Speaker: K.S. Sorokin (HSE St. Petersburg)

Title: Ricci flows in effective resistance metric for community detection in graphs

Date and time:  29.10.2025, 17:00 (GMT +03:00)

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.

Video of the talk
https://rutube.ru/video/3da3e87640b20b30b9dee45d39adbcf8/

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker: Dmitry Artamonov (MSU)

Title: Functional realization of the  Gelfand-Tselin base

Date and time:  15.10.2025, 17:00 (GMT +03:00)

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.

Video of the talk
https://rutube.ru/video/d4f429fff185e32ed243438270f9fe95/

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker: Dmitry Timashev (MSU)

Title: Maximal Poisson commutative subalgebras and 2-splittings of semisimple Lie algebras

Date and time:  1.10.2025, 17:00 (GMT +03:00)

Abstract: A fundamental problem in the theory of integrable systems is to construct a complete involutive set of first integrals for a Hamiltonian dynamical system. Algebraically the problem amounts to construction of a commutative subalgebra of maximal transcendence degree in a given Poisson algebra. A powerful method of constructing Poisson commutative subalgebras in the symmetric algebra S(g) of a semisimple Lie algebra g, known as the Lenard-Magri scheme, is based on including the Lie-Poisson bracket on S(g) in a pencil of compatible Poisson brackets. If all Poisson brackets in the pencil are linear, i.e., come from a pencil of Lie brackets on g, then the Lenard-Magri scheme yields a Poisson commutative subalgebra of maximal transcendence degree if and only if all values of the parameter in the pencil are regular, i.e. the indices of the Lie algebras corresponding to different values of the parameter are one and the same. Panyushev and Yakimova (2021) suggested an implementation of the Lenard-Magri scheme by considering a pencil of Lie brackets associated to a 2-splitting of g, i.e., a decomposition into a direct sum of Lie subalgebras g = f + h. The possible singular values of the parameter correspond to the Inönü-Wigner contractions of g along f and h. Extending the results of Panyushev and Yakimova, we prove a formula for the index of an Inönü-Wigner contraction of g and deduce that the resulting Poisson commutative subalgebra of S(g) has maximal transcendence degree if and only if both f and h are spherical Lie subalgebras.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker: Mikhail Chirkov (HSE University, YarSU)

Title: Quantisation ideals and canonical parametrisations of the unipotent group

Date and time:  17.09.2025, 17:00 (GMT +03:00)

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.

Video of the talk
https://rutube.ru/video/private/6ffa6d381300f213c7e5024e3fbf1c3c/?p=GIt1GsJfJbBgqJhQPAIgFg

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker: Yunhe Sheng (Jilin University, Changchun, China)

Title: Leibniz 2-algebras, linear 2-racks and the Zamolodchikov Tetrahedron equation

Date and time:  9.07.2025, 17:00 (GMT +03:00)

Abstract: First we show that a central Leibniz 2-algebra naturally gives rise to a solution of the Zamolodchikov Tetrahedron equation. Then we introduce the notion of linear 2-racks and show that a linear 2-rack also gives rise to a solution of the Zamolodchikov Tetrahedron equation. We show that a central Leibniz 2-algebra gives rise to a linear 2-rack if the underlying 2-vector space is splittable. Finally we discuss the relation between linear 2-racks and 2-racks, and show that a linear 2-rack gives rise to a 2-rack structure on the group-like category. A concrete example of strict 2-racks is constructed from an action of a strict 2-group.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker: Leonid Chekhov (Steklov Mathematical Institute and Michigan State University)

Title: Symplectic groupoid: geometry, networks, and moduli spaces of closed Riemann surfaces

Date and time:  25.06.2025, 17:00 (GMT +03:00)

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.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker: G.I. Sharygin (MSU, MIPT)

Title: Symmetries of the full symmetric Toda system and Li-Bianchi integrability

Date and time:  11.06.2025, 17:00 (GMT +03:00)

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).

Video of the talk
https://rutube.ru/video/private/7a03bddb8b8e25ca6c2bbb261532c550/?p=UtuEJV79owNeV2L6a2uh5w

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker: Pavlos Xenitidis (Liverpool Hope University, UK)

Title: Noncommutative discrete KdV equations, their symmetries and reductions

Date and time:  21.05.2025, 17:00 (GMT +03:00)

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.

Video of the talk:
https://rutube.ru/video/private/ed06ef18476635b102f18858696669de/?p=mAE0iTB1Ur5bM05L5hRtpQ

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker: D. Talalaev (MSU,YarSU)

Title: On a family of Poisson brackets on gl(n) compatible with the Sklyanin bracket

Date and time:  23.04.2025, 17:00 (GMT +03:00)

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/2502.16925

Video of the talk
https://rutube.ru/video/01eb3527f465571105f0ed1d5a93b716/

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker: 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.

Video of the talk
https://rutube.ru/video/9a48e95f99c8b4f2c004d3d7f129be74/

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com