BIMSA Integrable Systems Seminar

Many special functions appearing in the study of integrable systems have their finite field counterparts with extensive connections with number theory and algebraic geometry. For instance, it is well known that Gauss sums are finite field analogues of Gamma-functions and Kloosterman sums are finite field analogues of Bessel functions. In this talk I will present a new approach of studying certain special functions over finite fields using representation theory of finite Chevalley groups. Namely, I will first define finite field analogues of Gamma-functions and Whittaker functions and then identify them as matrix elements of representations of (subgroups of) general linear groups over a finite field and compare them with their counterparts defined over real groups.

For any abelian variety X with an action of a finite complex reflection group W, Etingof, Felder, Ma and Veselov constructed a family of integrable systems on T^*X. When X is a product of n copies of an elliptic curve E and W=S_n, this reproduces the usual elliptic Calogero­­-Moser system. Recently, together with Philip Argyres (Cincinnati) and Yongchao Lü (KIAS), we proposed that many of these integrable systems at the classical level can be interpreted as Seiberg­­-Witten integrable systems of certain super­symmetric quantum field theories. I will describe our progress in understanding this connection for the case X=E^n where E is an elliptic curve with the symmetry group Z_m (of order m=2,3,4,6), and W is the wreath product of Z_m and S_n. I will mostly talk about n=1 case, which is already rather interesting. Based on: arXiv 2309.12760.

The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a arrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by a family of differential equations on 2-torus depending on 3 parameters:  $B$ (abscissa), $A$ (ordinate), $\omega$ (frequency). We study its rotation number $\rho(B, A; \omega)$ as a function of  $(B, A)$ with fixed $\omega$. The phase-lock areas are those level sets $L_r:=\{\rho=r\}$ that have non-empty interiors. They exist only for integer rotation number values $r$: this is the rotation number quantization effect discovered by Buchstaber, Karpov and Tertychnyi. They are analogues of the famous Arnold tongues. Each $L_r$ is an infinite chain  of  domains going vertically to infinity and separated by points called constrictions (expect for those with $A=0$). See the phase-lock area portraits for $\omega=2$, 1,  0.3 at the presentation.

We show that: 1)  all constrictions in $L_r$ lie in the vertical line $\{ B=\omega r\}$;  2)  each constriction is positive, that is, some its punctured neighborhood in the vertical line lies in $\operatorname{Int}(L_r)$. These results, obtained in collaboration with Yulia Bibilo, confirm experiences of physicists (pictures from physical books of 1970-th) and two mathematical conjectures.

The proof  uses an equivalent description of model by linear systems of differential equations on $\oc$ (found by Buchstaber, Karpov and Tertychnyi), their isomonodromic deformations described by Painleve 3 equations and methods of the theory of slow-fast systems. If the time allows we will discuss new results and open questions.

In this talk, we mainly introduce some background of the structure, the representation and the quantization of the generalized intersection matrix algebras. Then we introduce a result on finite dimensional modules over indefinite type Kac-Moody Lie algebras. It is given in a joint work with Hongmei Hu and Yilan Tan.

The prominent role of matrix models in physics and mathematics is well known. It is especially interesting that some of those models are exactly solvable, meaning the one can find explicit formulas for correlation functions. This phenomenon has also been called superintegrability of matrix models. I will present some recent attempt to study it systematically and search for its algebraic origins. It leads to an interesting connection with the rapidly developing field of BPS algebras and their representations.

We discuss several results on work in collaboration (with V.Mishnyakov, A.Popolitov, F.Liu, R.Wang and with B. Kang, K.Wu, W.Z. Zhao). We construct W-representations for multi-character expansions, which involve a generic number of sets of time variables. We propose integral representations for such kind of partition functions which are given by tensor models and multi-matrix models with multi-trace couplings. In addition, we present the W-representation for a two-tensor model with order-3. We derive the compact expressions of correlators from the W-representation, and analyze the free energy in the large N limit. By establishing the correspondence between the two-color Dyck order in Fredkin spin chain and the tree operator on the ring, we prove that the entanglement scaling of Fredkin spin chain beyond the logarithmic scaling in ordinary critical systems.

The topological vertex, developed by Aganagic, Klemm, Marino and Vafa, provides an explicit algorithm to compute the open Gromov-Witten invariants of smooth toric Calabi-Yau threefolds in mathematics, as well as the A-model topological string amplitudes in physics. In this talk, I will introduce our recent work on the connection between the topological vertex and multi-component KP hierarchy.

This talk is based on a joint work with Zhiyuan Wang and Jian Zhou.

BCOV’s Feynman rule is a conjectural algorithm used to compute the higher genus Gromov-Witten invariants of Calabi-Yau threefolds. The Feynman graph that appears in BCOV’s rule can be interpreted as a form of geometric quantization. In this presentation, I will attempt to extract it from the A-model perspective and realize it as Givental’s R-matrix quantization action. Finally, I will explain how mixed field theory applies to this quantization formalism of the Feynman rule. This talk is based on a series of joint works with H.-L. Chang, J. Li, W.-P. Li, and Y. Zhou.

In the talk, I shall present our joint paper with A.Gerasimov and D.Lebedev. In this paper, we develop a representation theory approach to the study of generalized hypergeometric functions of Gelfand, Kapranov and Zelevisnky (GKZ). We show that the GKZ hypergeometric functions may be identified with matrix elements of non-reductive Lie algebras L(N) of oscillator type. The Whittaker functions associated with principal series representations of gl(n,R) being special cases of GKZ hypergeometric functions, thus admit along with a standard matrix element representations associated with reductive Lie algebra gl(n,R), another matrix element representation in terms of L(n(n-1)).

Weight systems, which are functions on chord diagrams satisfying certain 4-term relations, appear naturally in Vassiliev's theory of nite type knot invariants. In particular, a weight system can be constructed from any nite dimensional Lie algebra endowed with a nondegenerate invariant bilinear form. Recently, M. Kazarian suggested to extend the gl(N)-weight system from chord diagrams (treated as involutions without fixed point) to arbitrary permutations, which led to a recurrence formula allowing for an effective computation of its values, elaborated by Zhuoke Yang. In turn, the recurrence helped to unify the gl(N) weight systems, for N = 1, 2, 3, . . . , into a universal gl-weight system. The latter takes values in the ring of polynomials C[N][C1, C2, . . . ] in finitely many variables C1, C2, . . . (Casimir elements), whose coefficients are polynomials in N. The universal gl-weight system carries a lot of information about chord diagrams and intersection graphs. The talk will address the question which graph invariants can be extracted from it. We will discuss the interlace polynomial, the enhanced skew-characteristic polynomial, and the chromatic polynomial. In particular, we show that the interlace polynomial of the intersection graphs can be obtained by a specific substitution for the variables N, C1, C2, . . . . This allows one to extend it from chord diagrams to arbitrary permutations. Questions concerning other graph and delta-matroid invariants and their presumable extensions will be formulated. The talk is based on a work of the speaker and a PhD student Nadezhda Kodaneva.

Björner and Ekedahl [Ann. of Math. (2), 170(2): 799-817, 2009] pioneered the study of length-enumerating sequences associated with parabolic lower Bruhat intervals in crystallographic Coxeter groups. In this talk, we study the asymptotic behavior of these sequences in affine Weyl groups. We prove that the length-enumerating sequences associated with the dominant intervals corresponding to a dominant coroot lattice element are ``asymptotically'' log-concave. More precisely, we prove that a certain sequence of discrete measures naturally constructed from the length-enumerating sequences converges weakly to a continuous measure constructed from a certain polytope. Moreover, a certain sequence of step functions naturally constructed from the length-enumerating sequences uniformly converges to the density function of that continuous measure, which implies the weak convergence and that the sequences of numbers of elements in each layer of the dilated dominant interval converge to a sequence of volumes of hyperplane sections of the polytope. By the Brunn–Minkovski inequality, the density function is log-concave. Our approach relies on the ``dominant lattice formula'', which yields a new bridge between the discrete nature of Betti numbers of parabolic affine Schubert varieties and the continuous nature of the geometry of convex polytopes. Our technique can be seen as a refinement in our context of the classical Ehrhart's theory relating the volume of a polytope and the number of lattice points the polytope contains, by replacing the volume by volumes of transversal sections and the number the total lattice points by the number of lattice points of a given length. Joint with Gaston Burrull and Hongsheng Hu.

The q-deformed enveloping algebra U_q(\hat{\mathfrak{sl}}_2) and its positive part U_q^+ are studied in both mathematics and mathematical physics. The literature contains at least three PBW bases for U_q^+, called the Damiani, the Beck, and the alternating PBW bases. These PBW bases are related via exponential formulas. In this talk, we will introduce an exponential generating function whose argument is a power series involving the Beck PBW basis and an integer parameter m. The cases m = 2 and m = −1 yield the known exponential formulas for the Damiani and alternating PBW bases, respectively. We will give present two results on the generating function for an arbitrary integer m. The first result gives a factorization of the generating function. In the second result, we express the coefficients of the generating function in closed form.

We discuss several result on ongoing work in collaboration (with I. Gaiur & D. Van Straten and with V. Buchstaber & I. Gaiur) on interesting properties of multiplicative generalized Bessel kernels, which include the famous Clausen and Sonine –Gegenbauer formulas, examples of polynomials for Kontsevich discriminant locus given as addition laws for special 2-valued formal groups (Buchstaber–Novikov–Veselov) as well as connections with «period functions» solving some Picard–Fuchs type equations and associated with analogues of Landau–Ginzburg superpotentials. 

We discuss several result on ongoing work in collaboration (with I. Gaiur & D. Van Straten and with V. Buchstaber & I. Gaiur) on interesting properties of multiplicative generalized Bessel kernels, which include the famous Clausen and Sonine –Gegenbauer formulas, examples of polynomials for Kontsevich discriminant locus given as addition laws for special 2-valued formal groups (Buchstaber–Novikov–Veselov) as well as connections with «period functions» solving some Picard–Fuchs type equations and associated with analogues of Landau–Ginzburg superpotentials. 

Quantum intersection numbers were introduced through a natural quantization of the KdV hierarchy in a work of Buryak, Dubrovin, Guere, and Rossi. Because of the Kontsevich-Witten theorem, a part of the quantum intersection numbers coincides with the classical intersection numbers of psi-classes on the moduli spaces of stable algebraic curves. I will talk about our joint work in progress with Xavier Blot, where we relate the quantum intersection numbers to the stationary relative Gromov-Witten invariants of the Riemann sphere, with an insertion of a Hodge class. Using the Okounkov-Pandharipande approach to such invariants (with the trivial Hodge class) through the infinite wedge formalism, we then give a short proof of an explicit formula for the ``purely quantum'' part of the quantum intersection numbers, found before by Xavier, which in particular relates these numbers to the one-part double Hurwitz numbers.

We discuss several result on ongoing work in collaboration (with I. Gaiur & D. Van Straten and with V. Buchstaber & I. Gaiur) on interesting properties of multiplicative generalized Bessel kernels, which include the famous Clausen and Sonine –Gegenbauer formulas, examples of polynomials for Kontsevich discriminant locus given as addition laws for special 2-valued formal groups (Buchstaber–Novikov–Veselov) as well as connections with «period functions» solving some Picard–Fuchs type equations and associated with analogues of Landau–Ginzburg superpotentials. 

We introduce a new approach to Monge-Ampere geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge-Ampere geometry to the Poisson equation for the pressure that arises for incompressible Navier-Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via the (higher) Lagrangian submanifold it defines in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge-Ampere structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge-Ampere geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier-Stokes equations, such as Hill's spherical vortex and an integrable case of Arnol'd-Beltrami-Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction. https://arxiv.org/abs/2302.11604 

The eigenfunctions of the Ruijsenaars hyperbolic system were constructed by M. Hallnäs and S. Ruijsenaars in 2012. Recently in the joint work with S. Derkachov, S. Kharchev and S. Khoroshkin we proved some properties of these eigenfunctions using the so-called Baxter Q-operators. In the talk I will explain the motivation behind these operators, their key properties and how they are used to prove the bispectral symmetry, orthogonality and completeness of the eigenfunctions.

We discuss some recent progress on Baxter Q-operators for the XXZ spin chain with diagonal boundary conditions. A key tool is the universal K-matrix for affine quantum groups. Joint work with Alec Cooper and Robert Weston.

Given a series of WDVV or open-WDVV equation solutions satisfying the certain stabilization conditions, one can construct an infinite system of commuting partial differential equations. We illustrate these fact on the examples of A and D type Dubrovin--Frobenius manifolds and their "open extensions". These give KP, a reduction of a 2-component BKP and 2D Toda hierarchies respectively. Following D.Zuo to a B_n type Coxeter group one can associate n different WDVV solutions that are not necessarily polynomial. We will prove that these Dubrovin--Frobenius structures stabilize too and present the integrable systems associated to them.

Dispersionless integrable hierarchies are obtained as certain limits of classical integrable hierarchies such as the KP hierarchy and the Toda lattice hierarchy. They were introduced in 1990's and studied first, for example, in relation to string theory. In this century it was found that dispersionless hierarchies are closely related to the theory of conformal mappings. I shall talk about the relation of dispersionless hierarchies and the Loewner equations for conformal mappings.

Argument shift method is a construction that produces a commutative subalgebra of a Poisson algebra by differentiating its central elements along a suitable vector field. An important particular case of this situation is when the Poisson algebra is equal to the space of (polynomial) functions on a dual space of a Lie algebra $g$. In my talk I will discuss an attempt to raise this procedure to the universal enveloping algebra of $g$. Based on a joint work with Y.Ikeda and A.Molev

In my talk I’ll give an overview of the results obtained by me, as well as jointly with co-authors, related to the problem of classifying commuting (scalar) differential, or more generally, differential-difference or integral-differential operators in several variables. The problem, under some reasonable restrictions, essentially reduces to the description of projective algebraic varieties that have a non-empty moduli space of torsion-free sheaves with a fixed Hilbert polynomial.

More precisely, it turns out to be possible to classify the so-called quasi-elliptic rings, which describe a wide class of operator rings appeared in the theory of (quantum) integrable systems. They are contained in a certain non-commutative “universal” ring - a purely algebraic analogue of the ring of pseudodifferential operators on a manifold and admit (under some weak restrictions) a convenient algebraic-geometric description. This description is a natural generalization of the classification of rings of commuting ordinary differential or difference operators, described in the works of Krichever, Novikov, Drinfeld, Mumford, Mulase. Moreover, already in the case of dimension two there are significant restrictions on the geometry of spectral manifolds.

I will speak about a local analog of the Deligne-Riemann-Roch theorem for line bundles on a family of smooth projective curves. First, I recall the Deligne-Riemann-Roch theorem. Then I will speak about its local analog. The two parts for this local analog of the Deligne-Riemann-Roch theorem consist of the central extensions of the group that is the semidirect product of the group of invertible functions on the formal punctured disc and the group of automorphisms on this disc. These central extensions are by the multiplicative group. The theorem is that these central extensions are equivalent over the ground field of rational numbers. The talk is based on my reсent preprint arXiv:2308.0649.

It is well known that for a meromorphic linear system with only regular singularities, any formal solution is necessarily convergent. It is less well known that for meromorphic linear systems with irregular singularities, a prescribed asymptotics at an irregular singular point determine different fundamental solutions in different sectorial regions surrounding the singular point. The transition matrices between the preferred solutions in the different sectoral regions are known as the Stokes matrices. This talk shows a relation between Stokes matrices and various structures appearing in integrability. It then explains that how the theory of quantum groups, Yangians, crystal basis and so on can be used to study the Stokes phenomenon.

We discuss recent construction of Heun operators as bilinear combinations of two generators of the Askey-Wilson algebra (as well as of its degenerate cases). This construction is related to an important "band and time limiting" problem in Fourier analysis. Classical mechanical analogs of the Heun operators give rise to several families of dynamical systems having explicit solutions in terms of elliptic functions.

Let $\mathfrak{g}$ be an affine Lie algebra with associated Yangian $Y_h(\mathfrak{g})$. We prove the existence of two meromorphic $R$--matrices associated to any pair of representations of $Y_h(\mathfrak{g})$ in the category $\mathcal{O}$. They are related by a unitary constraint and constructed as products of the form $\mathcal R^{\uparrow/\downarrow}(s)=\mathcal R^+(s)\cdot\mathcal R^{0,\uparrow/\downarrow}(s)\cdot\mathcal R^-(s)$, where $\mathcal R^+(s) = \mathcal R^-_{21}(-s)^{-1}$. The factors $\mathcal R^{0,\uparrow/\downarrow}(s)$ are meromorphic, abelian $R$--matrices, with a WKB--type singularity in $\hbar$, and $\mathcal R^-(s)$ is a rational twist.  Our proof relies on two novel ingredients. The first is an irregular, abelian, additive difference equation whose difference operator is given in terms of the $q$--Cartan matrix of $\mathfrak g$. The regularisation of this difference equation gives rise to $\mathcal R^{0,\uparrow/\downarrow}(s)$ as the exponentials of the two canonical fundamental solutions. The second key ingredient is a higher order analogue of the adjoint action of the affine Cartan subalgebra $\mathfrak h\subset\mathfrak g$ on $Y_h(\mathfrak g)$. This action has no classical counterpart, and produces a system of linear equations from which $\mathcal R^-(s)$ is recovered as the unique solution. Moreover, we show that both $\mathcal R^{\uparrow/\downarrow}(s)$ give rise to the same rational $R$--matrix on the tensor product of any two highest--weight representations.

We investigate the high energy behavior of the SU(N) chiral Gross-Neveu model in 1 + 1 dimensions. The model is integrable and matrix elements of several local operators (form factors) are known exactly. The form factors show rapidity space clustering, which means factorization, if a group of rapidities is shifted to infinity. We analyze this phenomenon for the SU(N) model. For several operators the factorization formulas are presented explicitly.

Andronov's school began to take shape in 1931, when Alexander Alexandrovich himself, together with his wife E.A. Leontovich, moved from Moscow to Nizhny Novgorod. By the time of the move, A.A. Andronov was an established scientist. Even then, he introduced a number of new concepts into science, including self-oscillations, concepts of the roughness of the system, the bifurcation value of the parameter, the phase portrait, and so on. This is a long-lived school in which a unified scientific program has been actively developed by several generations of scientists. In my report, I will touch upon the scientific direction of the school, which is associated with rough (structurally stable) dynamic systems. The simplest of them - "Morse-Smale systems" got their name after the publication of S. Smale's work "On gradient dynamical system // Ann. Math. 74, 1961, P.199-206". He introduced a class of flows on manifolds of arbitrary dimension that copy the properties of coarse flows on the plane described in 1937 by A. Andronov and L. Pontryagin. For the introduced streams Smale proved the validity of inequalities similar to Morse inequalities for non-degenerate functions, after which such flows were called Morse-Smale flows. S. Smale considered it extremely important to study such flows, since he assumed that, by analogy with coarse flows on the plane, Morse-Smale flows exhaust the class of structurally stable flows on manifolds and are dense in the set all threads. Fortunately, it turned out that the multidimensional structurally stable world is much wider, and the Morse-Smale systems represent only its regular part - structurally stable systems with a non-wandering set consisting of a finite number of orbits. Due to the close connection of Morse-Smale systems with the carrier manifold, various topological objects, including wild ones, are realized as invariant subsets of such systems. This leads to a wide variety of Morse-Smale systems (especially on multidimensional manifolds) and, accordingly, complicates their topological classification.

I will formulate the finite-volume thermodynamics of a massive integrable QFT in terms of a has of relativistic loops. The loops interact through scattering factors associated with their intersections. For the doubly periodic spacetime, after decoupling the pairwise interactions by a Hubbard-Stratonovich transformation, the sum over loops can be performed explicitly. The resulting effective theory becomes mean field type in the limit when one of the periods becomes asymptotically large. The mean field obeys the Thermodynamical Bethe Ansatz equations.

One of the most fundamental problems in multidimensional chaos theory is the study of strange attractors which are robustly chaotic (i.e., they remain chaotic after small perturbations of the system). It was hypothesized in [1] that the robustness of chaoticity is equivalent to the pseudohyperbolicity of the attractor. Pseudohyperbolicity is a generalization of hyperbolicity. The main characteristic property of a pseudohyperbolic attractor is that each of its orbits has a positive maximal Lyapunov exponent. In addition, this property must be preserved under small perturbations. The foundations of the theory of pseudohyperbolic attractors were laid by Turaev and Shilnikov [2,3], who showed that the class of pseudohyperbolic attractors, besides the classical Lorenz and hyperbolic attractors, also includes wild attractors which contain orbits with a homoclinic tangency.​ ​ In this talk we give a review on the theory of pseudohyperbolic attractors arising in both systems with continuous and discrete time. At first, we explain what is meant under pseudohyperbolic attractors. Then, we describe our methods for the pseudohyperbolicity verification. We demonstrate the applicability of these methods for several well-known systems (with both pseudohyperbolic and non-pseudohyperbolic attractors). Finally, we present new examples of pseudohyperbolic attractors. ​ [1] Gonchenko, S., Kazakov, A., & Turaev, D. (2021). Wild pseudohyperbolic attractor in a four-dimensional Lorenz system. Nonlinearity, 34(4), 2018. [2] Turaev, D. V., & Shilnikov, L. P. (1998). An example of a wild strange attractor. Sbornik: Mathematics, 189(2), 291. [3] Turaev, D. V., & Shilnikov, L. P. (2008, February). Pseudohyperbolicity and the problem on periodic perturbations of Lorenz-type attractors. In Doklady Mathematics (Vol. 77, pp. 17-21).

 Let A be an arbitrary associative algebra. With the help of Olshanski’s centralizer construction one can define a sequence Y_1(A), Y_2(A),... of  "Yangian-type algebras" (they possess a number of properties of the Yangians of series A). I will discuss a link between these Yangian-type algebras and a class of double Poisson brackets on free associative algebras. The talk is based on the joint paper with Grigori Olshanski arXiv:2308.13325.

 The Yang-Baxter equation  is a fundamental equation in mathematical physics and statistical mechanics, it  has connections with knot theory, braid theory and some algebraic systems. 

In my talk I recall the definition of the Yang-Baxter,  Braid equation,  skew brace and relative Rota-Baxter operators on group. Further we discuss  connections between these objects, suggest some way for construction of relative Rota-Baxter operators, using known Rota-Baxter operators, describe some of these operators on 2-step nilpotent groups and construct some solutions to the Yang-Baxter equation on 2-step nilpotent groups. 

This is joint work with T. Kozlovskaya, P. Sokolov, K. Zimireva, and M. Zonov.

Tensor categories play an important role in theory of subfactors in operator algebras in connection to conformal field theory and condensed matter physics. A certain induction procedure called α-induction has been studied as a quantum version of the classical induction in group representation theory. I will present this without assuming knowledge on operator algebras.

There is a class of U_q(\widehat{sl}_2) models models where the infinite dimensional evaluation representations lead to Baxter's TQ=Q+Q equation where Q is an entire function rather than a polynomial. I will give a general introduction to the method of solving the Baxter equation in this case.

We present a brief history of the application of methods from symplectic geometry to fluid dynamics, and to geophysical systems in particular. The material will cover both analytical and numerical applications, and emphasize the importance of geometric concepts in operational weather prediction models. This seminar relates to others given recently in this series by Rubtsov and by D'Onofrio, and there will be a focus on the role of partial differential equations of Monge—Ampere type.

Rational Q-system is an efficient method for solving Bethe ansatz equations (BAE). One important feature of this method is that, unlike solving BAE directly, it gives only physical solutions of BAE. Therefore, it is intimately related to the completeness problem of Bethe ansatz. In this talk, I will first introduce the rational Q-system and discuss the completeness problem of the spin-1/2 Heisenberg spin chain. Then I will move to the discussion of the spin-s Heisenberg spin chain where the situation is more complicated. The key new feature here is that repeated roots are allowed. I will present the rational Q-system for the higher spin models and discuss the completeness problem for the spin-s Heisenberg spin chain. The solution of the proposed Q-system gives precisely the all the physical solutions required by completeness of Bethe ansatz.

After reviewing recent developments in the field of Schubert calculus, we'll describe two new "puzzle rules" – combinatorial rules for computing products of Schubert classes in the cohomology of partial flag varieties,  and more generally of Segre motivic classes. This is joint work with Allen Knutson.

We propose a new, infinite family of tensor fields, whose first representatives are the classical Nijenhuis and Haantjes tensors. We prove that the vanishing of a suitable higher-level Haantjes torsion is a sufficient condition for the integrability of the eigen-distributions of an operator field on a differentiable manifold. This new condition, which does not require the explicit knowledge of the spectral properties of the considered operator, generalizes the celebrated Haantjes theorem, because it provides us with an effective integrability criterion applicable to the generic case of non-Nijenhuis and non-Haantjes tensors. We also propose a tensorial approach to the theory of classical Hamiltonian integrable systems, based on the geometry of Haantjes tensors. We introduce the family of symplectic-Haantjes manifolds as a natural setting where the notion of integrability can be formulated. In particular, the theory of separation of variables for classical Hamiltonian systems can also be formulated in the context of our new geometric structures.
References: P. Tempesta, G. Tondo, Contemporary Mathematics, AMS (2023) (to appear) D. Reyes, P. Tempesta, G. Tondo, J. Nonlinear Science 33, 35 (2023) P. Tempesta, G. Tondo, Communications in Mathematical Physics 389, 1647-1671 (2022) P. Tempesta, G. Tondo, Annali Mat. Pura Appl. 201, 57-90 (2022) P. Tempesta, G. Tondo, J. Geometry and Physics 160, 103968 (2021)

The Haldane–Shastry spin chain has long-range interactions and remarkable properties including Yangian symmetry at finite length and explicit highest-weight wave functions featuring Jack polynomials. This stems from the trigonometric spin-Calogero–Sutherland model, which is intimately related to affine Hecke algebras, already enjoys these properties from affine Schur–Weyl duality and reduces to the Haldane–Shastry chain in the ‘freezing’ limit. I will present some new results for these models, including Heisenberg-like symmetries whose spectrum can be characterised by Bethe ansatz.

Based on recent work with D. Serban and ongoing work with G. Ferrando, F. Levkovich-Maslyuk and D. Serban.

We propose commuting set of matrix-valued difference operators in terms of the elliptic Baxter-Belavin R-matrix in the fundamental representation of GL(M). In the scalar case M = 1 these operators are the elliptic Ruijsenaars-Macdonald operators, while in the general case they can be viewed as anisotropic versions of the quantum spin Ruijsenaars Hamiltonians. We show that commutativity of the operators for any M is equivalent to a set of R-matrix identities and prove them for the elliptic Baxter-Belavin R-matrix. We show that the Polychronakos freezing trick can be applied to this model. It provides the commuting set of Hamiltonians for long-range spin chain. We also discuss the trigonometric degenerations based on the XXZ R-matrix. The talk is based on joint work with Andrei Zotov arXiv:2201.05944 arXiv:2202.01177

The semigeostrophic equations are a mathematical model representing atmospheric motion on a subcontinental scale. Their remarkable mathematical features enable the equations to model singular behaviours like weather fronts. This talk presents a new approach to classifying these singular structures using the geometry of Monge-Ampère equations.

In the geometrical view, solutions are understood as Lagrangian submanifolds of a suitably defined phase space equipped with a pseudo-Riemannian metric. We show the interplay between solution singularities, elliptic-hyperbolic transitions of the Monge-Ampère operator, and the degeneracies of the metric on a few examples.

For an integrable hierarchy which possesses a bihamiltonian structure with semisimple hydrodynamic limit, we prove that the linear reciprocal transformation with respect to any of its symmetry transforms it to another bihamiltonian integrable hierarchy. Moreover, we show that the central invariants of the bihamiltonian structure are preserved under the reciprocal transformation. The main tools that we use to obtain this result are the bihamiltonian and variational bihamiltonian cohomologies defined for a bihamiltonian structure of hydrodynamic type. We also apply this result to study the problem of classification of bihamiltonian integrable hierarchies.

I am going to present an introduction into the geometric approach to Monge–Ampère operators and equations based on contact and symplectic structures of cotangent and the 1st jet bundles of a smooth manifold. This approach was developed by V. Lychagin and goes back to the ideas of E.Cartan and his successor T. Lepage. I shall try to make my talk self-contained. I also plan to discuss various applications and links with important geometric structures.

We study some natural representations of current Lie algebras g \otimes k[t], called Weyl modules. They are natural analogues of irreducible representations of simple Lie algebras. There are several current analogues of classical theorems about Lie algebras where these modules «play role» of irreducible modules. In my talk I will explain analogues of duality theorems, namely Peter-Weyl theorem, Schur-Weyl duality etc.

Rota-Baxter operators on Lie algebras were first studied by Belavin, Drinfeld and Semenov-Tian-Shansky as operator forms of the classical Yang-Baxter equation.

As a fundamental tool in studying integrable systems, the factorization theorem of Lie groups by Semenov-Tian-Shansky was obtained by integrating a factorization of Lie algebras from solutions of the modified Yang-Baxter equation. Integrating the Rota-Baxter operators on Lie algebras, we introduce the notion of Rota-Baxter operators on Lie groups and more generally on groups. Then the factorization theorem can be achieved directly on groups. As the underlying structures of Rota-Baxter operators on groups, the notion of post-groups was introduced. The differentiation of post-Lie groups gives post-Lie algebras. Post-groups are also related to Lie-Butcher groups, and give rise to solutions of Yang-Baxter equations.

The talk is based on the joint work with Chengming Bai, Li Guo, Honglei Lang and Rong Tang.

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.

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.

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.

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).

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.).

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).

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.

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.

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.

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 my talk is based on several joint works with A. Sorin, Yu. Chernyakov and G. Sharygin.

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.