---

Видео на тему: Центр интегрируемых систем - видео

Title: Classifications on soliton solutions of KP type systems Abstract: In this talk, we will give a classification of the regular soliton solutions of the KP hierarchy, referred to as the KP solitons, under the Gel'fand-Dickey reductions in terms of the permutation of the symmetric group. As an example, we show that the regular soliton solutions of the (good) Boussinesq equation as the 3-reduction can have at most one resonant soliton. In particular, we show that the non-crossing permutation gives the regularity condition for the soliton solutions. Meanwhile the classification on singular soliton solutions of the Boussinesq equation is also interesting.Title: Quantum Dilogarithms and New Integrable Lattice Models in Three Dimensions Abstract: In this talk I will describe a class of integrable 3D lattice models (related to solution of Zamolodchikov tetrahedron equation) related to a class of so called quantum dilogarithic functions. The partition function per site in these models can be exactly calculated in the limit of an infinite lattice by using the functional relations, symmetry and factorization properties of the transfer matrix. The results of such calculations for 3D models associated with the Faddeev modular quantum dilogarithm are briefly presented.Title: Lagrangian formulation of the Darboux system Abstract: The classical Darboux system governing the rotation coefficients of three-dimensional metrics of diagonal curvature admits an equivalent formulation as a sixth-order partial differential equation for a scalar potential associated with the corresponding τ-function. In this talk, we show that this equation possesses a Lagrangian structure and can be interpreted as an explicit scalar representation of the generating PDE of the KP hierarchy, in the sense recently proposed by Frank Nijhoff within the Lagrangian multiform framework. We further construct scalar Lagrangian formulations for differential-difference and fully discrete analogues of the Darboux system. In the continuous and semi-discrete settings, the Lagrangians can be written in terms of elementary functions, specifically logarithms, whereas in the fully discrete case they naturally involve special functions, notably dilogarithms. An additional outcome of this approach is that the dispersionless limits of these Lagrangians yield a complete classification of three-dimensional second-order integrable Lagrangians of certain form.Title: Integrability of supersymmetric Calogero Moser models Abstract: We analyze the integrability of the N-extended supersymmetric Calogero-Moser model. We explicitly construct the Lax pair for this system, which properly reproduces all equations of motion. After adding a supersymmetric oscillator potential we reduce the latter to solving dU/dt = AU for the time evolution operator U(t). The bosonic variables, however, evolve independently of U on closed trajectories, as is required for superintegrability. To visualize the structure of the conserved currents we derive the complete set of Liouville charges up to the 5-th power in the momenta, for the N=2 supersymmetric model. The additional, non-involutive, conserved charges needed for a maximal superintegrability of this model are also found.Title: Spin models and flag manifolds Abstract: We will consider a one-dimensional sigma model with the target space being an SU(3) full flag manifold, equipped with an arbitrary invariant metric. We explicitly describe all geodesics in terms of elliptic functions and demonstrate that the spectrum of the Laplace-Beltrami operator may be found by solving polynomial equations of a special type. These results are based on the previously discovered connection between sigma models and Gaudin models, which also holds in the SU(n) case. The talk is based on joint works with D. Bykov: - D. Bykov and A. Kuzovchikov. “The classical and quantum particle on a flag manifold”. arXiv:2404.15900 [hep-th] - D. Bykov and A. Kuzovchikov. “Sigma models from Gaudin spin chains”. arXiv:2508.20889 [hep-th]Title: Applications of associative Yang-Baxter equation for constructing integrable systems Abstract: We review different applications of the associative Yang-Baxter equation (AYBE) to integrable systems. Namely, we study a class of quantum R-matrices in the fundamental representation which satisfy not only the standard quantum Yang-Baxter equation but also the quadratic relation called AYBE. It allows to propose constructions of the classical Lax pairs for integrable tops, quadratic r-matrix structures of Sklyanin type, classical spin chains and continuous 1+1 integrable field theories of Landau-Lifshitz type. One of the most general is the model of interacting tops. Another construction is an R-matrix valued Lax pair. With its help one can define a quantization for the model of interacting tops. By proceeding to half-quantum (hybrid) Lax equations we obtain a family of quantum long-range spin chains of the Haldane-Shastry type. Finally, we briefly discuss extension of the AYBE to BC_N root system, which involves the boundary K-matrices.Title: Discrete Painlevé equations from geometric deautonomization of QRT maps Abstract: In this talk we consider some examples of discrete Painlevé equations that can be obtained from a given QRT map using the technique of geometric deautonomization. One common interesting feature of such equations is that they often correspond to quasi-translations, or the elements of infinite order in the corresponding affine Weyl group whose certain power is a translation. Such elements often become translations if one considers a smaller affine Weyl subgroup, the phenomena that is known as the projective reduction.Title: Boost Automorphic Symmetry and AdS Integrable Deformations Abstract: We address the new structures arising in quantum and string integrable theories, as well as construct a method to find them. Initially we implement the automorphic symmetries on periodic lattice systems and exploit properties of an integrable hierarchy. This prescription is first applied for the new sl_2 sector, Generalised Hubbard type classes and more. We then construct a boost recursion for systems with R-/S-matrices that exhibit arbitrary spectral dependence, which is also an apparent property of the scattering in string theory integrable backgrounds. The generalised bottom-up approach based on coupled differential systems is derived to resolve for exact form of the associated R-matrices. In addition, we single out a class of quantum integrable models whose two-body S-matrices are of non-difference type and which induce a new integrable structure on AdS string backgrounds. These models can be rigorously realised as deformations of the AdS_3 and AdS_2 string worldsheet theories. We demonstrate that their R-matrices, among braiding unitarity and crossing symmetry, satisfy the free-fermion condition and give rise to deformed Hopf (super)algebra structures. The corresponding deformed algebraic structures closely parallel with integrable string models on AdS_3 x S^3b x M^4 and AdS_2 x S^2 x T^6, thereby providing a unified framework for their non-difference-form deformations.Title: Parametric Korteweg-de Vries hierarchy, polynomial dynamical systems and hyperelliptic sigma functions Abstract: In our works with V. M. Buchstaber we have solved the problem of constructing dynamical systems corresponding to a hyperelliptic curve of infinite genus. In the talk the details of this construction will be presented. The dynamical systems in question are closely related to the famous Korteweg-de Vries hierarchy and a series of solutions of this hierarchy in terms of Kleinian hyperelliptic sigma functions. There are commutative diagrams giving embeddings of universal Jacobian bundles of hyperelliptic curves of any genus into a polynomial map of complex spaces of infinite dimension. The dynamical systems are explicit polynomial dynamical systems on these complex spaces, and the commutative diagrams bring them to differentiations of Abelian functions on the Jacobians of hyperelliptic curves.Title: Ricci flows in effective resistance metric for community detection in graphs Abstract: Community detection in complex networks is a fundamental problem, open to new approaches in various scientific settings. We introduce a novel community detection method, based on Ricci flow on graphs. Our technique iteratively updates edge weights (their metric lengths) according to their (combinatorial) Foster version of Ricci curvature computed from effective resistance distance between the nodes. The latter computation is known to be done by pseudo-inverting the graph Laplacian matrix. At that, our approach is alternative to one based on Ollivier-Ricci geometric flow for community detection on graphs, significantly outperforming it in terms of computation time. In our proposed method, iterations of Foster-Ricci flow that highlight network regions of different curvature -- are followed by a Gaussian Mixture Model (GMM) separation heuristic. That allows to classify edges into ``strong'' (intra-community) and ``weak'' (inter-community) groups, followed by a systematic pruning of the former to isolate communities. We benchmark our algorithm on synthetic networks generated from the Stochastic Block Model (SBM), evaluating performance with the Adjusted Rand Index (ARI). Our results demonstrate that proposed framework robustly recovers the planted community structure of SBM-s, establishing Ricci-Foster Flow with GMM-clustering as a principled and computationally effective new tool for network analysis, tested against alternative Ricci–Ollivier flow coupled with spectral clustering.Title: Functional realization of the Gelfand-Tselin base Abstract: Consider a realization of finite-dimensional irreducible representations of gl(n) in the space of functions on the group GL(n). The question is: which functions on the group correspond to the Gelfand-Tsetlin basis vectors? The answer has been known since the 1960s for the first non-trivial case, n=3. In this case, the corresponding functions are written as the result of a substitution of a certain expression involving determinants on the group into the Gauss hypergeometric function. In the talk a generalization of this result to the case of arbitrary n will be presented. To obtain such a generalization certain classes of functions and systems of hypergeometric type (close to GKZ functions) will be constructed.Title: Quantisation ideals and canonical parametrisations of the unipotent group Abstract: Quantisation ideals for dynamical systems on the free associative algebra have proven to be an effective tool for solving the problem of deformation quantisation, as well as for obtaining non-deformation quantisation. This talk will provide a brief overview of the history and motivation behind this approach. In our joint work with A.V. Mikhailov and D.V. Talalaev, we generalize this approach to the case of discrete dynamics on the free associative algebra A. This dynamics is defined by a well-known solution of the tetrahedron equation (Case α from Sergeev's list), which is related to the problem of re-parameterisation of the unipotent group N(3, A). As a result, we construct several families of quantisations, analyze their classical limit and obtain canonical integrable systems compatible with re-parameterisations. We have to mention that the charts and re-parameterisations (mutations) form a cluster-like structure, with Poisson brackets that represent a deformation of the log-canonical type.Title: Symplectic groupoid: geometry, networks, and moduli spaces of closed Riemann surfaces Abstract: I will describe the Bondal's symplectic groupoid: a set of pairs (B,A) with A unipotent upper-triangular matrices and B an element of GL(n) being such that the matrix B A B^T is itself unipotent upper triangular. Since works of J.Nelson, T.Regge, B.Dubrovin, and M.Ugaglia it was known that entries of A can be identified with geodesic functions on a Riemann surface with holes; these entries then enjoy a closed Poisson algebra (reflection equation) expressible in the r-matrix form. In our recent work with M.Shapiro, we solved the symplectic groupoid in terms of planar networks; we used this solution to construct a complete set of geodesic functions for a closed Riemann surface of genus 2; all geodesic functions are elements of the upper cluster algebra whereas Dehn twists are described by cluster mutations. This is a joint work with M.Shapiro.Title: Symmetries of the full symmetric Toda system and Li-Bianchi integrability Abstract: The full symmetric Toda system is an integrable Hamiltonian system on the space of symmetric real matrices, similar to the open Toda chain. In this report, I will talk about how to build vector fields that preserve this system. In particular, it follows that this system is integrable in the sense of the Li-Bianchi theorem (that is, it has a solvable algebra of symmetries of maximum dimension).Title: Noncommutative discrete KdV equations, their symmetries and reductions Abstract: Employing the Lax pairs of the discrete noncommutative Hirota's Korteweg-de Vries (KdV) and the potential KdV equations, we derive differential-difference equations consistent with these equations which play the role of generalised symmetries of the latter. Miura transformations map them to a noncommutative modified Volterra equation and its master symmetry are given. The use of the symmetries for the reduction of the potential KdV equation is demonstrated and the reductions to a noncommutative discrete Painleve equation and a system of partial differential equations generalising the Ernst equation and the Neugebauer-Kramer involution are presented. A Darboux and an auto-Backlund transformation for the Hirota KdV are presented and their relation to the noncommutative Yang-Baxter map is given.Title: On a family of Poisson brackets on gl(n) compatible with the Sklyanin bracket Abstract: The talk is focused on the family of compatible quadratic Poisson brackets on gl(n), generalizing the Sklyanin one. For any of the brackets in the family, the argument shift determines the compatible linear bracket. I will describe the application of the bi-Hamiltonian formalism for some pencils from this family, namely a method for constructing involutive subalgebras for a linear bracket starting by the center of the quadratic bracket. I will provide some interesting examples of families of this type. An important ingredient of the construction is the family of antidiagonal principal minors of the Lax matrix. A crucial but quite unbiguous condition of the log-canonicity of brackets of these minors with all the generators of the Poisson algebra establishes a relation of our families with cluster algebras, a similar property arises in the context of Poisson structures consistent with mutations. The talk is based on the recent joint paper with V.V. Sokolov https://arxiv.org/abs/2502.16925Speaker: Pavlos Kassotakis (University of Patras, Greece) Title: On pentagon and entwining tetrahedron maps Date and time: 9.04.2025, 17:00 (GMT +03:00) Abstract: In this talk, we present equivalence classes of rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, up to Möbius transformations, we find quadrirational one-component maps of rational functions in two arguments that serve as solutions of the pentagon equation. Also, provided that a pentagon map admits at least one partial inverse, we obtain entwining pentagon set theoretical solutions. Furthermore, we show how to obtain Yang–Baxter and entwining tetrahedron maps from pentagon maps.Title: On pentagon and entwining tetrahedron maps Abstract: In this talk, we present equivalence classes of rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, up to Möbius transformations, we find quadrirational one-component maps of rational functions in two arguments that serve as solutions of the pentagon equation. Also, provided that a pentagon map admits at least one partial inverse, we obtain entwining pentagon set theoretical solutions. Furthermore, we show how to obtain Yang–Baxter and entwining tetrahedron maps from pentagon maps.Title: Rota-Baxter operators and Yang-Baxter equation Abstract: In 1960, G. Baxter introduced the notion of Rota-Baxter operator. In 1960-70s, such operators were actively studied on Banach algebras. In 1980s, A.A. Belavin, V.G. Drinfeld and M.A. Semenov-Tian-Shansky rediscovered Rota-Baxter operators while studying classical and modified Yang-Baxter equations. In 2000, M. Aguiar found even deeper connection between Rota-Baxter operators defined on associative algebras and associatie Yang-Baxter equation. In 2020, Rota-Baxter operators on a group were defined by L. Guo, H. Lang, and Y. Sheng, such operators are connected to quantum Yang-Baxter equation.Title: KP integrability in topological recursion (continuation) Abstract: Topological recursion of Chekhov-Eynard-Orantin is a remarkable universal recursive procedure that has been found in many enumerative geometry problems, from combinatorics of maps, to random matrices, Gromov-Witten invariants, Hurwitz numbers, Mirzakhani's hyperbolic volumes of moduli spaces, knot polynomials. A recursion needs an initial data: a spectral curve, and the recursion defines the sequence of invariants of that spectral curve. Kadomtsev-Petviashvili hierarchy is an integrable hierarchy of nonlinear PDEs. Except for many important properties, it quite often appears in the applications: a lot of functions from combinatorics, mathematical physics, theory of moduli spaces and Gromov-Witten theory are solutions to the KP hierarchy. In the talk I will define the KP integrability property for the topological recursion invariants and show that TR invariants are KP integrable if and only if the corresponding spectral curve is rational. If time permits I will discuss the construction of the KP tau function on the TR spectral curve of any genus which can be seen as a non-perturbative generalization of the Krichever's construction of the KP tau function on any elliptic curve. The talk is based on the series of joint works with A. Alexandrov, P. Dunin-Barkowski, M. Kazarian and S. Shadrin ( https://arxiv.org/abs/2309.12176, https://arxiv.org/abs/2406.07391, https://arxiv.org/abs/2412.18592).