Speaker:  Tassos Bountis (P.G. Demidov Yaroslavl State University, Russia)

Date and time:  27.10.2021, 17:00 (GMT +03:00)

Title: Integrable and nonintegrable Lotka-Volterra systems

Abstract:  In recent years, there has been renewed interest in the study of anti-symmetric Lotka-Volterra Hamiltonian (LVH) systems of competing species. In particular, it is interesting to add linear (or nonlinear) terms to these systems, and either seek to preserve integrability, or investigate the dynamics of ''nearby'' nonintegrable systems in the n-dimensional phase space.

In this talk, I will first show how new integrable classes of LVH systems were discovered applying the Painlevé property, and then demonstrate that ''nearby'' non-integrable systems typically continue to possess very simple dynamics. Finally, I will discuss some very recent findings revealing possible connections between the Painlevé property and Brenig's method of integrating polynomial systems of ODEs by reduction to canonical form.

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

Speaker:  Alexander Shapiro (University of Edinburgh, Great Britain)

Date and time:  29.09.2021, 17:00 (GMT +03:00)

Title: Cluster structure on quantum Toda chain

Abstract:  Double Bruhat cells serve as phase spaces for a family of (degenerately) integrable systems, the generalized open Toda lattices. From cluster algebraic perspective double Bruhat cells were studied by Berenstein–Fomin–Zelevinsky and Fock–Goncharov, while the Toda lattice was investigated by Gekhtman–Shapiro–Vainshtein. In this talk I will describe cluster Poisson coordinates on double Bruhat cells, and use them to recover the quantum Toda lattice. If time permits, I will also briefly discuss our joint work with Gus Schrader on the eigenfunctions of the quantum Toda Hamiltonians.

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

Speaker:  Vladimir Dragović (UT Dallas)

Date and time:  14.07.2021, 17:00 (GMT +03:00)

Title: Integrable billiards, extremal polynomials, and combinatorics

Abstract:  A comprehensive study of periodic trajectories of the billiards within ellipsoids in the d-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of d intervals on the real line. Classification of periodic trajectories is based on a new combinatorial object: billiard partitions.

The case study of trajectories of small periods T, d ≤ T ≤ 2d is given. In particular, it is shown that all d-periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates d + 1-periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for d = 3.

The talk is based on the following papers:

V. Dragović, M. Radnović, Periodic ellipsoidal billiard trajectories and extremal polynomials, Communications. Mathematical Physics, 2019, Vol. 372, p. 183-211.

G. Andrews, V. Dragović, M. Radnović, Combinatorics of the periodic billiards within quadrics, arXiv: 1908.01026, The Ramanujan Journal, DOI: 10.1007/s11139-020-00346-y.

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

Speaker:  Adam Doliwa (University of Warmia and Mazury, Poland)

Date and time:  30.06.2021, 17:00 (GMT +03:00)

Title: Non-commutative birational maps satisfying Zamolodchikov's tetrahedron equation from projective geometry over division rings II

Abstract:  The report will be a continuation of the talk on June 16, which already introduced the general concepts of compatibility conditions in higher dimensions and the interpretation of the corresponding maps in terms of Desargues configurations   https://cis.uniyar.ac.ru/node/369

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

Speaker:  Andrei Zotov (Steklov Mathematical Institute RAS)

Date and time:  19.05.2021, 17:00 Moscow time (GMT +03:00) / 15:00 UK time

Title: On dualities in integrable systems

Abstract:  I will review main ideas underlying dualities in integrable systems including p-q (or Ruijsenaars) duality, spectral duality, quantum-classical duality and some other interrelations between integrable many-body systems and spin chains.

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

Speaker:  Adam Doliwa (University of Warmia and Mazury, Poland)

Date and time:  16.06.2021, 17:00 (GMT +03:00)

Title: Non-commutative birational maps satisfying Zamolodchikov's tetrahedron equation from projective geometry over division rings

Abstract:  The notion of multidimensional consistency is an important ingredient of the contemporary theory of integrable systems. In my talk I will focus on geometric origin of the multidimensional consistency of Hirota's discrete KP equation. Because the relevant geometric theorem is valid in projective geometries over division rings, we are led to non-commutative version of the equation, which is due to Nimmo. I will show how four-dimensional consistency of the discrete KP system gives the corresponding solution to Zamolodchikov's tetrahedron equation (generalization of the Yang-Baxter equation to more dimensions). In particular, different algebraic descriptions of the same geometric theorem lead to different (but of course equivalent) solutions of the equation. Finally, I will discuss how natural ultra-locality condition imposed on the solution gives Weyl commutation relations. The talk is based on joint works with Sergey Sergeev and Rinat Kashaev. 

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

Speaker:  Frank Nijhoff (University of Leeds, UK)

Date and time:  05.05.2021, 17:00 Moscow time (GMT +03:00) / 15:00 UK time

Title: Elliptic integrable systems on the space-time lattice

Abstract:  I will review the status of various integrable lattice equations (partial difference equations) of elliptic type, together with their integrability aspects, associated Lax systems, explicit solutions and higher-rank generalisations.

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

Speaker:  Vladimir Sokolov (Landau Institute for Theoretical Physics, Chernogolovka, Russia and UFABC, Sao Paulo, Brazil)

Date and time:  21.04.2021, 17:00 (GMT +03:00)

Title: Non-Abelian generalizations of integrable PDEs and ODEs

Abstract:  A general procedure for non-abelinization of а given integrable polynomial differential equation is described. We are considering NLS type equations as an example. We also find non-abelinating Euler's top. Results related to the Painlevé-2 and Painlevé-4 equations are discussed.

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

Speaker:  Pavlos Xenitidis (Liverpool Hope University, UK)

Date and time:  07.04.2021, 17:00 (GMT +03:00)

Title: Symmetries and Integrability of Difference Equations

Abstract:  Symmetries provide arguably the most reliable means to test and prove the integrability of a given equation. They are used in the analysis and classification of partial differential equations and differential-difference equations since 1970’s and 1980’s, and only very recently this approach has been extended to partial difference equations. In this talk I will consider a class of partial difference equations in two dimensions and discuss the general form of their symmetries. I will derive necessary integrability conditions for these equations and explain how they lead to symmetries and conservation laws. I will also demonstrate how symmetries can be used to find solutions and reduce a partial difference equation to discrete Painlevé type equations. Finally, I will discuss several extensions of the theory to other classes of scalar equations and to systems of difference equations.

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

Speaker:  Anton Dzhamay (University of Northern Colorado)

Date and time:  24.03.2021, 17:00 (GMT +03:00)

Title: Discrete Schlesinger equations and difference Painlevé equations

Abstract:  The theory of Schlesinger equations describing isomonodromic dynamic on the space of matrix coefficients of a Fuchsian system w.r.t. continuous deformations is well-know. In this talk we consider a discrete version of this theory. Discrete analogues of Schlesinger deformations are Schlesinger transformations that shift the eigenvalues of the coefficient matrices by integers. By discrete Schlesinger equations we mean the evolution equations on the matrix coefficients describing such transformations. We derive these equations, show how they can be split into the evolution equations on the space of eigenvectors of the coefficient matrices, and explain how to write the latter equations in the discrete Hamiltonian form. We also consider some reductions of those equations to the difference Painlevé equations, again in complete parallel to the differential case. 

This is a joint work with H. Sakai (the University of Tokyo) and T. Takenawa (Tokyo Institute of Marine Science and Technology).

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