise to an associative algebra isomorphism that quantizes the (classical) Riemann-Hilbert-Birkhoff Poisson map.