curriqulum 24/25

Research seminar – Modern Problems of Mathematical Physics (Term 1-4) / DG120268 / 24-26

Instructors: Andrei Marshakov, Alexey Litvinov

Classical groups, their invariants and representations (Term 1-2) / MA060690 / 24-26

Instructor: Grigori Olshanski

The title of the course is deliberately copied from the famous book by Hermann Weyl (1939; 1946). The material in the book forms the core of representation theory. For this reason, working through this material is useful for everyone who wants to deal with any problems in representation theory or apply its results. The purpose of the course is to introduce students to the main ideas and results of Weyl’s book, as well as to their further development. Of course, in addition to Weyl’s book, we will use other, more modern sources.

Geometric representation theory (Term 1-2) / DA060271 / 24-26

Instructor: Michael Finkelberg

Geometric representation theory applies algebraic geometry to the problems of representation theory. Some of the most famous problems of representation theory were solved on this way during the last 40 years. The list includes the Langlands reciprocity for the general linear groups over the functional fields, the Langlands-Shelstad fundamental Lemma, the proof of the Kazhdan-Lusztig conjectures; the computation of the characters of the finite groups of Lie type. We will study representations of the affine Hecke algebras using the geometry of affine Grassmannians (Satake isomorphism) and Steinberg varieties of triples (Deligne-Langlands conjecture). This is a course for master students knowing the basics of algebraic geometry, sheaf theory, homology and K-theory.
Full Syllabus

Introduction in algebraic topology (Term 1-2) / MA060691 / 24-26

Instructor: Maxim Kazarian

Symplectic geometry is a kind of skew-symmetric analogue of Riemannian geometry. This domain of differential geometry serves as a geometric basis of calculus of variations, quantum and classical mechanics, geometric optics and thermodynamics. The language of symplectic geometry is used everywhere in modern mathematics: in the Lie group theory, the theory of differential equations, integrable systems, singularity theory, topology. The course will cover both basic concepts of the theory and some specific problems needed for particular applications.

Cluster varieties and integrable systems (Term 1-2) / MA060597 / 24-26

Instructor: Andrei Marshakov

We start with definition of the cluster Poisson varieties from the angle of view of Hamiltonian mechanics and geometry of Poisson manifolds. Then we turn to description of the basic features of the Fock-Goncharov construction for the Lie groups, being (together with the spaces of mould of flat connections) an important example of Poisson manifold, possessing cluster structure. Then we formulate main properties of integrable systems on Poisson submanifolds in Lie groups. Finally, we plan to discuss the equivalence of the class of such systems for A-series with their combinatorial construction of Goncharov and Kenyon, and possibly their deautonomization and relation with supersymmetric gauge theories.

Cluster integrable systems (Term 1-2) / MA060692 / 24-26

Instructor: Mikhail Bershtein

Characteristic classes (Term 1-2) / MA060693 / 24-26

Instructor: Alexander Gaifullin

The course will include introduction to theory of characteristic classes, namely, the Stiefel-Whitney, Chern and Pontryagin classes of vector bundles and the Miller-Morita-Mumford classes of fibre bundles with fibre an oriented surface. The exposition will be based on the splitting principle and on the computation of the cohomology rings of the Grassmann manifolds. We will also discuss the connection with theory of cohomological operations (Steenrod squares), including the formulae due to Wu and Thom.

Path integral: stochastic processes and basics of quantum mechanics (Term 1-2) / MA060542 / 24-26

Instructor: Andrei Semenov

One of the most powerful methods of modern theoretical physics is the method of functional integration or path integration. The foundations of this approach were developed by N. Wiener at the beginning of the 20th century, but it spread widely after R. Feynman, who applied this approach in quantum mechanics. At present, the functional integral has found its application in the theory of random processes, polymer physics, quantum and statistical mechanics, and even in financial mathematics. Despite the fact that in some cases its applicability has not yet been mathematically rigorous proven, this method makes it possible to obtain exact and approximate solutions of various interesting problems with surprising elegance. The course is devoted to the basics of this approach and its applications to the theory of random processes and quantum mechanics. In the first part of the course, using the example of stochastic differential equations, the main ideas of this approach will be described, as well as various methods for exact and approximate calculation of functional integrals. Further, within the framework of the course, the main ideas of quantum mechanics will be considered, and both the operator approach and the approach using functional integration will be considered. It will be demonstrated that, from the point of view of formalism, the description of random processes and the description of quantum mechanical systems are very similar. This will make it possible to make a number of interesting observations, such as, for example, the analogy between supersymmetric quantum mechanics and the diffusion of a particle in an external potential. In the final part of the course, depending on the interests of the audience, various applications of the functional integration method will be discussed, such as polymer physics, financial mathematics, etc.
Full Syllabus

Introduction to two-dimensional conformal field theory (Term 1-2) / MA060694 / 24-26

Instructor: Alexey Litvinov

Conformal field theories are relatively simple quantum field theories that serve as starting points for perturbation theory for more generic quantum field theories with the mass gap. In two dimensions, unlike higher dimensions, the algebra of conformal transformations is infinite-dimensional. As a result, it has been possible to exactly solve certain nontrivial two-dimensional conformal field theories. This course provides introduction to basic concepts of two-dimensional conformal field theory. We will review basic ideas of the bootstrap approach to quantum field theory and describe the mathematical structures that appear in conformal field theory: representation theory of the Virasoro algebra, differential equations of correlation functions, conformal blocks etc.

Integrable systems of classical mechanics (Term 1-2) / MA060695 / 24-26

Instructor: Vadim Prokofev

The course is devoted to the study of integrable systems, their general properties and methods applied to their study them. We will use the example of integrable systems of classical mechanics. Using the example of such models as the Calogero-Moser, Ruijenaars-Schneider, Toda systems, we will talk about the Lax representation, r-matrix, Bäcklund transformations – methods used for integrable systems in general. In addition, attention will be paid to the relationship of integrable systems with each other and with other branches of mathematical physics.

Derived equivalences of Satake type (Term 1-2) / MA060696 / 24-26

Instructor: Artem Prikhodko

Hamiltonian mechanics (Term 1-2) / MA060697 / 24-26

Instructor: Vladimir Poberezhny

Hamiltonian mechanics is one of basic disciplines in mathematical physics. The scope of our course is to introduce students to modern views on the fundamentals of the theory of integrable systems and mathematical physics. Mastering its program makes it possible the further study of advanced courses of mathematical physics.
The mathematics of the modern theory of Hamiltonian systems includes methods of the theory of differential equations and dynamical systems, Lie groups and algebras and their representations, symplectic and Poisson geometry, analysis on manifolds and many others. Acquiring practical skills in applying the techniques and structures of these branches of mathematics, the ability to combine them to solve problems in mechanics is one of the main goals of this course. The course can be recommended not only to students of mathematical physics programm, but also to those planning to specialize in pure mathematics or its applications.

Around the Ising model in 20 hours (Term 3-4) / MMA060698 / 24-26

Instructor: Semen Shlosman

Methods of conformal field theory for quantum field theory and string theory (Term 3-4) / MA060699 / 24-26

Instructor: Alexey Litvinov

The course assumes a basic knowledge of two-dimensional conformal field theory. The course will emphasize various methods and approaches of two-dimensional conformal field theory with application in quantum field theory and string theory. In particular, such issues as: conformal bootstrap and classification of partition functions in CFT, conformal field theory with the boundary, counting the number of physical states in string theory, D-branes in string theory, renormalization group and Zamolodchikov’s c-theorem, integrable perturbations of two-dimensional conformal theory will be considered.

Modern dynamical systems (Term 3-4) / MA060257 / 24-26

Instructors: Aleksandra Skripchenko, Sergei Lando

Dynamical systems in our course will be presented mainly not as an independent branch of mathematics but as a very powerful tool that can be applied in geometry, topology, probability, analysis, number theory and physics. We consciously decided to sacrifice some classical chapters of ergodic theory and to introduce the most important dynamical notions and ideas in the geometric and topological context already intuitively familiar to our audience. As a compensation, we will show applications of dynamics to important problems in other mathematical disciplines. We hope to arrive at the end of the course to the most recent advances in dynamics and geometry and to present (at least informally) some of results of A. Avila, A. Eskin, M. Kontsevich, M. Mirzakhani, G. Margulis.
In accordance with this strategy, the course comprises several blocks closely related to each other. The first three of them (including very short introduction) are mainly mandatory. The decision, which of the topics listed below these three blocks would depend on the background and interests of the audience.
Full Syllabus

Integrable many-body systems and nonlinear equations (Term 3-4) / MA060602 / 24-26

Instructor: Anton Zabrodin

This course is devoted to many-body integrable systems of classical mechanics such as Calogero-Moser, Ruijsenaars-Schneider and their spin generalizations. These systems play a significant
role in modern mathematical physics. They are interesting and meaningful from both mathematical and physical points of view and have important applications and deep connections with different problems in mathematics and physics. The history of integrable many-body systems starts in 1971 from the famous Calogero-Moser model which exists in rational, trigonometric or hyperbolic and (most general) elliptic versions. Later it was discovered that there exists a one-parametric deformation of the Calogero-Moser system preserving integrability, often referred to as relativistic extension. This model is now called the Ruijsenaars-Schneider system. In its most general version the interaction
between particles is described by elliptic functions.
The integrable many-body systems of Calogero-Moser type have intimate connection with nonlinear integrable equations such as
Korteveg-de Vries and Kadomtsev-Petviashvili (KP) equations. Namely, they describe dynamics of poles of singular solutions (in general, elliptic solutions) to the nonlinear integrable partial differential equations. The Ruijsenaars-Schneider system plays the same role for singular solutions to the Toda lattice equation.
In this course the algebraic structure of the integrable many-body systems will be presented. The Lax representation and interrals of motion will be obtained using the corerspondence with the equations of the KP type. The necessary material about the latter will be given in the course. The construction of the spectral curves will be discussed.

Geometry in field theory, first step (Term 3-4) / MA060724 / 24-26

Instructor: Alexei Rosly

This is a continuation of the subject started in “Some Uses of Twistors in Field Theory”, 2024.
The subject of this course will be mainly twistors and the complex geometry. The word ‘twistor’ should be properly understood as ‘twistor transform’. The latter in short means the following.
To certain 4-dimansional Riemannian (or pseudo-Riemannian, but Riemannian case is still more convenient) manifolds one can associate a 3-dimensional complex manifold. In this setting, the points of a 4-fold M are in 1:1 correspondence with “real” lines in the complex 3-fold P. For example, to the Riemannian 4-fold M=S⁴ one gets associated complex 3-fold P=CP³, while for M=R⁴=S⁴\{∞} one gets M=CP³\{a line}. In this context M plays the role of a (Euclidian version of) the space-time, whereas P is called the twistor space. As a matter of fact, this construction can be carried out for certain 4n-dimensional Riemannian metrics, yielding a complex manifold the twistor space) of dimension 2n+1. This correspondence allows one to rewrite some interesting equation of Mathematical Physics for fields on M in terms of complex geometry of P. This transition can be called the twistor transform.
In this course we shall consider a number of examples of such a character.