Цель семинара — знакомство участников с различными идеями и конструкциями, связанными с деформационным квантованием, теорией квантовых групп и (более общо) некоммутативной геометрии. Темы, которые мы постараемся обсудить даны внизу, но порядок их обсуждения зависит от вкусов и запросов участников; при желании некоторые из темы могут быть отложены на потом, а вместо них будут рассмотрены новые темы, предложенные участниками семинара. Студентам, которые примут участие в работе семинара, будет предложено сделать один или несколько докладов на заинтересовавшую их тему (что будет учтено при простановке оценок). Участие в работе семинара в первом семестре не является необходимым для того, чтобы присоединиться к нам весной!
21 ноября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Anton Savin from People's Friendship University in Moscow (RUDN)
Тема: Index formulas for elliptic operators and cyclic cohomology
Abstract:
The famous theorem of Atiyah and Singer (1963) gives a formula for the Fredholm index of an elliptic operator on a smooth closed manifold in topological terms. Namely, the Fredholm index (an integer) is equal to the integral over the cotangent bundle of the manifold of the product of the cohomology class defined by the operator (the Chern character of the main symbol of the operator) and the Todd class of the cotangent bundle of the manifold. Later, index formulas were obtained for elliptic operators in many geometric situations.
In this talk, I will explain a formula for the index of elliptic operators associated with the actions of discrete groups on smooth manifolds. The index formula in this situation is also stated in terms of the Todd class. However, the Todd class in this case is defined in terms of the noncommutative geometry of Alain Connes and turns out to be an element of the cyclic cohomology of the symbol algebra.
Same in Russian:
Докладчик: Антон Савин из РУДН
Название: Формулы индекса эллиптических операторов и циклические когомологии
Аннотация:
Знаменитая теорема Атьи и Зингера (1963) даёт формулу для фредгольмова индекса эллиптического оператора на гладком замкнутом многообразии в топологических терминах. А именно, фредгольмов индекс (целое число) равен интегралу по кокасательному расслоению многообразия произведения класса когомологий, определяемого оператором (характера Черна главного символа оператора) и класса Тодда кокасательного расслоения многообразия. Позже формулы индекса были получены для эллиптических операторов во многих геометрических ситуациях.
В докладе будет рассказано о формуле индекса эллиптических операторов, ассоциированных с действиями дискретных групп на гладких многообразиях. Формула индекса в этой ситуации также формулируется в терминах класса Тодда. Однако, класс Тодда в данном случае определяется в терминах некоммутативной геометрии Алена Конна и оказывается элементом циклических когомологий алгебры символов.
14 ноября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Sotiris Konstantinou-Rizos from Yaroslavl State University
Тема: A method for solving integrable nonlinear PΔEs
Abstract:
It has become understood over the past few decades that integrable systems of partial difference equations (PΔEs) are interesting in their own right. On one hand, they model many processes in nature, the industry and the IT sector, and, on the other hand, they have many interesting algebro-geometric properties, as well as they are related to many important equations of Mathematical Physics such as the Yang-Baxter equation and the Zamolodchikov tetrahedron equation.
In this talk we present a discrete Darboux-Lax scheme for constructing solutions to quad-graph equations that do not necessarily possess the 3D-consistency property. As an illustrative example we use an Adler-Yamilov type of system that is the compatibility condition of two Darboux transformations for the nonlinear Schroedinger (NLS) equation. For this Adler-Yamilov system we construct 1- and 2- soliton solutions, starting from a simple seed one.
Moreover, we present integrable discretisations of a noncommutative NLS equation.
This talk is based on some recent results obtained in collaboration with P. Xenitidis and X. Fisenko.
7 ноября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Sotiris Konstantinou-Rizos from Yaroslavl State University
Тема: Integrable Systems, Darboux Transformations and Yang-Baxter maps
Abstract:
Darboux and Baecklund transformations are very important tools in the theory of Integrable Systems. On one hand they can be used to construct interesting solutions to integrable nonlinear PDEs starting from trivial ones, and on the other hand they constitute a bridge between integrable PDEs and integrable PΔEs. At the same time, the Yang-Baxter equation is one of the most fundamental equations of mathematical physics, and it has applications in a very wide range of fields of Mathematics and Physics; from geometry and representation theory, to statistical and quantum mechanics.
In this talk, I will present several Darboux transformations of NLS type and I will explain what integrable discretisation of PDEs via Darboux-Baecklund transforms means. Then, I will demonstrate the relation between discrete integrable systems and Yang-Baxter maps and will show how to use Darboux matrices for constructing Yang-Baxter maps. I will show commutative and noncommutative (on division rings) examples of Yang-Baxter maps of KdV, NLS and derivative NLS type.
This talk is based on some old results in collaboration with A.V. Mikhailov and P. Xenitidis and some new results with P. Xenitidis, X. Fisenko and A. Nikitina.
24 октября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Jean-Pierre Magnot from LAREMA, Angers and Lycee Jeanne d'Arc, Clermont-Ferrand
Тема: A zoo of well-posed generalized Kadomtsev-Petviashvili hierarchies and their $(t_2,t_3)$-Zakharov-Shabat equations
Abstract:
We first give in this exposition a summary of the techniques that actually prove well-posedness of the Kadomtsev-Petviahvili hierarchy understood as a system on an algebra of formal pseudo-differential operators with coefficients in a diffeological algebra. In a second part of this paper, we propose a guided tour of the multiple algebras to which this result can be applied, examining which PDEs can be derived from the Kadomtsev-Petviashvili hierarchy through the $(t_2,t_3)-$Zakharov-Shabat zero curvature equation. This talk is based on recent works with E.G. Reyes and V.Rubtsov.
17 октября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Oleg Sheinman from Steklov Mathematical Institute
Тема: Separation of Variables for Hitchin systems
Abstract:
There exist basically two methods of exact solution of finite dimensional integrable systems. These are the classical method of Separation of Variables (SoV), and Inverse Spectral Method which is a great modern achievement. Both of them apply to Hitchin systems. In this talk we focus on the method of Separation of Variables. It goes back to Hamilton and Jacobi, its modern form is due to Arnold and Sklyanin. Majority of classical (finite-dimensional) integrable systems had been resolved by means of SoV. As for Hitchin systems, Separation of Variables gives also a simplest way to define them.
In the talk, I shall define Hitchin systems by means of SoV and prove their integrability. By means of the method of generating functions (of symplectic geometry) I'll derive a fundamental fact that Hitchin trajectories are straight lines (windings) on certain Abelian varieties replacing Liouville tori in this context. Every Hitchin system by definition is related with a certain complex reductive group G referred to as the structure group. In the case G=GL(n) I'll give an explicit theta function formula for solutions. Then I'll explain that in the cases G=SO(n), G=Sp(2n) there emerges a certain obstruction for a similar theta function solution related to the peculiarities of the inversion problem for Prim varieties. If the time admits, I'll argue that the above definition of Hitchin systems is equivalent to the conventional one.
10 октября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Георгий Щарыгин
Тема: Deformation quantisation of the argument shift method n Ugl_n
Abstract:
In my talk, based on a joint work with Yasushi Ikeda and Alexander Molev I will explain the main idea of the argument shift method of Mischenko and Fomenko, which gives maximal commutative Poisson subalgebras in the symmetric algebras S(g), where g is a Lie algebra. After that I will describe the recent results concerning the way it can be transferred to the universal enveloping algebras.
3 октября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Nikita Safonkin from University of Reims Champagne-Ardenne
Тема: Yangian-type algebras and double Poisson brackets
Abstract:
Let A be an arbitrary associative algebra. With the help of Olshanski's centralizer construction one can define a sequence Y_1(A), Y_2(A),... of "Yangian-type algebras" (they possess a number of properties of the Yangians of series A). I will discuss a link between these Yangian-type algebras and a class of double Poisson brackets on free associative algebras. The talk is based on the joint paper with Grigori Olshanski arXiv:2308.13325.
26 сентября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Yuancheng Xie from Peking University
Тема: Space curves and solitons of the KP hierarchy
Abstract:
It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta (or Klein sigma) functions associated with hyperelliptic curves, and soliton solutions can be obtained by rational limits of the corresponding curves.
In this talk, I will associate a family of singular space curves indexed by the numerical semigroups〈l, lm+1, . . . , lm+k〉where m ≥ 1 and 1 ≤ k ≤ l−1 with a class of generalized KP solitons. Some of these curves can be deformed into smooth "space curves", and they provide canonical models for the l-th generalized KdV hierarchies (KdV hierarchy corresponds to the case l = 2). If time permits, we will also see how to construct the space curves from a commutative ring of differential operators in the sense of the well-known Burchnall-Chaundy theory.
This talk is based on a joint work with Professor Yuji Kodama.
19 сентября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Grigoriy Papayanov from University of Haifa
Тема: Formal geometry and deformation quantizations
Abstract:
Every smooth manifold, of any kind --- C-infinity, algebraic, complex analytic etc. is endowed with a natural algebra bundle together with a flat connection: the jet bundle. Every geometric structure on a manifold, understood in a broad sense, for example, as a differential operator between some tensor bundles, defines the corresponding structure on the jet bundle. It follows that if the geometric structure has a unique local form (like Darboux form for the symplectic structure), it is uniquely defined by the reduction of the structure group of the jet bundle, together with the structure Lie algebra of the flat connection. This observation is known as Gelfand-Kazhdan formal geometry. I want to explain in what sense the (symplectic) deformation quantization is a structure to which formal geometry mechanism is applicable, how to use it to give a conceptual explanation of the results of Fedosov-Nest-Tsygan-Bezrukavnikov-Kaledin on the classification of quantizations, and to which calculations this method might lead.
12 сентября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Andrey Smilga from the University of Nantes
Тема: Dynamical systems with higher time derivatives
Abstrac:
Mechanical and field theory systems with higher time derivatives in the Lagrangian are "ghost-ridden". This means that the spectra of the corresponding Hamiltonians have no bottom and include states with arbitrarily low energies. In most such nonlinear systems, these ghosts are "malignant" — the classical equations of motion have blow-up solutions (which run into a singularity at finite time), which lead to breaking of unitarity in quantum theory.
There are, however, some special nonlinear higher-derivative systems (in particular, exactly soluble systems) where the ghosts are "benign" and, in spite of the absence of the ground state in the spectrum, there is no blow-up and unitarity is preserved. We discuss in detail one of such systems.
We consider also the simplest nontrivial exactly soluble higher-derivative field systems with the equation of motion
u_{ttt} + 6uu_t + u_x = 0,
and
u_{ttt} + 6u^2u_t + u_x = 0.
These are the "rotated" KdV and the modified KdV system where the spatial coordinate and time change their roles. Higher time derivatives bring about the ghosts.
In the rotated KdV, the ghosts are "malignant" and there are blow-up solutions. But in the modified rotated KdV, such solutions seem to be absent: we present arguments that the ghosts are benign in this case, i.e. the classical dynamics of this system does not involve a blow-up. In addition, we solved the Cauchy problem for the rotated modified KdV numerically and did not see a trace of blow-up. Unfortunately, at some point, the solutions become unstable and there is a horizon, beyond which we could not go with the numerical methods that we used.