curriqulum 23/24

Research Seminar “Modern Problems of Mathematical Physics” (Term 1-8) / DG120268 / 23-25

Instructor: Andrei Marshakov

Course “Modern problems of mathematical physics” is a student seminar, so participants are expected to give talks based on the modern research papers. Current topic of the seminar can vary from time to time. Topics that were already covered, or can be covered in the future, are:
classical integrable equations, complex curves and theta-functions,
quantum integrable models (quantum-mechanical and field-theoretical),
models of statistical physics, stochastic integrability,
quantum/classical duality, supersymmetric gauge theories, cluster algebras etc.

Representations of Finite Groups (Term 1-2) / MA060595 / 23-25

Instructor: Grigori Olshanski

Representation theory is used in many areas of mathematics (algebra, topology, algebraic groups, Lie groups and Lie algebras, quantum groups, algebraic number theory, combinatorics, probability theory, …), as well as in mathematical physics. Therefore, mastering the basic technique of representation theory is necessary for mathematicians of various specialties. The aim of the course is to give an introduction to representation theory on the material of finite groups. Particular attention will be paid to representations of the symmetric groups.

Tentative program:
1. Reminder of the basics from the algebra course: group algebra of a finite group, irreducible representations, Schur’s lemma, characters, orthogonality relations, Maschke’s theorem, Burnside’s theorem
2. Representations of finite Abelian groups, duality for finite Abelian groups, Fourier transform, biregular representation
3. Intertwining operators, induced representations, Frobenius duality
4. Mackey machine, projective representations, coverings over symmetric groups
5. Functional equation for characters, Gelfand pairs, spherical functions, connection with orthogonal polynomials
6. Representations of the symmetric group: various approaches to the classification and construction of irreducible representations
7. If time permits: principal series representations for the group GL(N) over a finite field, Hecke algebra, Harish-Chandra theory

Cluster Varieties and Integrable Systems (Term 1-2) / MA060597 / 23-25

Instructor: Andrei Marshakov

Geometric Representation Theory (Term 1-2) / DA060271 / 23-25

Instructors: 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

Differential Topology (Term 1-2) / MA060599 / 23-25

Instructor: Alexander Gaifullin

The course will cover two topics, which are central in topology of smooth manifolds, the de Rham cohomology theory and Morse theory. The course will culminate in two famous results of differential topology: Smale’s h-cobordism theorem and Milnor’s discovery of exotic smooth structures on the 7-dimensional sphere. The h-cobordism theorem proved by S. Smale in 1962 is the main (and almost the only) tool for proving that two smooth manifolds (of dimension greater than or equal to 5) are diffeomorphic. In particular, this theorem implies the high-dimensional Poincare conjecture (for manifolds of dimensions 5 and higher). Milnor’s discovery of exotic smooth structures on the 7-dimensional sphere and further results of Kervaire and Milnor were the first steps towards surgery theory, which is the most powerful tool for classifying smooth manifolds.

Topics
= = Homology and cohomology //
De Rham cohomology. Singular homology. Pairing between homology and cohomology. Multiplication in cohomology and intersection of cycles. Poincare duality
= = Morse theory //
Morse functions. Cobordisms corresponding to critical points. Morse inequalities. Lefschetz theorem on hyperplane sections. Smale’s h-cobordism theorem. High-dimensional Poincare conjecture
= = Characteristic classes //
Principal bundles and their characteristic classes. Chern-Weil theory. Chern classes and Pontryagin classes. Integral Chern classes and Pontryagin classes. Smooth structures on the 7-dimensional sphere

Phase Transitions, Rigorous (Term 3-4) / MA060600 / 23-25

Instructor: Semen Shlosman

This is a course on rigorous results in statistical mechanics, random fields and percolation theory. Some of it will be dedicated to the theory of phase transitions, uniqueness or non-uniqueness of the lattice Gibbs fields. We will also study the models at the criticality, where one hopes to find (in dimension 2) the onset of conformal invariance. We will see that it is indeed the case for the percolation (and the Ising model, if time permits).
The topics will include:
= Crossing probabilities as a characteristic of sub-, super- and at- criticality.
= Critical percolation and its power-law behavior.
= The Russo-Seymour-Welsh theory of crossing probabilities – a cornerstone of critical percolation
= Cardy’s formula for crossing probabilities
= Parafermionic observables and S. Smirnov theory
= Conformal invariance of two-dimensional percolation a la Khristoforov.
= Ising model in 1D, 2D and 3D.
= Ising model on Cayley trees.
= Conformal invariance of two-dimensional Ising model
= O(N)-symmetric models
= Continuous symmetry in 2D systems: The Mermin–Wagner Theorem and the absence of Goldstone bosons.
= The Berezinskii–Kosterlitz–Thouless transition
= Reflection Positivity and the chessboard estimates in statistical mechanics
= Infrared bounds and breaking of continuous symmetry in 3D

Introduction to quantum field theory (Term 3-4) / MA060505 / 23-25

Instructors: Vladimir Losyakov, Petr Dunin-Barkowski

As you know, the modern theory of fundamental physics (the “standard model of elementary particle physics”) is a quantum field theory (QFT). In addition to this central role in modern physics, quantum field theory also has many applications in pure mathematics (for example, from it came the so-called quantum knot invariants and Gromov-Witten invariants of symplectic manifolds).
“Ordinary” quantum mechanics deals with systems with a fixed number of particles. In QFP, the objects of study are fields (not in the sense of a “field of complex numbers”, but in the sense of an “electromagnetic field”), whose elementary perturbations are analogs of quantum mechanical particles, but can appear and disappear (“born” and “die”); at the same time, the number of degrees of freedom turns out to be infinite.

Within the framework of this course, the basic concepts of QFT will be introduced “from scratch”. The Fock space and the formalism of operators on it, as well as the formalism of the “continuum integral” will be defined. The main example under consideration will be the quantum scalar field theory. A scalar field in physical terminology is a field that, at the classical level, is defined by one number at each point (i.e., in fact, its state at a given time is just a numerical function on space), unlike a vector field (an example of which, in particular, is an electromagnetic field). However, considering the quantum theory of a scalar field (even separately, and simpler than for the Higgs field) is in any case very useful, since it allows you to get acquainted with the apparatus and phenomena of QFT on a simpler example than vector and spinor fields. The course will consider the “perturbation theory” (that is, in fact, a method for calculating the first orders of smallness in a small parameter expansion) for a scalar field and describe ways to calculate various probabilities of events with particles.
Full Syllabus

Symplectic Geometry (Term 1-2) / MA060596 / 23-25

Instructor: Maxim Kazarian

Introduction to Quantum Groups (Term 1-2) / MA060426 / 23-25

Modern Dynamical Systems (Term 3-4) / MA060257 / 23-25

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

Some Uses of Twistors in Field Theory (Term 3-4) / MA060601 / 23-25

Instructor: Alexei Rosly

The subject of this course will be mainly the complex geometry. The choice of topics, however, is determined by their uses in Field Theory and Theory of Integrable Systems

Integrable Many-Body Systems and Nonlinear Equations (Term 3-4) / MA060602 / 23-25

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.

Path integral: stochastic processes and basics of quantum mechanics (Term 1-2) / MA060542 / 23-25

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