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!
19 ноября 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.310 и дистанционно в
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Идентификатор конференции: 89343866699
Код доступа: 127069 )
Докладчик: Vsevolod Chistyakov (Lomonosov Moscow State University)
Title: Chiral Anomaly and Non-conservation of Fermion Numbers in Instanton Fields
Киральная аномалия и несохранение фермионных чисел в полях инстантонов
Abstract:
In field theory, there are symmetries of the classical action that are violated after quantization. In this talk we will consider a particular case of such symmetry, chiral symmetry in the theory of massless spinor fields interacting with a gauge field. The violation of chiral symmetry leads to the non-conservation of fermion numbers on Euclidean classical solutions of gauge fields of a special kind, known as instantons. This effect is closely related to the index theorem.
Аннотация:
В теории поля существуют симметрии классического действия, которые нарушаются после квантования. В докладе будет рассмотрен частный случай такой симметрии - киральной симметрии в теории безмассовых спинорных полей, взаимодействующих с калибровочным полем. Нарушение киральной симметрии приводит к несохранению чисел фермионов на евклидовых классических решениях калибровочных полей специального вида, называемых инстантонами. Этот эффект тесно связан с теоремой об индексе.
12 ноября 2024 (вторник), 16:00, ОЧНО в НМУ в ауд.310 и дистанционно в
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Докладчик: Alexey Ilin (Skoltech and HSE Moscow)
Title: Bethe algebras and the argument shift algebras
Алгебры Бете и алгебры сдвига аргумента
Abstract:
I will speak about the relation of Bethe subalgebras in Yangians on one hand and the Gaudin/ argument shift subalgebras on the other hand. In particular this relation allows one to quantize the argument shift subalgebras for the Lie algebras of arbitrary type. The talk is based on a joint work with Leonid Rybnikov, arXiv2009.06934.
Я расскажу про связь подалгебр Бете в янгианах с одной стороны и подалгебр Годена/сдвига аргумента с другой. В частности, эта связь позваляет квантовать алгебры сдвига аргумента для алгебры Ли произвольного типа. Доклад основан на совместной работе с Леонидом Рыбниковым, arXiv2009.06934.
5 ноября 2024 (вторник), 18:30, ОЧНО в НМУ в ауд.310 и дистанционно в
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Докладчик: Pavel Saponov (HSE Moscow)
Title: Some recent results on quantum matrices: the spectrum and quantum derivatives
(gl(k), gl(n))-двойственность и гомоморфизм Ольшанского
Abstract:
Two quantum integrable systems, one related to the Lie algebra gl(k) and the other to the algebra gl(n) are called (gl(k), gl(n))-dual to each other, if the images of higher Hamiltonian functions of these systems coincide if we pass to a suitable representation. I will describe a previously unknown example of such duality, closely related with G.I.Olshanski's construction that describes the centralizers of the universal enveloping algebra U(gl(k)) inside U(gl(n+k)) as the image of the Yangian Y(gl(n)).
Две квантовые интегрируемые системы, одна связанная с алгеброй Ли gl(k), другая - с алгеброй Ли gl(n), называются (gl(k), gl(n))-двойственными, если образы алгебр высших гамильтонианов этих систем в каком-то представлении совпадают. Я расскажу о неизвестном ранее примере такой двойственности, связанном с конструкцией Г. И. Ольшанского, описывающей централизатор алгебры U(gl(k)) в алгебре U(gl(n+k)) как образ Янгиана Y(gl(n)).
22 октября 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.310 и дистанционно в
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Идентификатор конференции: 89343866699
Код доступа: 127069 )
Докладчик: Pavel Saponov (HSE Moscow)
Title: Some recent results on quantum matrices: the spectrum and quantum derivatives
Abstract:
I will give a brief overview of recent interesting results on quantum matrix algebras (QMA). In general, quantum matrix algebras are a wide class of associative algebras including such well known objects as the universal enveloping algebras and quantum groups. I'll talk about the notion of quantum spectrum and how it parameterize the center of some QMA including the universal enveloping algebra U(gl_n). Besides, the properties of quantum differential operators will be discussed, in particular, the matrix generalization of the Capelli identities.
15 октября 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.310 и дистанционно в
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Идентификатор конференции: 89343866699
Код доступа: 127069 )
Докладчик: Георгий Шарыгин
Title: Deformation quantisation of integrable systems: a naive approach
Abstract:
Recall that by Kontsevich's theorem every Poisson manifold M admits deformation quantisation: an associative but in general noncommutative *-product on the space of formal series in h with coefficients in smooth functions C(M), induced from the Poisson structure on M. Let now f_1,...,f_n be an involutive set of functions on M (i.e. {f_i,f_j}=0 for all pairs). One can ask if these functions can be extended to elements in the deformed algebra which would commute with respect to the *-product? In my talk (based on an old joint paper with D.Talalaev) I will describe a rather naive construction that gives a set of obstructions which vanish iff the answer to the question is positive: the last time I recalled the Hochschild cohomology construction and described (very briefly) the classes with values in this cohomology theory. This time I will briefly recall that result and continue to construct the obstructions with values in Lichnerowicz-Poisson cohomology groups.
8 октября 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.310 и дистанционно в
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Идентификатор конференции: 89343866699
Код доступа: 127069 )
Докладчик: Rostislav Potapov (ITMF, MSU Moscow)
Title: Moyal star product and functional integral
Abstract:
Recall that the Moyal star product provides deformation quantization of the standard symplectiс bracket on T^{*}R^{n}. I will show how to use correlators in the free theory to construct associative star product on functions, which coincides with the Moyal product.
This talk is based on the lectures by Nima Moshayedi: https://arxiv.org/abs/2012.14662
Название: Произведение Мойяла и функциональный интеграл
Аннотация:
Известно, что звездочка-произведение Мойяла дает деформационное квантование стандартной симплектической скобки на T^{*}R^{n}. Я расскажу о том, как с помощью корреляторов в свободной теории можно получить ассоциативное умножение функций, которой совпадает с произведением Мойяла.
Доклад основан на лекциях Нимы Мошайеди: https://arxiv.org/abs/2012.14662
1 октября 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.310 и дистанционно в
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Идентификатор конференции: 89343866699
Код доступа: 127069 )
Докладчик: Георгий Шарыгин
Title: Deformation quantisation of integrable systems: a naive approach
Abstract:
Recall that by Kontsevich's theorem every Poisson manifold M admits deformation quantisation: an associative but in general noncommutative *-product on the space of formal series in h with coefficients in smooth functions C(M), induced from the Poisson structure on M. Let now f_1,...,f_n be an involutive set of functions on M (i.e. {f_i,f_j}=0 for all pairs). One can ask if these functions can be extended to elements in the deformed algebra which would commute with respect to the *-product? In my talked (based on an old joint paper with D.Talalaev) I will describe a rather naive construction that gives a set of obstructions which vanish iff the answer to the question is positive.
17 сентября 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.310 и дистанционно в
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Идентификатор конференции: 89343866699
Код доступа: 127069 )
Докладчик: Alexander Molev (University of Sydney, Australia)
Title: Quantum Casimir elements and higher Capelli identities
Abstract:
We consider central elements in the quantized enveloping algebra of type A. A family of such elements (quantum Gelfand invariants) was constructed in a landmark paper by Reshetikhin, Takhtadzhyan and Faddeev (1989). We will calculate the eigenvalues of the invariants acting in irreducible highest weight representations. We will also construct q-immanant polynomials whose coefficients are central elements of the quantized enveloping algebra parameterized by arbitrary Young diagrams. The constant terms coincide with the elements arising from the general construction of Drinfeld (1989) and Reshetikhin (1989) as their Harish-Chandra images are the classical Schur polynomials. For another special value of the parameter, we get q-analogues of Okounkov's quantum immanants (1996). We also consider the action of the associated reflection equation algebra in the braided Weyl algebra of Gurevich, Pyatov and Saponov (2012) and prove q-analogues of the higher Capelli identities generalizing the work of Gurevich, Petrova and Saponov (2022). This is joint work with Naihuan Jing and Ming Liu.
10 сентября 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.310 и дистанционно в
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Идентификатор конференции: 89343866699
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Докладчик: Rostislav Potapov (ITMF, MSU Moscow)
Title: Poisson-Lie groups and quantisation of constant r-matrices
Abstract:
One of the Drinfeld's theorems allows to construct quantisation of constant skew symmetric r-matrices.
I'll tell about appearance of such r-matrices in Poisson-Lie groups and how this quantisation is connected with the deformation quantisation
Название: Группы Пуассона-Ли и квантование постоянных r-матриц
Аннотация:
Одна из теорем Дринфельда позволяет построить квантование постоянных антисимметричных r-матриц.
Я расскажу о том, как такие r-матрицы возникают в группах Пуассона-Ли и о том, как это связано с деформационным квантованием
11 июня 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.401 и дистанционно в
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Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Alexander Mikhailov (University of Leeds, UK)
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 stucture (symplectic or Poisson). From the deformation quantisation theory we know that a formal deformation of a commutative algebra $\cA$ leads to a Poisson bracket on $\cA$ and that the classical limit of a derivation on the deformation leads to a Hamiltonian derivation on $\cA$
defined by the Poisson bracket. In this talk I present a generalisation of it for formal deformations of an arbitrary noncommutative algebra $\cA$. I will show that a deformation leads to a commutative Poisson algebra structure on $\Pi(\cA):=Z(\cA)\times(\cA/Z(\cA))$ and to the structure of a $\Pi(\cA)$-Poisson module on $\cA$, where $Z(\cA)$ denotes the centre of $\cA$. The limiting derivations are then still derivations of $\cA$, but with the Hamiltonians belonging to $\Pi(\cA)$, rather than to $\cA$. 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.
This talk is based on a joint work with Pol Vanhaecke.
4 июня 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.401 и дистанционно в
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Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Armen Sergeev (Steklov Mathematics Institute, Moscow, Russia)
Title: Mathematical questions in the theory of topological dielectrics
Abstract:
The subject of this talk is the topological insulator theory . In addition to it being an important part of theoretical Physics, it is closely connected to various branches of pure Mathematics, such as Homotopy theory, K-theory, noncommutative Geometry and Clifford Algebras.
Topological insulators are characterised by a wide energy gap, stable with respect to small perturbations. This observation implies that topological methods can be used to study them.
Symmetries of these objects play the key role in their study. One can trace the description of the possible symmetry types to the works of Kitaev, who suggested a classification of topological insulators based on the description of their symmetry groups and their representations.
In my talk I will pay special attention to topological insulators invariant with respect to the inversion of time. An important example of such systems is the quantum spin Hall dielectric, that has a nontrivial topological mod-2 invariant, introduced by Kane and Mele. Mathematical theory of such insulators is related with the study of topological involutive spaces and quaternion bundles.
28 мая 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.401 и дистанционно в
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Идентификатор конференции: 899 4326 1657
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Докладчик: Dimitry Gourevitch (Valenciennes, France)
Название: Алгебра Уравнения Отражений vs Квантовые группы
Аннотация:
Я предполагаю сравнить свойства и приложения этих объектов, в том числе
к интегрируемым системам. Также хочу обсудить возможные приложения к весовым системам.
21 мая 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.401 и дистанционно в
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Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Rostislav Potapov (ITMP, Lomonosov Moscow State University, Russia)
Тема: Classical r-matrix and Babelon-Viallet theorem/Классическая r-матрица и теорема Бабелона-Виалле
Abstract:
Classical r-matrix arises in the problem of Poisson structures for integrable models in classical mechanics. I will tell about Babelon–Viallet theorem, which ensures the existence of the classical r-matrix for models with diagonalizable Lax matrix. Several examples will be provided, including Poisson–Lie groups. If there is time left, I will talk about the deformation quantization of the r-matrix bracket on the Poisson-Lie group, leading to the quantum Yang-Baxter equation.
Аннотация:
Классическая r-матрица возникает в задачах о Пуассоновой структуре интегрируемых систем классической механики. Я расскажу о теореме Бабелона-Виалле, гарантирующей существование классической r-матрицы для систем с диагонализируемой матрицей Лакса. Будет приведено несколько примеров r-матричных Пуассоновых структур, включая группы Пуассона-Ли. Если останется время, я расскажу о деформационном квантовании r-матричной скобки на группе Пуассона-Ли, приводящему к квантовому уравнению Янга-Бакстера
14 мая 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.401 и дистанционно в
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Идентификатор конференции: 899 4326 1657
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Докладчик: Daria Maksimenko (ITMP, Lomonosov Moscow State University, Russia)
Тема: Clifford algebras and their applications/Алгебры Клиффорда и их приложения
Abstract:
In my talk I will tell about Clifford algebras, their definitions and basic properties, as well as about important subgroups in the group of their invertible elements. After this I will tell about their applications in Geometry (Cartan-Dieudonne's theorem, and the study of vector fields on spheres), and in Physics (Majorana and Weyl spinors).
Аннотация:
Я расскажу про алгебры Клиффорда, дам основные определения и опишу важные подгруппы в их группах единиц. После этого я опишу их приложения в геометрии (теорему Картана-Дьёдонне и описание векторных полей на S^N) и в физике (майорановские и вейлевские спиноры).
7 мая 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.401 и дистанционно в
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Идентификатор конференции: 899 4326 1657
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Докладчик: Alexei Aleshin (Lomonosov Moscow State University, Russia)
Тема: Beable-guided measurement theory
Абстракт:
Measurement model in de Broglie-Bohm theory is developed. Following von Neumann principles, both the measurement device and the target system are described by the same quantum laws. This investigation includes basic models of coordinates and momentum measurements. Equations for pilot waves are solved analytically, and de Broglie particles trajectories are calculated numerically. The resulting model resolves the issues related to probability distribution in momentum space that were addressed in the works \cite{kurt,naun,heim}. Origin of quantum contextuality in the de Broglie-Bohm theory is discussed. Further, a new problem is considered: whether the de Broglie-Bohm theory admits Heisenberg uncertainty principle in special thought experiment, where momentum and coordinate are measured in turns several times. By direct calculation, we show that uncertainty relation is restored due to turbulence-like perturbations. We consider also the measurement models which do not admit this mechanism and thus may open up new possibilities for experimental testing of the de Broglie-Bohm theory.
В представленной работе моделируется и анализируется поведение квантовой системы при измерении в теории де Бройля-Бома.
Теория де Бройля-Бома - математическое обобщение квантовой теории, включающее помимо уравнения Шредингера для волновой функции, также и уравнение для траекторий элементарных частиц - уравнение ведения. Устанавливаются и решаются следующие проблемы:
1) Какого взаимоотношение пространства наблюдаемых состояний (пространства Гильберта) и пространства скрытых траекторий Бомовских частиц (которое в классическом пределе деформируется до фазового пространства)?
2) Как решается проблема измерения в аксиоматике данной теории?
3) Возможно ли нарушение принципа неопределенностей Гейзенберга и принципа контекстуальности в теории с явными траекториями частиц?
Для решения поставленных задач моделируется простейшее проективное измерение координаты и импульса. Полученные результаты воспроизводят правило Борна в импульсном представлении, позволяют понять происхождение квантовой контекстуальности и необратимости в процессе измерения. Анализируется ряд моделей измерений с возможным нарушением принципа неопределенностей Гейзенберга. Обнаруживается целый класс моделей измерения, в которых данный принцип нарушается. Устанавливается связь между наблюдаемыми и скрытыми переменными.
9 апреля 2024 (вторник), 18:00, дистанционно в
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Идентификатор конференции: 899 4326 1657
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Докладчик: Christofer Brav (MIPT, Dolgoprudnyi, Russia)
Тема: Poisson brackets and the string Lie algebra of a Calabi-Yau category
Абстракт:
Goldman defined a symplectic structure on the moduli space of local systems on a closed oriented surface, constructed a collection of natural Hamiltonians on the moduli space by taking trace of monodromy around loops on the surface, and computed Poisson brackets among these Hamiltonians in terms of what is now called the Goldman bracket on free homotopy classes of loops. Chas and Sullivan generalized the Goldman bracket to a string bracket on the degree-shifted equivariant homology of the free loop space of a closed oriented manifold of any dimension, but the compatibility with the corresponding shifted-symplectic geometry on the moduli space of local systems remained mostly conjectural. We generalize these results of Goldman and of Chas-Sullivan to higher dimensional 'non-commutative' closed oriented manifolds in the form of smooth Calabi-Yau categories. Our main results are the description of a chain-level 'string Lie bracket' on cyclic chains of a smooth Calabi-Yau category and the intertwining of this string Lie bracket on cyclic chains with the shifted Poisson bracket on functions on the moduli space of objects in the category.
26 марта 2024 (вторник), 18:00, дистанционно в
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Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Alexander Chervov (Institute Curie, Paris, France)
Тема: Применение методов машинного обучения к задачам конечных групп (диаметр, рост, короткий путь)
Абстракт:
Задача оценки диаметров (расстояния между наиболее удаленными вершинами) графов Кэли конечных групп - привлекает усилия лучших математиков, включая Т.Тао. Пример нерешенной гипотезы - при любом выборе образующих в S_n диаметр будет не более O(n^2). Мы расскажем о применении методов машинного обучения к подобным задачам. На "Kaggle" ( принадлежит Google ) недавно был конкурс - построение наиболее коротких путей на графах Кэли групп (группы кубика Рубика и близкие). Конкурс показал, что методы машинного обучения пока работают хуже чем классические солверы кубиков - тема еще не сильно исследована специалистами по машинному обучению. Что создает отличную возможность - кто хочет присоединиться к исследованиям в этой области - дайте знать.
19 марта 2024 (вторник), 18:00, дистанционно в
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Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Vladimir Rubtsov (Universite Angers, France)
Тема: Higher Bessel kernels, Buchstaber and Kontsevich polynomials and beyond, part 2
Abstract:
I describe two families of polynomials in C[x,y,z] which are:
1) S_3 -invariant;
2) quadratic in each single variable;
3) discriminant type and have in its turn split discriminants in each variable;
I shall discuss importance and appearance of these two families in different domains of mathematics and shall try to show some of their applications.
5 марта 2024 (вторник), 18:00, дистанционно в
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Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Vladimir Rubtsov (Universite Angers, France)
Тема: Higher Bessel kernels, Buchstaber and Kontsevich polynomials and beyond
Abstract:
I describe two families of polynomials in C[x,y,z] which are:
1) S_3 -invariant;
2) quadratic in each single variable;
3) discriminant type and have in its turn split discriminants in each variable;
I shall discuss importance and appearance of these two families in different domains of mathematics and shall try to show some of their applications.
27 февраля 2024 (вторник), 18:00, дистанционно в
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Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Alexander Zheglov (MSU, Moscow)
Тема: Theory of normal forms for ODOs, II
Abstract:
This is a continuation of my talk. I'll discuss necessary conditions on Schur operators of ODOs and on normal forms of non-commuting ODOs. After that I'll talk about some applications of the theory to the study of commuting operators in Weyl algebra and its connections to the Dixmier conjectures.
20 февраля 2024 (вторник), 18:00, ОЧНО в НМУ в ауд.401 и дистанционно в
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Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Alexander Zheglov (MSU, Moscow)
Тема: Theory of normal forms for ODOs
Abstract:
Normal forms of differential operators are obtained after conjugation by some invertible operator («Schur operator»), calculated using one of the operators in a ring. Normal forms of commuting operators are polynomials with constant coefficients in the differentiation, integration and shift operators, which have a finite order in each variable, according to a theorem from https://arxiv.org/abs/2205.06790.
Normal forms can be calculated for any given commuting operators. If the operators do not commute, their normal forms will in general be given by a series, for which, however, it is possible to calculate any given number of terms. It is expected that the study of normal forms of commuting operators can subsequently help to solve the problem of finding explicit examples of commuting differential or difference operators of arbitrary rank, in the study of commuting operators in Weyl algebra, as well as in the description of the structure of the moduli space of torsion free sheaves with a fixed Hilbert polynomial on the spectral manifold of a ring of commuting operators of arbitrary dimension. I'll talk about recent investigations in the theory of normal forms. The talk is based on an ongoing work with Junhu Guo.
13 февраля 2024 (вторник), 16:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Irina Bobrova (MPI, Leipzig)
Тема: Towards non-Abelian recurrences
Abstract:
Similar to the continuous case, some famous O∆Es are results of a reduction of integrable P∆Es. From this point of view, the discrete Hirota equation also known as the discrete KP or two-dimensional discrete Toda lattice (2d dTL) is a master equation. By using the plane-wave reduction, it reduces to the two-term version of the Gale-Robinson sequence sometimes called the Somos-N like equation (arXiv:1705.01094). The latter is closely connected with q-difference Painlevé equations (arXiv:1405.5379). In this talk, we will discuss a non-abelian version of this scheme: 2d dTL -> Somos-N -> q-Painlevé. The talk is based on arXiv:2311.11124.
22 декабря 2023 (пятница), 17:30, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Piotr Grinevich (Landau Institute)
Тема: KP equations, M-curves and nonnegative Grassmannians
Abstract:
In the theory of KP2 equations there exist two methods to determine real nonsingular multi-soliton solutions (whose geometrical properties are quite intriguing and closely related to Tropical Geometry). The first method is by doing Darboux' transformations in the points of a completely nonnegative Grassmannians. The second method is by degenerating M-curves with properly chosen divisors. Both objects (i.e. nonnegative Grassmanians and M-curves) show up in many different problems in various branches of Mathematics. The purpose of our work was to investigate the way in which these two objects are related with each other. In more formal terms, to find a method to associate a degenerate M-curve with a divisor on it to a point in nonnecgative Grassmannian, so that the corresponding KP2-solutions are the same.
Based on joint papers with S.Abenda
Same in Russian:
Докладчик: П.Г.Гриневич из ИТФ им Л.Д.Ландау РАН
Название: Уравнения КП, М-кривые и неотрицательные грассманианы
Аннотация:
В теории уравнения КП2 есть два способа строить вещественные регулярные многосолитонные решения (кстати, их геометрия сама по себе очень интересна и связана с тропической геометрией). Первый -- с помошью преобразования Дарбу по точкам вполне неотрицательного грассманиана. Второй -- вырождая M-кривые с правильно расположенными дивизорами. И вполне неотрицательные грассманианы и M-кривые возникают в во многих задачах и совершенно различных областей математики. Целью наших работ было дать ответ на вопрос -- как эти два объекта связаны. Более формально -- как по точке вполне неотрицательного грассманиана построить вырожденную M-кривую и дивизор на ней, дающие то же решение?
Рассказ основан на совместных работах с Симонеттой Абендой.
Please note, that THIS TIME OUR SEMINAR WILL BE HELD OFFLINE, in IUM (11, B.Vlassievski pereulok, room 303), at 17:30.
5 декабря 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Yasushi Ikeda (Moscow State University)
Тема: Second order quantum argument shifts in general linear Lie algebras
Abstract:
Let g be a Lie algebra. Argument shift subalgebras in Sg is the Poisson commutative subalgebra (with respect to the Lie-Poisson structure), spanned by the iterated derivations of the central elements in Sg with respect to a constant vector field. Inspired by quantum partial derivative operators on Ugl proposed by Gurevich, Pyatov, and Saponov, I and Georgiy Sharygin showed that the quantum argument shift algebras are generated by the corresponding iterated quantum shifts of the central elements in Ugl (later Alexander Molev wrote the computations in an advanced manner). Thus the quantum argument shift operator gives us an alternative way to find generators of the quantum argument shift subalgebra. In this talk, I introduce a general formula for calculating the second order quantum argument shift of an arbitrary central element in Ugl and obtain generators of the quantum argument shift subalgebra (up to the second order). Behind this there is some interesting combinatorial observation.
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 Шарыгин
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.