Beijing Institute of Mathematical Sciences and Applications

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.

Let $G$ be a simple complex Lie group. I will review the classical Hitchin integrable system on the cotangent bundle to the moduli space ${\rm Bun}_G(X)$ 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 $G^\vee$), 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 $L^2({\rm Bun}_G(X))$, 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 $H_{x,V}$ called Hecke operators (depending on a point $x\in X$ and a representation $V$ of $G^\vee$), whose eigenvalues on the quantum Hitchin eigenfunction $\psi_L$ corresponding to a real oper $L$ are real analytic solutions $\beta(x,\overline x)$ of certain differential equations $D\beta=0$, $\overline D\beta=0$ 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.

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.

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.<br>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.

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.

Elliptic genus, and its various generalizations, is one of the simplest numerical invariants of a scheme that one can consider in elliptic cohomology. I will present a topological condition which implies that elliptic genus is invariant under wall-crossing. It is related to Krichever—Höhn’s elliptic rigidity. Many applications are possible; I will focus on elliptic Donaldson—Thomas theory for this talk.

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.

We develop a general approach to dimer models analogous to Krichever’s scheme in the theory of integrable systems. This leads to dimer models on doubly periodic bipartite graphs with quasiperiodic positive weights. Dimer models with periodic weights and Harnack curves are recovered as a special case. This generalization from Harnack curves to general 𝑀-curves, which are in the focus of our approach, leads to transparent algebro-geometric structures. In particular, the Ronkin function and surface tension are expressed as integrals of meromorphic differentials on 𝑀-curves. Based on Schottky uniformization of Riemann surfaces, we compute the weights and dimer configurations. The computational results are in complete agreement with the theoretical predictions. The talk is based on joint works with N. Bobenko and Yu. Suris.

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.

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.

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.

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 ${\rm sl}(2)$, 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 ${\rm sl}(2)$- and ${\rm gl}(N)$-weight system, for arbitrary $N$. These methods are based on an idea, due to M.Kazarian, which suggests a recurrence for ${\rm gl}(N)$-weight system extended to permutations. The recurrence immediately leads to a construction of a universal ${\rm gl}$-weight system taking values in the ring of polynomials $\mathbf{C}N,C_1,C_2,C_3,\dots]$ in infinitely many variables and allowing for a specialization to ${\rm gl}(N)$- and ${\rm sl}(N)$-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 ${\rm gl}(N|M)$, and, together with M.~Kazarian, to other classical series of Lie algebras. It happened that certain specializations of the universal ${\rm gl}$-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.

I will explain a construction which for every finite poset $X$ (such as a Young diagram) produces a square matrix $M^X$ . 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 $M^X$ are monomials in variables $x_i$ . Our main result is that the eigenvalues of $M^X$ are polynimials in $x_i$ with integer coefficients. Joint work with Richard Kenyon, Maxim Kontsevich, Oleg Ogievetsky, Cosmin Pohoata and Will Sawin.

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 $q\leftrightarrow t^{-1}$, which is related to the functional equation of the Hasse-Weil zetas for curves, the symmetry $\epsilon_1 \leftrightarrow \epsilon_2$ 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 $q$-hypothesis in type $A_1$, in full detail.

I will give a survey of the series of my joint works with Finkelberg and Nakajima giving a mathematical construction of the so called Coulomb branches of 3𝐷 𝑁=4 super-symmetric gauge theories (no knowledge of any of these words will be needed). I will also explain its connection with the (purely mathematical subject) of symplectic duality.

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.

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.

The harmonic locus consists of the monodromy-free Schroedinger operators with rational potential quadratically growing at infinity. It is known after Duistermaat and Gr\"unbaum that in the multiplicity-free case the poles z_1, ..., z_N of such potentials satisfy the following algebraic system, describing the complex equilibriums of the corresponding Calogero-Moser system. Oblomkov proved that the harmonic locus can be identified with the set of all partitions via Wronskian map for Hermite polynomials. We show that the harmonic locus can also be identified with the subset of the Calogero-Moser spaces introduced by Wilson, which is invariant under a natural symplectic action of C^*. As a corollary, for the multiplicity-free part of the locus we effectively solve the inverse problem for the Wronskian map by proving that the spectrum of Moser's matrix coincides with the set of contents of the corresponding Young diagram. We also compute the characters of the C^*-action at the fixed points, proving a conjecture of Conti and Masoero. The talk is based on a joint work with Giovanni Felder.

Even before the introduction of Conformal Field Theory by Belavin, Polyakov and Zamolodchikov, it appeared indirectly in the work of den Nijs and Nienhuis using Coulomb gas techniques. The latter postulate (unrigorously) that height functions of lattice models converge to the Gaussian Free Field, allowing to derive many exponents and dimensions of 2D lattice models. This convergence is in many ways mysterious, in particular it was never formulated in the presence of a boundary, but rather pn a torus or a cylinder. We will discuss possible formulations on general domains or Riemann surfaces and their relations to CFT, SLE and conformal invariance of critical lattice models. Interestingly, new objects in complex geometry and potential theory seem to arise.

A beta ensemble (or log-gas system) on the real line is a random collection of $N$ point particles $x_1,\ldots, x_N$ whose joint probability distribution has a special form containing the Vandermonde raised to the power $\beta>0$. I will survey results related to some discrete analogs of beta ensembles, which live on $q$-lattices, and large-$N$ limit transitions.

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.

We begin by an overview of how geometric ideas entered the theory of differential Painlevé equations in the work of of K.Okamoto, which led to the better understanding of their symmetries (Backlünd transformations) in terms of affine Weyl groups. These ideas were then extended by H.Sakai to the discrete (elliptic, multiplicative, and additive) Painlevé equations and resulted in the beautiful Sakai classification scheme for both differential and discrete Painlevé equations. In the latter case, it is the symmetry group that is the source of a discrete dynamics. In the second part of the talk we discuss the notion of an abstract discrete Painlevé equation and its various concrete realizations. This leads to the study of a refined identification problem, which is a classification of different orbits for the same abstract discrete Painlevé dynamic, and results in the appearance of special symmetry groups that are not a part of the general (i.e., generic) Sakai classification scheme. We illustrate this by an example of a discrete Painlevé-II equation and its symmetry group. This is based on a joint work with Yang Shi, Alex Stokes, and Ralph Willox.