Speaker:  Alexander Mikhailov (Leeds University)

Date and time:  5.06.2024, 17:00 (GMT +03:00)

Title: Deformations of noncommutative algebras and non-Abelian Hamiltonian systems

Abstract: By a well-known procedure, usually referred to as ``taking the classical limit'', quantum systems become classical systems, equipped with a Hamiltonian structure (symplectic or Poisson). From the deformation quantisation theory we know that a formal deformation of a commutative algebra A leads to a Poisson bracket on A and that the classical limit of a derivation on the deformation leads to a  Hamiltonian derivation on A defined by the Poisson bracket. In this talk I present a generalisation of it for formal deformations of an arbitrary noncommutative algebra A [1]. A deformation leads in this case to a commutative Poisson algebra structure on П(A):=Z(A)\times(A/Z(A)) and to the structure of a $П(A)-Poisson module on A, where Z(A) denotes the centre of A. The limiting derivations are then still derivations of A, but with the Hamiltonians belong to П(A), rather than to A. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the  Kontsevich integrable map, the quantum plane the quantised Grassmann algebra and quantisations of the  Volterra hierarchy [2], [3], [4].

This talk is based on a joint work with Pol Vanhaecke [1].

[1] Alexander V. Mikhailov and Pol Vanhaecke.  Commutative Poisson algebras from deformations of noncommutative algebras. arXiv:2402.16191v2, 2024.

[2] Alexander V. Mikhailov, Quantisation ideals of nonabelian integrable systems. Russ. Math. Surv., 75(5):199, 2020.

[3] Sylvain Carpentier, Alexander V. Mikhailov and Jing Ping Wang. Quantisation of the Volterra hierarchy. Lett. Math. Phys., 112:94, 2022.

[4] Sylvain Carpentier, Alexander V. Mikhailov and Jing Ping Wang. Hamiltonians for the quantised Volterra hierarchy. arXiv:2312.12077, 2023.

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

Speaker:  Vladimir Roubtsov (Universit'e d'Angers, LAREMA, ITTP RAS)

Date and time:  13.12.2023, 17:00 (GMT +03:00)

Title: Kontsevich and Buchstaber polynomials, multiplication kernels and differential operators of the Calabi-Yau type

Abstract: We are discussing several recent results of ongoing work (in collaboration with I. Gayur and D. By Van Straten and with V. Buchstaber and I. Gayur) about interesting properties of multiplicative generalized Bessel kernels, which include the well-known Clausen and Sonin-Gegenbauer formulas, examples of Kontsevich discriminant locus polynomials given as addition laws for special two-valued formal groups (Buchstaber-Novikov-Veselov), as well as the connection with "period functions" solving some Picard-Fuchs-type equations for the Calabi-Yau cases and related to analogues of Landau-Ginzburg superpotentials.

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

Speaker: P. Grinevich (Moscow State University, MIAN)

Date and time: 11.05.2022, 17:00 (GMT +03:00)

Title: Signatures on plabic graphs and completely non-negative Grassmannians II

Abstract: This is a continuation of the talk on 13.04.2022.

As shown by A. Postnikov the cells of completely non-negative Grassmannians can be rationally parametrized by graphs embedded in a disk with positive weights on the edges. In this case the matrix elements representing the Grassmannian points are given as sums along all possible paths from the boundary sources to the boundary sinks. An alternative approach is to define the Grassmannian points by solving a system of linear equations corresponding to the vertices of the graph. In this case positivity is achieved only with the correct choice of signs on the edges called a signature. T. Lam proved the existence of a signature consistent with the property of complete positivity without presenting it explicitly. We give an explicit construction and prove the uniqueness of such a signature up to the action of the natural gauge group.

The report is based on joint work with S. Abenda.

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

Speaker: M. Feigin (University of Glasgow)

Date and time: 16.02.2022, 17:00 (GMT +03:00)

Title: Rational and trigonometric solutions of WDVV and related equations 

Abstract:  I am going to discuss a class of solutions of WDVV equations which are constructed in terms of special collections of vectors with multiplicities. These solutions can be viewed as the trigonometric version of a class of rational solutions introduced by Veselov about 20 years ago. In the case of root systems rational solutions are almost dual to Frobenius manifold structure on the space of orbits of the Coxeter group. The class of rational solutions is closed under the natural operations of taking subsystems and restrictions. We show that similar operations can be applied in the trigonometric settings. In the case of root systems trigonometric solutions are expected to be almost dual to Frobenius manifold structures on the orbit spaces of the extended affine Weyl groups, and in the simply-laced cases they are known to describe quantum cohomology of the resolutions of simple singularities.

I am also going to discuss a very close relation between WDVV equations and the commutativity equations F_i F_j = F_j F_i. These equations appeared in the supersymmetric mechanics and they also admit special trigonometric and rational solutions. 

The talk is based on joint works with M. Alkadhem. 

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

Стартует новый набор в кружок по машинному обучению. 
Занятия будут проходить в центре, лаборатория Делоне, Комсомольская 3, по средам в 19.30. 

6 октября знакомимся с языком Python. 
Ждем студентов разных курсов, и всех интересующихся, главное желание и готовность трудиться. Будет интересно!

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:  Rinat Kashaev (University of Geneva,  Switzerland)

Date and time: 10.03.2021, 17:15 (GMT +03:00)

Title: The local Yang-Baxter and tetrahedron equations

Abstract:  I will review few constructions of solutions of Zamolodchikov’s tetrahedron equations, starting from Maillet-Nijhoff local Yang-Baxter equations in Korepanov’s form. In particular, I will discuss a set-theoretical solution on the group manifold of SL(2,R) which underlies the q-oscillator quantum solution of Bazhanov-Mangazeev-Sergeev and which implies the unitarity of the latter.

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