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:  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:  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:  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

Speaker:  Andrei Vesnin (Tomsk State University)

Date and time: 24.02.2021, 17:00 (GMT +03:00)

Title: Hyperbolic polyhedra and hyperbolic knots: the right-angled case

Abstract:  A polyhedron is said to be right-angled if all its dihedral angles are equal to pi/2. Three-dimensional hyperbolic manifolds constructed from bounded right-angled polyhedra have many interesting properties [1]. Inoue [2,3] initiated enumerating bounded right-angled hyperbolic polyhedra by their volumes. Atkinson obtained low and upper bounds of volumes via vertex number [4]. In [5] we enumerate ideal (with all vertices at infinity) right-angled hyperbolic polyhedra. The obtained results imply that the right-angled knot conjecture from [6] holds for knots with small crossing number. Atkinson’s upper bounds were improved in [7,8] for bounded and ideal cases both. Finally, we will discuss the relation of results from [5] with the maximum volume theorem from [9].

The talk is based on joint results with Andrey Egorov [5,7,8].

References.

[1] A. Vesnin, Right-angled polyhedra and hyperbolic 3-manifolds, Russian Mathematical Surveys 72 (2017), 335-374.

[2] T. Inoue, Organizing volumes of right-angled hyperbolic polyhedra, Algebr. Geom. Topol. 8 (2008), 1523-1565.

[3] T. Inoue, Exploring the list the smallest right-angled hyperbolic polyhedra, Experimental Mathematics 2019, published online.

[4] C. Atkinson, Volume estimates for equiangular hyperbolic Coxeter polyhedra, Algebr. Geom. Topol. 9 (2009), 1225-1254.

[5] A. Vesnin, A. Egorov, Ideal right-angled polyhedra in Lobachevsky space, Chebyshevskii Sbornik 21 (2020), 65-83.

[6] A. Champanerkar, I. Kofman, J. Purcell, Right-angled polyhedra and alternating links, arXiv:1910.13131.

[7] A. Egorov, A. Vesnin, Volume estimates for right-angled hyperbolic polyhedra, Rendiconti dell’Instituto di Matematica dell’Universita di Trieste 52 (2020), 565-576.

[8] A. Egorov, A. Vesnin, On correlation of hyperbolic volumes of fullerenes with their properties, Comput.  Math. Biophys. 8 (2020), 150-167.

[9] G. Belletti, The maximum volume of hyperbolic polyhedral, Trans. Amer. Math. Soc. 374 (2021), 1125-1153.

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

Speaker:  Igor G. Korepanov

Date and time: 23.12.2020, 17:00 (GMT +03:00)

Title: Functional tetrahedron equation and related structures

Abstract: We show how the functional tetrahedron equation appears, together with its solutions, from almost nothing -- lines in a plane and simple linear algebra. Then, a discrete-time dynamical system arises naturally, with an algebraic curve conserved in time, and divisors moving linearly along their Picard group. Finally, a reduction of this system is constructed, for which the conserved quantities turn into the partition function of an inhomogeneous six-vertex free-fermion model.

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