Speaker:  Evgeny Smirnov (HSE University, Independent University of Moscow)

Date and time: 30.09.2020, 17:00 (GMT +03:00)

Title: Schubert polynomials for the classical groups

Abstract: A classical result of Borel states that the cohomology ring of the full flag variety GL(n)/B is isomorphic to the polynomial ring in n variables modulo the ideal generated by the symmetric polynomials with zero constant term. On the other hand, this ring has a remarkable basis formed by the Schubert cycles, i.e. the classes of orbit closures of a Borel subgroup in GL(n). In 1970s-80s J. Bernstein, I. Gelfand and S. Gelfand and independently A. Lascoux and M.-P. Schützenberger constructed an explicit collection of representatives of the Schubert cycles, known as the Schubert polynomials. These polynomials have many nice combinatorial properties; they are obtained as the generating functions of certain diagrams (configurations of pseudolines), known as pipe dreams. In particular, their coefficients are positive integers.

The same problem can be considered for generalized flag varieties G/B of other classical groups: G=SO(n) and Sp(2n). The Schubert polynomials for the classical groups of types B/C/D were defined by S. Billey and M. Haiman in 1995; in 2011 T. Ikeda, L. Mihalcea and H. Naruse have studied their T-equivariant analogues, i.e. some “nice” representatives of Schubert classes in the T-equivariant cohomology ring of G/B. I am planning to describe analogues of pipe dreams for these cases, obtained in our recent joint work with Anna Tutubalina.

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

В осеннем семестре 2020 года запланирован спецкурс центра интегрируемых систем. 

Про что? 

Спецкурс будет состоять из двух модулей: введение в тензорный анализ и краткое введение в теорию алгебр и групп Ли.

Данные разделы глубоко встроены в язык и методы современной математики, многие задачи теории интегрируемых систем неразрывно связаны с этими областями. Будет уделяться особое внимание приложениям этих сюжетов в нелинейной динамике, математической физике, численных методах и других областях математики.

Для кого?

Приглашаются студенты 2-4 курса, аспиранты, магистранты, все заинтересованные.

Кто?

Преподавателями курса будут Маргарита Михайловна Преображенская (введение в тензорный анализ) и Дмитрий Валерьевич Талалаев (введение в теорию алгебр и групп Ли). 

Когда?

Спецкурс будет проходить по средам на четвертой паре (15:00).

Первая лекция состоится 16 сентября 2020 г.

Где?

7-й корпус, ауд. 309.

Тематический план

     1.  Тензорная алгебра

          a.  Двойственное пространство

          b.  Тензоры

          c.  Тензорное произведение

          d.  Внешнее произведение

          e.  Определитель и пфаффиан

     2.  Введение в теорию алгебр и групп Ли

          a.  Алгебры Ли, определение, первые примеры

          b.  Разрешимость, нильпотентность

          c.  Полупростые алгебры Ли

          d.  Форма Киллинга

          e.  Системы корней

          f.   Универсальная обертывающая алгебра

Speaker: Leonid Rybnikov (HSE, Moscow)

Date and time: 16.09.2020, 17:00 (GMT +03:00)

Title: Quantization of Drinfeld zastava spaces

Abstract: Drinfeld zastava is a certain closure of the moduli space of based maps from the projective line to the flag variety of the Lie algebra sl_n (also known as the monopole space). The natural (Atiyah-Hitchin) Poisson structure on the space of maps extends to the zastava space. We describe it in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The quantum Hamiltonian reduction of the corresponding quotient of its universal enveloping algebra produces a quantization Y of the coordinate ring of the zastava space . The same quantization was obtained by Gerasimov, Kharchev, Lebedev and Oblezin in 2004 as a subquotient of the Yangian Y(gl_n). We also generalize our construction to the case of the affine Kac-Moody algebra sl_n^.

(The talk  is based on the joint work with Michael Finkelberg arXiv:1009.0676)

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

Speaker: B. Bychkov (Higher School of Economics,  Moscow)

Date and time: 3.06.2020, 18:00 (GMT +03:00)

Title: Star-triangle transformation of the Potts model partition function as a solution for the tetrahedron equation and related combinatorial topics

Abstract: The report is devoted to several well-known functional relations in the theory of polynomial graph invariants, as well as some new interpretations. The identification of Tutte polynomials with partition functions of the Potts-type models seems to be an ideal possibility to apply the methods from the theory of integrable models of statistical physics to the combinatorics of graphs and vice versa.

I will present at least one proof of the fact that the parameter transformation, defining the invariance of the n=2 Potts model partition function under the star-triangle transformation, gives an orthogonal solution for the local Yang-Baxter equation and for the tetrahedron equation.

Using the Biggs duality on the space of Potts model partition functions I will present several results about the connection between the chromatic and flow polynomials and as a consequence obtain shifting order formulas on the space of Potts model partition functions.

The talk is based on the recent joint work with A.Kazakov and D.Talalaev.

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

Speaker: Michael Gekhtman (University of Notre Dame, USA)

Date and time: 20 May, 18.00 (Moscow time)

Title: Poisson-Lie Groups and Cluster Structures

Abstract: Coexistence of diverse mathematical structures supported on the same variety often leads to deeper understanding of its features. If the manifold is a Lie group, endowing it with a Poisson structure that respects group multiplication (Poisson– Lie structure) is instrumental in a study of classical and quantum mechanical systems with symmetries. 

On the other hand, the  ring of regular functions on certain  Poisson varieties can have a structure of a cluster algebra. I will discuss natural cluster structures in the rings of regular functions on simple complex Lie groups and their homogenous spaces and Poisson–Lie/Poisson-homogeneous structures compatible with these cluster structures. Much of this talk is based on an ongoing collaboration with M. Shapiro and     A. Vainshtein.

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

Speaker: Andrew N. W. Hone (University of Kent, UK)

Date and time: 22.04.2020, 16:00

Talk title: Efficient ECM factorization in parallel with the Lyness map  

Abstract:  The Lyness map is a birational map in the plane which provides one of the simplest discrete analogues of a Hamiltonian system with one degree of freedom, having a conserved quantity and an invariant symplectic form. As an example of a symmetric Quispel-Roberts-Thompson (QRT) map, each generic orbit of the Lyness map lies on a curve of genus one, and corresponds to a sequence of points on an elliptic curve which is one of the fibres in a pencil of biquadratic curves in the plane. Here we present a version of Lenstra's elliptic curve method (ECM) for integer factorization, which is based on iteration of the Lyness map with a particular choice of initial data. More precisely, we give an algorithm for scalar multiplication of a point on an elliptic curve, which is represented by one of the curves in the Lyness pencil. We explain how this might be implemented efficiently, using suitable projective coordinates, by performing the calculations in parallel using four processors, and make a brief comparison with state of the art methods. This is an introductory talk: no prior knowledge of factorization algorithms, or Lyness maps, is assumed.

To access the online seminar please contact a.tolbey@uniyar.ac.ru

Speaker: Georgy Sharygin (Moscow State University)

Date and time: 08.04.2020, 16:00

Talk title: On a noncommutative cross-ratio

Abstract: Abstract: the cross-ratio of four points on a projective line is one of the main projective invariants that finds the most unexpected applications in modern mathematics, from geometry and topology to the theory of integrable systems. I will talk about how we can extend the definition of the cross-ratio to the case when the projective line is replaced by its "noncommutative analog", that is, instead of projecting pairs of real or complex numbers, we consider "projectivization of a noncommutative algebra". It turns out that there is a way to do this so that most of the useful properties are preserved. Exploring the applications of the resulting expression is an interesting open problem. The report is based on joint work with V. Retakh and V. Rubtsov https://arxiv.org/abs/1905.01366.

Speaker: Anton Dzhamay, University of Northern Colorado, USA.
Date and time: 25.03.2020, 16:00.
Talk title: Discrete orthogonal polynomials and discrete Painlevé equations

Abstract: Suppose that in some discrete set of points on a line, say on natural numbers, a certain weight function is given, and we want to construct a set of polynomials orthogonal with respect to a given weight. The standard Gram-Schmidt procedure is not effective. A faster approach is to use a recursive procedure based on the so-called three-term linear relationship. But for many weights, the coefficients of this relation in a complex way depend on the recursion step. We will consider one such example where this dependence turns out to be given by a discrete Painlevé equation, and show how the general algebraic-geometric theory of Painlevé equations helps to work effectively with problems of this type. It turns out that the class of important applied problems in which discrete Painlevé equations arise is sufficiently large. One of the objectives of the talk is to show how to recognize and bring to a standard form equations of this type (joint work with Galina Filipuk (Warsaw) and Alexander Stokes (London)). (Based on https://arxiv.org/abs/1910.10981).

Докладчик: Горбунов Василий Геннадьевич (Абердинский университет, Шотландия; ВШЭ)
Тема: Полная положительность и кластерные структуры
Дата: 11 марта 2020 года (среда)

Аннотация:
Идея того, что у хороших математических объектов должна существовать "положительная часть" была явно оформлена в работах Люстига.  Изучая конкретный пример, положительную часть группы Gl(n), Беренштейн, Фомин и Зелевинский обнаружили чрезвычайно интересную структуру, ответственную за "положительность", которую сейчас называется кластерная алгебра. Как это часто бывает, кластерные алгебры очень скоро обнаружили во многих, совершенно не относящихся к первоначальной, областях математики. В докладе мы дадим обзор этих результатов.

Место проведения
7-й корпус ЯрГУ, аудитория 422

Докладчик: Ольга Витальевна Починка  (д.ф.-м.н., зав. лаб. динамических систем и приложений НИУ ВШЭ - Нижний Новгород)

Тема доклада: О классификации структурно устойчивых систем

Место: 418 ауд., 7-й корпус (Союзная, 144)

Аннотация: Нижний Новгород (ранее Горький) по праву считается местом рождения гиперболической теории. Основополагающая работа А.А. Андронова и Л.С. Понтрягина "Rough systems" положила начало исследованиям структурно устойчивых систем. В рамках лекции будут изложены классические и современные результаты по топологической классификации регулярных и хаотических структурно устойчивых систем.

Доклад проходит в рамках объединенного семинара Нелинейной динамики и Интегрируемых систем.