Seminar (online): A. Vesnin "Hyperbolic polyhedra and hyperbolic knots: the right-angled case"

Submitted by A.Tolbey on Fri, 02/19/2021 - 11:01

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].


[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

Event date
Wed, 02/24/2021 - 17:00

Seminar (online): I. G. Korepanov "Functional tetrahedron equation and related structures"

Submitted by A.Tolbey on Sat, 12/19/2020 - 10:05

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

Event date
Wed, 12/23/2020 - 17:00

Seminar (online): V. Roubtsov "Quantum uniformisation and CY algebras"

Submitted by A.Tolbey on Sat, 12/05/2020 - 12:40

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

Date and time: 9.12.2020, 17:00 (GMT +03:00)

Title: Quantum uniformisation and CY algebras

Abstract: In this talk, I will discuss a special class of quantum del Pezzo surfaces. In particular I will introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations. I will try to explain (at least a part of) terminology above.

To access the online seminar please contact  Anna Tolbey

Event date
Wed, 12/09/2020 - 17:00

Seminar (online): Anton Izosimov "The pentagram map, Poncelet polygons and commuting difference operators"

Submitted by A.Tolbey on Thu, 04/30/2020 - 17:52

Speaker: Anton Izosimov (University of Arizona, USA)

Date and time: 6.05.2020, 18.00 (Moscow time)

Title:  The pentagram map, Poncelet polygons and commuting difference operators

Abstract: The pentagram map is a discrete integrable system on the space of projective equivalence classes of planar polygons. By definition, the image of a polygon P under the pentagram map is the polygon P' whose vertices are the intersection points of consecutive shortest diagonals of P, i.e. diagonals connecting second nearest vertices. In the talk, I will discuss the problem of describing the fixed points of the pentagram map. In other words, the question is: which polygons P are projectively equivalent to their "diagonal" polygons P'? It is a classical result of Clebsch that all pentagons have this property. Furthermore, in 2005 R.Schwartz proved that this property is also enjoyed by all Poncelet polygons, i.e. polygons that are inscribed in a conic section and circumscribed about another conic section. In the talk I will argue that in the convex case the converse is also true: if P is convex and projectively equivalent to its diagonal polygon P', then P is a Poncelet polygon. The proof is based on properties of commuting difference operators, real elliptic curves, and theta functions.

To access the online seminar please contact  Anna Tolbey

Event date
Wed, 05/06/2020 - 18:00

Seminar: S.V. Smirnov "Discretisation of two-dimensional Toda Chains"

Submitted by skonstantin on Fri, 02/08/2019 - 17:05

Lecturer: Sergei Smirnov, MSU
Talk title: Discretisation of two-dimensional Toda Chains 
Time: February 20, at 16:00 
Venue: 7th building YarSU, seminar room 427 (144 Soyuznaya str., 150007)

It is well known that, in the continuous case, the two-dimensional Toda chains corresponding to the Cartan matrices of simple Lie algebras are Darboux integrable, that is, integrable in an explicit form, and the chains corresponding to the generalised Cartan matrices are integrable by the inverse scattering method. 

Although discrete versions of particular cases were considered earlier, in 2011 I.T. Khabibullin proposed a systematic method for the discretisation of the so-called exponential type systems (generalisation of Toda chains): the idea was to find a discretisation in which the characteristic integrals during the transition from the continuous to the semi-discrete model (and from the semi-discrete to the purely discrete) retain their form. The articles by Khabibullin et al. demonstrated that this method works for Toda chains of small length.

I will explain why this method works for discretising chains of arbitrary lengths of the series A and C and what is the progress in the question of the integrability of these discretisations in the general case.


Event date
Wed, 02/20/2019 - 16:00

A.V. Mikhailov "Polynomial integrable Hamiltonian systems on symmetric powers of plane curves"

Submitted by skonstantin on Sat, 12/01/2018 - 19:26

Abstract: We have found a quite general construction of commuting vector fields on the $k$-th symmetric power of $\mathbb{C}^{m}$ and tangent vector fields to the $k$-th symmetric power of an affine variety $V\subset\mathbb{C}^{m}$. Application of this construction to the $k$-th symmetric power of a plane algebraic curve $V_g$ of genus $g$ leads to $k$ integrable Hamiltonian systems on $\mathbb{C}^{2k}$ (or on $\mathbb{R}^{2k}$, if the base field is $\mathbb{R}$). In the case $k=g$, the symmetric power ${\rm Sym}^k(V_g)$ is birationally isomorphic to the Jacobian of the curve $V_g$, and our system is equivalent to well-known Dubrovin's system, which was derived and studied in the theory of finite-gap solutions (algebro-geometric integration) of the Korteweg–de Vries equation. We have found coordinates in which the obtained systems and their Hamiltonians are polynomial. For $k=2,\ g=1,2,3$ we present these systems explicitly and discuss the problem of their integration.

Venue: 7th corpus YarSU, lecture theatre 419

Event date
Wed, 12/05/2018 - 16:00

Lectures by S. Igonin "English for mathematicians"

Submitted by skonstantin on Mon, 11/26/2018 - 14:23

Sergei Igonin will give lectures on "English for mathematicians" for everyone.

The lectures will be taking place on Thursdays (November 29, December 6December 13) from 14:15 to 15:45.
The lectures are also planned to be continued in the next semester.

Venue: Lecture theatre 317, 7th building YarSU (144 Soyuznaya str.).

Learning materials can be found on the website.

Discussion of the following topics is planned.
1. Effective ways to learn English.
2. Features of the use of English in communicating about mathematics.
3. How to make reports and write mathematical texts in English.

Materials from leading universities in Moscow and the UK will be used.
The lecturer has working experience in English at the University of Leeds (United Kingdom).

Event date
Thu, 11/29/2018 - 14:15

Boris Bychkov "Symmetric Polynomials"

Submitted by skonstantin on Fri, 11/23/2018 - 11:16

Boris Bychkov (Higher School of Economics) will give a lecture on "Symmetric polynomials" for 1st and 2nd year students and school students.

The lecture will take place on November 21, 2018, at 17:30.

Venue: 8/10 Kirova st. (2nd building of YarSU), aud. 202.

Abstract: A polynomial in several variables is called symmetric if it is invariant with respect to any permutations of variables. The main theorem on symmetric polynomials asserts that any symmetric polynomial can be expressed in terms of elementary ones, and in a unique way. We shall obtain several more such sets of polynomials, and then we shall define Schur polynomials, a basis in the space of symmetric polynomials, parametrised by partitions (Young diagrams), and will discuss some interesting properties of this basis.

Event date
Wed, 11/21/2018 - 17:30

Mini-course by Vladimir Rubtsov for 1st and 2nd year students and school students

Submitted by skonstantin on Mon, 11/19/2018 - 17:09

Vladimir Nikolaevich Rubtsov from the University of Angers will give lectures for first and second year students and school students.

1nd and 2nd lecture: November 22, 2018, at 16:00

3rd and 4th lecture: November 23, 2018, at 16:00

Venue: 2nd corpus, YarSU, aud. 106.

Mini-course title: "Complex numbers, their generalisations, geometry and applications."

Abstract: In my lectures, I will talk about various extensions of natural and real numbers, the basic algebraic structures and operations associated with these extensions, as well as the conditions of complexity and their applications.

Event date
Thu, 11/22/2018 - 16:00