Beijing Summer Workshop in Mathematics
and Mathematical Physics


The theme for 2024 workshop:
Integrable Systems and Algebraic Geometry

(Dedicated to the memory of Igor Krichever)

BIMSA, June 24— July 5, 2024

Course Descriptions

  • Pavel Etingof (Massachusetts Institute of Technology, USA)

    The Hitchin System and its Quantization

    Let G be a simple complex Lie group. I will review the classical Hitchin integrable system on the cotangent bundle to the moduli space Bun G(X) of principal G-bundles on a smooth complex projective curve X (possibly with punctures), as well as its quantization by Beilinson and Drinfeld using the loop group LG. I will explain how this system unifies many important integrable systems, such as Toda, Calogero-Moser, and Gaudin systems. Then I'll discuss opers (for the dual group G 𝗏 ), which parametrize the (algebraic) spectrum of the quantum Hitchin system. Finally, I will discuss the analytic problem of defining and computing the spectrum of the quantum Hitchin system on the Hilbert space L 2 (Bun G (X)), and will show that (modulo some conjectures, known in genus 0 and 1) this spectrum is discrete and parametrized by opers with real monodromy. Moreover, we will see that the quantum Hitchin system commutes with certain mutually commuting compact integral operators H L ,v called Hecke operators (depending on a point x∈X and a representation V of G 𝗏 ), whose eigenvalues on the quantum Hitchin eigenfunction ψ L corresponding to a real oper L are real analytic solutions β (x , x ) of certain differential equations D β = 0 , D β = 0 associated to L and V. This constitutes the analytic Langlands correspondence, developed in my papers with E. Frenkel and Kazhdan following previous work by Braverman-Kazhdan, Kontsevich, Langlands, Nekrasov, Teschner, and others. I will review the analytic Langlands correspondence and explain how it is connected with arithmetic and geometric Langlands correspondence.

  • Samuel Grushevsky (Simons Center for Geometry and Physics, Stony Brook University, USA)

    Integrable systems approach to the Schottky problem and related questions

    We will review the integrable systems approach to the classical Schottky problem of characterizing Jacobians of Riemann surfaces among all principally polarized complex abelian varieties. Starting with the Krichever's construction of the spectral curve from a pair of commuting differential operators, we will proceed to show that theta functions of Jacobians satisfy the KP hierarchy, and will review Novikov's conjecture (proven by Shiota) solving the Schottky problem by the KP equation. We will finally discuss some of the motivation for Krichever's proof of Welters' trisecant conjecture, and related characterizations for Prym varieties.

  • Nikita Nekrasov (Simons Center for Geometry and Physics, Stony Brook University, USA)

    Integrable many-body systems and gauge theories

    Elliptic Calogero-Moser and Toda systems, Gaudin and other spin chains are algebraic integrable systems which have intimate connections to gauge theories in two, three, and four dimensions. I will explain two such connections: first, classical, through Hamiltonian reduction and second, quantum, through dualities of supersymmetric gauge theories.

  • Andrei Okounkov (Columbia University, USA)

    From elliptic genera to elliptic quantum groups

    This course will be an example-based introduction to elliptic cohomology, Krichever elliptic genera, rigidity, and related topics. We will work our way towards the geometric construction of elliptic quantum groups.