# BIMSA Integrable Systems Seminar

**Time: **

17:00-18:30, Friday, Jan. 20

**Organizers: **

Niсolai Reshetikhin, Andrey Tsiganov, Ivan Sechin

**Online: **

Zoom ID：815 4690 4797

Passcode：BIMSA

**Schedule**

** 2023-03-24 Fri**

**Title:** On the Geometry of Landau-Ginzburg Model

**Speaker:** Huijun Fan (School of Mathematical Sciences, Peking University)

**Password: **BIMSA

**Direct link:**

https://us02web.zoom.us/j/81546904797?pwd=T1hjVFNzNEU4V3hiMUpGbVpqbE9SUT09

**Abstract: **

An LG model (M, f) is given by a noncompact complex manifold M and the holomorphic function f defined on it, which is an important model in string theory. Because of the mirror symmetry conjecture, the research on the geometric structure and quantization theory of LG model has attracted more and more attention. Given a Calabi-Yau (CY) manifold, we can define Gromov-Witten theory (A theory) on it, and also study the variation of Hodge structure on its mirror manifold (B theory). Accordingly, LG model includes A theory - FJRW theory and Hodge structure variational theory. This report starts with some examples, gives the geometric and topological information contained by a LG model, and derives the relevant Witten equation (nonlinear) and Schrodinger equation (linear). The study of the solution space of these two sets of equations will lead to different quantization theories. Secondly, we give our recent correspondence theorem of Hodge structures between LG model and CY manifold. Finally, we will discuss some relevant issues.

**Speaker Info: **

Huijun Fan is the director of the Key Laboratory of Mathematics and Applied Mathematics of the Ministry of Education of Peking University and the deputy director of the Sino-Russian Math Center. He has won national outstanding youth grant, Changjiang Distinguished Professor of the Ministry of Education, and the second prize of the National Natural Science Award. He was the plenary speaker of the 2021 annual meeting of the Chinese Mathematical Society.

** 2023-03-17 Fri**

**Title:** The centralizer construction and Yangian-type algebras

**Speaker:** Grigori Olshanski (IITP, Skoltech, and HSE Univ., Moscow)

**Password: **BIMSA

**Direct link:**

https://us02web.zoom.us/j/81546904797?pwd=T1hjVFNzNEU4V3hiMUpGbVpqbE9SUT09

**Abstract: **

In the 1980s, Vladimir Drinfeld introduced and studied the notion of Yangian Y(g) associated with an arbitrary simple complex Lie algebra g. The Yangian Y(g) is a deformation of U(g[x]), the universal enveloping algebra for the Lie algebra of polynomial currents g[x]. The general definition of Yangian is radically simplified for the classical series A, and it is even more convenient to work with the reductive algebra g=gl(n).

In the same 1980s, it was discovered that the Yangian Y(gl(n)) can be constructed in an alternative way, starting from some centralizers in the universal enveloping algebra U(gl(n+N)) and then letting N go to infinity. This "centralizer construction" was then extended to the classical series B, C, D, which lead to the so-called twisted Yangians. The theory that arose from this is presented in Alexander Molev's book "Yangians and classical Lie algebras", Amer. Math. Soc., 2007.

I will report on the recent work arXiv:2208.04809, where another version of the centralizer construction is proposed. It produces a new family of algebras and reveals new effects and connections.

** 2023-03-10 Fri**

**Title:** Bethe subalgebras and Kirillov-Reshetikhin crystals

**Speaker:** Leonid Rybnikov (Higher School of Economics, Moscow)

**Password: **BIMSA

**Direct link:**

https://us02web.zoom.us/j/81546904797?pwd=T1hjVFNzNEU4V3hiMUpGbVpqbE9SUT09

**Abstract: **

Bethe subalgebras form a family of maximal commutative subalgebras of the Yangian of a simple Lie algebra, parametrized by regular elements of the corresponding adjoint Lie group. We introduce an affine (Kirillov-Reshetikhin) crystal structure on the set of eigenlines for a Bethe subalgebra in a representation of the Yangian (under certain conditions on the representation, satisfied by all tensor products of Kirillov-Reshetikhin modules in type A). This helps to describe the monodromy of solutions of Bethe ansatz for the corresponding XXX Heisenberg magnet chain.

This is a joint project with Inna Mashanova-Golikova and Vasily Krylov.

** 2023-03-03 Fri**

**Title:** Geometry of Discrete Integrable Systems: QRT Maps and Discrete Painlevé Equations

**Speaker:** Anton Dzhamay, BIMSA, Beijing

**Password: **BIMSA

**Direct link:**

https://us02web.zoom.us/j/81546904797?pwd=T1hjVFNzNEU4V3hiMUpGbVpqbE9SUT09

**Abstract: **

Many interesting examples of discrete integrable systems can be studied from the geometric point of view. In this talk we will consider two classes of examples of such system: autonomous (QRT maps) and non-autonomous (discrete Painlevé equations). We introduce some geometric tools to study these systems, such as the blowup procedure to construct algebraic surfaces on which the mappings are regularized, linearization of the mapping on the Picard lattice of the surface and, for discrete Painlevé equations, the decomposition of the Picard lattice into complementary pairs of the surface and symmetry sub-lattices and construction of a birational representation of affine Weyl symmetry groups that gives a complete algebraic description of our non-linear dynamic.

This talk is based on joint work with Stefan Carstea (Bucharest) and Tomoyuki Takenawa (Tokyo).

** 2023-02-24 Fri**

**Title:** Different approaches for constructing non-abelian Painlevé equations

**Speaker:** Irina Bobrova, National Research University Higher School of Economics, Moscow.

**Password: **BIMSA

**Direct link:**

https://us02web.zoom.us/j/81546904797?pwd=T1hjVFNzNEU4V3hiMUpGbVpqbE9SUT09

**Abstract: **

The famous Painlevé equations play a significant role in modern mathematical physics. The interest in their non-commutative extensions was motivated by the needs of modern quantum physics as well as by natural attempts of mathematicians to extend ‘’classical’’ structures to the non-commutative case.

In this talk we will consider several approaches that are useful for detecting non-commutative analogs of the Painlevé equations. Namely, the matrix Painlevé-Kovalevskaya test, integrable non-abelian auxiliary autonomous systems, and infinite non-commutative Toda equations. All of these methods allow us to find a finite list of non-abelian candidates for such analogs. To provide their integrability, one can present an isomonodromic Lax pair.

This talk is based on a series of papers joint with Vladimir Sokolov and on arXiv:2205.05107 joint with Vladimir Retakh, Vladimir Rubtsov, and George Sharygin (publ. in J. Phys. A: Math. Theor.).

** 2023-02-17 Fri**

**Title:** Elliptic van Diejen difference operators and elliptic hypergeometric integrals of Selberg type

**Speaker:** Masatoshi Noumi (Rikkyo University, Tokyo)

**Password: **BIMSA

**Direct link:**

https://us02web.zoom.us/j/81546904797?pwd=T1hjVFNzNEU4V3hiMUpGbVpqbE9SUT09

**Abstract: **

In this talk, I propose a class of eigenfunctions for the elliptic van Diejen operators (Ruijsenaars operators of type BC) which are represented by elliptic hypergeometric integrals of Selberg type. They are constructed from simple seed eigenfunctions by integral transformations, thanks to gauge symmetries and kernel function identities of the van Diejen operators.

Based on a collaboration with Farrokh Atai (University of Leeds, UK).

** 2023-02-10 Fri**

**Title:** Semifinite harmonic functions on Bratteli diagrams

**Speaker:** Pavel Nikitin (BIMSA)

**Password: **BIMSA

**Direct link:**

https://us02web.zoom.us/j/81546904797?pwd=T1hjVFNzNEU4V3hiMUpGbVpqbE9SUT09

**Abstract: **

Locally semisimple algebras (LS-algebras) are inductive limits of semisimple algebras, and can be fully characterized by their Bratteli diagrams ($\mathbb{N}$-graded graphs). (Finite) harmonic functions on Bratteli diagrams are a standard tool in the representation theory of LS-algebras and semifinite harmonic functions are a natural generalization. We plan to give an overview of the subject, starting with the classical results for the infinite symmetric group, followed by the recent results for the infinite symmetric inverse semigroup. Joint work with N.Safonkin

** 2023-02-03 Fri**

**Title:** On the full Kostant-Toda lattice and the flag varieties

**Speaker:** Yuancheng Xie (Peking University)

**Password: **BIMSA

**Direct link:**

https://us02web.zoom.us/j/81546904797?pwd=T1hjVFNzNEU4V3hiMUpGbVpqbE9SUT09

**Abstract: **

In 1967, Japanese physicist Morikazu Toda proposed an integrable lattice model to describe motions of a chain of particles with exponential interactions between nearest neighbors. Since then, Toda lattice and its generalizations have become the test models for various techniques and philosophies in integrable systems and wide connections are built with many other branches of mathematics. In this talk, I will characterize singular structure of solutions of the so-called full Kostant-Toda (f-KT) lattices defined on simple Lie algebras in two different ways: through the τ -functions and through the Kowalevski-Painlevé analysis. Fixing the spectral parameters which are invariant under the f-KT flows, we build a one to one correspondence between solutions of the f-KT lattices and points in the corresponding flag varieties.

** 2023-01-27 Fri**

**Title:** Random growth of Young diagrams with uniform marginals

**Speaker:** Yuri Yakubovich

**Password: **BIMSA

**Direct link:**

https://us02web.zoom.us/j/81546904797?pwd=T1hjVFNzNEU4V3hiMUpGbVpqbE9SUT09

**Abstract: **

Many (random) growth procedures for integer partitions/Young diagrams has been introduced in the literature and intensively studied. The examples include Pitman's `Chinese restaurant' construction, Kerov's Plancherel growth and many others. These procedures amount to insertion of a new box to a Young diagram on each step, following certain Markovian procedure. However, no such procedure leading to the uniform measure on partitions of $n$ after $n$ steps is known.

I will describe a Markiovian procedure of adding a rectangular block to a Young diagram with the property that given the growing chain visits some level $n$, it passes through each partition of $n$ with equal probabilities, thus leading to the uniform measure on levels. I will explain connections to some classical probabilistic objects. Also I plan to discuss some aspects of asymptotic behavior of this Markov chain and explain why the limit shape is formed.

** 2023-01-20 Fri**

**Title:** The full Toda system, QR decomposition and geometry of the flag varieties

**Speaker:** Dmitry Talalaev (MSU, YarSU, ITEP)

**Password: **BIMSA

**Direct link:**

https://us02web.zoom.us/j/81546904797?pwd=T1hjVFNzNEU4V3hiMUpGbVpqbE9SUT09

**Abstract: **

The full Toda system is a generalization of an open Toda chain, which is one of the archetypal examples of integrable systems. The open Toda chain illustrates the connection of the theory of integrable systems with the theory of Lie algebras and Lie groups, is a representative of the Adler-Kostant-Symes scheme for constructing and solving such systems. Until recently, only some of the results from this list were known for the full Toda system. I will talk about the construction, the commutative family, quantization and solution of the full Toda system by the QR decomposition method, as well as about the application of this system to the geometry of flag varieties.

The material of this talk is based on several joint works with A. Sorin, Yu. Chernyakov and G. Sharygin.

** 2023-01-13 Fri **

**Title: **Skew Howe duality, limit shapes of Young diagrams and universal fluctuations.

**Speaker: **Anton Nazarov (Saint Petersburg University)

**Password: **BIMSA

**Direct link:**

https://us02web.zoom.us/j/81546904797?pwd=T1hjVFNzNEU4V3hiMUpGbVpqbE9SUT09

**Abstract: **

Schur-Weyl, Howe and skew Howe dualities in representation theory of groups lead to multiplicity-free decompositions of certain spaces into irreducible representations and can be used to introduce probability measures on Young diagrams that parameterize irreducible representations. It is interesting to study the behavior of such measures in the limit, when groups become infinite or infinite-dimensional. Schur-Weyl duality and GL(n)-GL(k) Howe duality are related to classical works of Anatoly Vershik and Sergey Kerov, as well as Logand-Schepp, Cohn-Larsen-Propp and Baik-Deift-Johannson. Skew GL(n)-GL(k) Howe duality was considered by Gravner, Tracy and Widom, who were interested in the local fluctuations of the diagrams, the limit shapes were studied Sniady and Panova. They demonstrated that results by Romik and Pittel on limit shapes of rectangular Young tableaux are applicable in this case.

We consider skew Howe dualities for the actions of classical Lie group pairs: GL(n)-GL(k), Sp(2n)-Sp(2k), SO(2n)-O(2k) on the exterior algebras. We describe explicitly the limit shapes for probability measures defined by the ratios of dimensions and demonstrate that they are essentially the same for all classical Lie groups. Using orthogonal polynomials we prove central limit theorem for global fluctuations around these limit shapes. Using free-fermionic representation we study local fluctuations for more general measures given by ratios of representation characters for skew GL(n)-GL(k) Howe duality. These fluctuations are described by Tracy-Widom distribution in the generic case and in the corner by a certain discrete distribution, first obtained in papers by Gravner, Tracy and Widom. Study of local fluctuations for other classical series remains an open problem, but we present numerical evidence that these distributions are universal.

Based on joint works with Dan Betea, Pavel Nikitin, Olga Postnova, Daniil Sarafannikov and Travis Scrimshaw. See arXiv:2010.16383, 2111.12426, 2208.10331, 2211.13728.