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

Invited Talks

  • Mikhail Bershtein (The University of Edinburgh, UK)

    Chiralization of cluster structures

    The chiralization in the title denotes a certain procedure which turns cluster X-varieties into q−W algebras. Many important notions from cluster and q−W worlds, such as mutations, global functions, screening operators, R-matrices, etc. emerge naturally in this context. In particular, we discover new bosonizations of q−W algebras and establish connections between previously known bosonizations. If time permits, I will discuss potential applications of our approach to the study of 3d topological theories and local systems with affine gauge groups. This talk is based on a joint project with J. Shiraishi, J.E. Bourgine, B. Feigin, A. Shapiro, and G. Schrader.

  • Alexei Borodin (Massachusetts Institute of Technology, USA )

    Geometry of dimer models

    Random dimer coverings of large planar graphs are known to exhibit unusual and visually apparent asymptotic phenomena that include formation of frozen regions and various phases in the unfrozen ones. For a specific family of subgraphs of the (periodically weighted) square lattice known as the Aztec diamonds, the asymptotic behavior of dimers admits a precise description in terms of geometry of underlying Riemann surfaces. The goal of the talk is to explain how the surface structure manifests itself through the statistics of dimers. Based on joint works with T. Berggren and M. Duits.

  • Thomas Bothner (University of Bristol, UK)

    What is ... a Riemann-Hilbert problem?

    In its classical setting, the Riemann-Hilbert problem refers to Hilbert’s 21st problem of constructing a Fuchsian ODE system with prescribed poles and a given monodromy group. Using singular integral equation techniques, Plemelj presented a solution to this problem in 1908 which became widely accepted. However, Kohn, Arnold and Il’yashenko noticed in the mid 1980s that Plemelj had actually worked on a problem similar to Hilbert’s 21st for so-called regular ODE systems rather than Fuchsian ones. These new investigations resulted eventually in a negative answer to Hilbert’s original problem given by Bolibruch in 1989 with further developments by Bolibruch and Kostov soon after.
    Tangentially to the solution of Hilbert’s classical problem, the singular integral equation techniques used therein, a.k.a. analytic factorizations of given functions defined on curves, gave rise to a class of modern Riemann-Hilbert factorization problems. In fact nowadays we view such problems as part of a broad analytical toolbox that is useful in the analysis of problems in mathematics and physics, for instance the Wiener-Hopf methods in hydrodynamics and diffraction. The goal of this talk is to first review some facts of the classical Riemann-Hilbert theory and then present a few recent developments of its modern counterpart. Special attention in the second part will be given to matrix- and operator-valued Riemann-Hilbert problems that arise in random matrix theory and integrable probability.

  • Ivan Cherednik (University of North Carolina, USA)

    Q-zeta revisited

    The fundamental feature of practically all zeta-functions and L-functions is that their meromorphic continuations to complex s provide a lot of information about the corresponding objects. However, complex values of s have generally no direct arithmetic/geometric meaning, and occur as a powerful technical tool. We will discuss the refined theory, which is basically the replacement of the terms 1 / n s by the invariants of lens space L ( n , 1 ) , certain q , t , a -series. One of their key properties is the superduality q ↔ t − 1 , which is related to the functional equation of the Hasse-Weil zetas for curves, the symmetry ϵ 1 ↔ ϵ 2 of Nekrasov’s instantons and to other refined theories in mathematics and physics. These invariants have various specializations, including Rogers-Ramanujan identities and the topological vertex. We will begin the talk with the Riemann q-zeta-hypothesis in type A 1 , in full detail.

  • Sergei Lando (HSE University and Krichever Center for Advanced Studies, Russia)

    Weight systems associated to Lie algebras

    V. A. Vassiliev's theory of finite type knot invariants allows one to associate to such an invariant a function on chord diagrams, which are simple combinatorial objects, consisting of an oriented circle and a tuple of chords with pairwise distinct ends in it. Such functions are called “weight systems”. According to a Kontsevich theorem, such a correspondence is essentially one-to-one: each weight system determines a certain knot invariant.
    In particular, a weight system can be associated to any semi-simple Lie algebra. However, already in the simplest nontrivial case, the one for the Lie algebra sl( 2 ), computation of the values of the corresponding weight system is a computationally complicated task. This weight system is of great importance, however, since it corresponds to a famous knot invariant known as the colored Jones polynomial.
    Last few years was a period of significant progress in understanding and computing Lie algebra weight systems, both for sl( 2 )- and gl( N )-weight system, for arbitrary N. These methods are based on an idea, due to M. Kazarian, which suggests a recurrence for gl( N )-weight system extended to permutations. The recurrence immediately leads to a construction of a universal gl-weight system taking values in the ring of polynomials ℂ [ N , C 1 , C 2 , C 3 , … ] in infinitely many variables and allowing for a specialization to gl( N ) and sl( N )-weight systems for any given value of N. A lot of new explicit formulas were obtained.
    Simultaneously, Zhuoke Yang extended the construction to the Lie superalgebras gl ( N | M ), and, together with M. Kazarian, to other classical series of Lie algebras. It happened that certain specializations of the universal gl( N )-weight system lead to well-known combinatorial invariants of graphs, allowing thus to extend these invariants to permutations.
    Certain integrability properties of the Lie algebra weight systems will be discussed.
    The talk is based on work of M. Kazarian, the speaker, and N. Kodaneva, P. Zakorko, Zhuoke Yang, and P. Zinova.

  • Andrei Marshakov (Krichever Center for Advanced Studies, Russia)

    Krichever tau-function: basics and perspectives

    I plan to start with the definition of quasiclassical tau-function, introduced by Igor Krichever in 1992, formulate its main properties with some simple proofs, and discuss certain particular cases, which include the Seiberg-Witten prepotentials, matrix models etc. Then I am going to turn to certain modern developments, related with this object, which include the relation with instanton partition functions, isomonodromic tau-dunctions and even some unexpected relations with other famous relations in mathematical physics.

  • Grigori Olshanski (Krichever Center for Advanced Studies and HSE University, Russia)

    Macdonald-level extension of beta ensembles and multivariate hypergeometric polynomials (online)

    A beta ensemble (or log-gas system) on the real line is a random collection of N point particles x 1 , . . . , x N whose joint probability distribution has a special form containing the Vandermonde raised to the power β > 0 . I will survey results related to some discrete analogs of beta ensembles, which live on q-lattices, and large-N limit transitions.

  • Senya Shlosman (Krichever Center for Advanced Studies, Russia and BIMSA, China)

    Pedestals matrices: Polynomial matrices with polynomial eigenvalues

    I will explain a construction which for every finite poset X (such as a Young diagram) produces a square matrix MX. Its matrix elements are indexed by pairs P, Q of linear orders on X (pairs of standard tableaux in case of Young diagrams). The entries of MX are monomials in variables xi. Our main result is that the eigenvalues of MX are polynimials in xi with integer coefficients. Joint work with Richard Kenyon, Maxim Kontsevich, Oleg Ogievetsky, Cosmin Pohoata and Will Sawin.

  • Alexander Veselov (Loughborough University, UK)

    Integrability in topology

    The fruitful interaction of integrable systems and algebraic geometry is well-known and goes back to XIX-th century, when the theta functions were introduced and used to integrate the geodesic flow on ellipsoids. The relation of integrability with topology is less known. I will discuss several concrete examples of integrability (understood in a wide sense) in topology, including Morse theory and theory of complex cobordisms.

  • Paul Wiegmann (University of Chicago, USA and BIMSA, China)

    Peierls phenomenon via Bethe Ansatz: reflection of Krichever's works on Peierls model

    In the 1930s Rudolf Peierls argued that the one-dimensional electrons interacting with phonons undergo an instability, leading to the formation of a periodic structure known as an electronic crystal. Peierls's instability stands in a short list of major phenomena of condensed matter physics.
    From a mathematical perspective, a comprehensive solution to the Peierls problem was given in papers by Igor Krichever and co-authored by Natasha Kirova, Sergei Brazovski, and Igor Dzyaloshinsky In the early 80’s. It was found that electronic crystals are periodic solutions of soliton equations, falling within the framework of Krichever-Novikov's theory of finite-gap potentials.
    The Peierls phenomenon also emerges as a limiting case of models of interacting fermions, such as Gross-Neveu models with a large rank symmetry group when the rank of the group tends to infinity. These models are solvable by the Bethe Ansatz for finite rank groups. The talk presents the result of a recent paper co-authored by Konstantin Zarembo, Valdemar Melin, and Yoko Sekiguchi, where Krichever’s finite-gaps solutions of soliton equations were obtained as a singular large rank limit of the Bethe Ansatz solution of models with Lie group symmetry.

  • Anton Zabrodin (Krichever Center for Advanced Studies, Russia)

    Deformed Ruijsenaars-Schneider model: integrability and time discretization

    We will discuss the recently introduced deformed Ruijsenaars-Schneider (RS) many-body system. One the one hand, it is the dynamical system for poles of elliptic solutions to the Toda lattice with constraint of type B. On the other hand, equations of motion for this system coincide with those for pairs of RS particles which stick together preserving a special fixed distance between the particles. We prove integrability of the deformed RS system by finding the integrals of motion explicitly. We also obtain Backlund transformations and integrable time discretization of the deformed RS system.

  • Da-jun Zhang (Shanghai University, China)

    • Title: Elliptic solitons related to the Lamé functions

    In this talk I will report recent progress on the elliptic solitons related to the Lamé functions. Apart from the classical solitons that are composed by usual exponential type plane wave factors, there exist “elliptic solitons” which are composed by the Lamé-type plane wave factors and expressed using Weierstrass functions. Recently, we found vertex operators to generate tau functions for such type of solitons. We also established an elliptic scheme of direct linearization approach.

  • Youjin Zhang (Tsinghua University, China)

    Bihamiltonian integrable systems and their classification

    Bihamiltonian structure plays an important role in the theory of integrable systems. For a system of evolutionary PDEs with one spatial variable which possesses a bihamiltonian structure, one is able to find, under a certain appropriate condition, infinitely many conservation laws of the system from the bihamiltonian recursion relation and to arrive at its integrability. In the case when the bihamiltonian structure of the system of evolutionary PDEs possesses a hydrodynamic limit, one can further obtain from it a flat pencil of metrics, and relate it to Frobenius manifold structures or their generalizations under a certain condition, such a relationship may help one to find applications of the integrable system in different research areas of mathematical physics. In this talk, we will recall the notion of bihamiltonian integrable systems, explain their relationship with Frobenius manifold structures or their generalizations, and review the results on the classification of bihamiltonian integrable hierarchies which possess semisimple hydrodynamic limits.