Curriculum 18/19

Research seminar “Modern Problems of Mathematical Physics” (Term 1-8) / MA12268 / 18-19

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: now it is devoted to the study of N=2 supersymmetric gauge theory and its links with random matrix models, ABJM theory, localization, complex curves, and integrable systems. Other topics that were already covered, or can be covered in the future, are: classical integrable equations, complex curves and their theta-functions, quantum integrable models (quantum-mechanical and field-theoretical), models of statistical physics

Research seminar “Strings and Cluster Varieties” (Term 1-8) / MA12176 / 18-19

Instructor: Andrei Marshakov

The course is directed to substantive work of the master and PhD students in order to understand recently found relations among supersymmetric gauge theories, refined topological strings, cluster varieties and integrable systems. The plan of wrk on the course consists of several introductive lectures on the various consistuents of the subject as well as student talks on recent original papers and results of their own investigation. The core topics include relation between cluster varieties and Painleve equations and approaches to the Seiberg-Witten theories with fundamental matter based on Toda systems and spin chains

Lie Groups and Lie Algebras, and their Representations / MA06173 / 18-19

Instructor: Grigori Olshanski

We shall begin with the basics of the theory of Lie groups and Lie algebras. Then we shall provide an accessible introduction to the theory of finite-dimensional representations of classical groups on the example of the unitary groups U(N).

Tentative plan:
— linear Lie groups and their Lie algebras;
— universal enveloping algebras;
— Haar measure on a linear Lie group;
— general facts about representations of compact groups and their characters;
— radial part of Haar measure;
— Weyl’s formula for characters of the unitary groups;
— Weyl’s unitary trick;
— classification and realization of representations;
— symmetric functions

Hamiltonian Mechanics / MA06174 / 18-19

Instructor: Andrei Marshakov

This is the first among the base courses in the theoretical physics, aimed for the master students. Matematical methods of modern theory of Hamiltonian systems are based on the concepts, arosen in different fields of mathematics:
== differential equations and dynamical systems,
== Lie groups and algebras,
== differential geometry on manifolds.
Many modern directions in mathematics (e.g. symplectic geometry) got their origin from the problems of classical mechanics.This course is recommended to all students, interested in mathematical physics, and it does not imply any special preliminary education in physics.
The preliminary program of the course includes:
1. Lagrangian formalism: minimal action principle, Euler-Lagrange equations, symmetries and integrals of motion, Noether theorem.
2. Simplest examples: dynamics for a single degree of freedom, Kepler’s problem etc.
3. Basis of the Hamiltonian formalism: phase space, Legebdre transform, Hamilton equations, the Poisson and symplectic structures, Darboux theorem.
4. The Hamilton-Jacobi equation, canonical transform, Liouville theorem.
5. Integrable systems: separation of variables, Liouville integrability. Systems with Lax representation.
6. Examples of integrable systems: Toda and Calogero problems, integrable systems on Lie groups, geometry of spectral curves etc.

Differential and Symplectic Geometry / MA06175 / 18-19

Instructor: Maxim Kazaryan

Symplectic geometry is the mathematical language of classical mechanics a hamiltonian dynamic. The course will cover the base of symplectic and contact geometry: symplectic forms, symplectomorphisms, Hamilton vector fields, as well as contact counterparts of these objects. Some applications in differential geometry will include geometry of caustics and wave fronts and their singularities. Among topological applications we will discuss different aspects of Arnold’s conjecture on fixed points of symplectomorphisms and its variations

Geometric Representation Theory / MA06271 / 18-19

Instructors: Mikhail Finkelberg, Alexander Braverman, assistant: Alexei Litvinov

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.

Dynamical Systems and Ergodic Theory / MA06257 / 18-19

Instructors: Aleksandra Skripchenko, Anton Zorich

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.

Gauge Theory and Gravitation / MA06178 / 18-19

Instructor: Alexei Rosly

The present course could be also entitled ‘Classical Field Theory’, which menas it deals with all basic material needed in a study of fields preceeding to a study of their quantum properties. This requires in particular understanding such tools as Lagrangian, action functional, field equations (Euler-Lagrange equations). We shall also learn what are the most important symmetry principles which put certain constraints on a field theory. With this are related conservation laws. Typical important symmetries to mention are Lorentz and Poincare symmetry, conformal symmetry, gauge symmetry, general coordinate covariance.
A traditional approach to Classical Field Theory has a perfect base in the 2nd volume of Landau-Lifshitz’ course. However, since that prominent book was written, new elements came forward, which required more knowledge of differential geometry and topology. In our lecture course, we shall get familiar with most important basic facts from these branches of mathematics with application to field theory. For example, understanding instantons (even at a classical level) requires good knowledge of a number of notions from modern math courses, such as vector bundles, connections, homotopy groups. Therefore our course has to go beyond the reach of Landau-Lifshitz’ volume 2.

Quantum Integrable Systems / MA06315 / 18-19

Instructor: Anton Zabrodin

The course is devoted to quantum integrable systems. The history of quantum integrable systems starts from 1931 when H.Bethe managed to construct exact eigenfunctions of the Hamiltonian of the Heisenberg spin chain with the help of a special substitution which became famous since that time (ansatz Bethe). In one or another form this method turns out to be applicable to many spin and field-theoretical integrable models. From the mathematical point of view, Bethe’s method is connected to representation theory of quantum algebras (q-deformations of universal enveloping algebras and Yangians).
Here is the list of topics which will be discussed in the course.
== Coordinate Bethe ansatz on the example of the Heisenberg model and one-dimensional Boe gas with point-like interaction between particles.
== Bethe ansatz in exactly solvable models of statistical mechanics on the lattice.
== Calculation of physical quantities in integrable models in thermodynamic limit, thermodynamic Bethe ansatz.
== Bethe equations and the Yang-Yang function, caclulation of norms of Bethe vectors.
== Quantum inverse scattering method and algebraic Bethe ansatz, quantum R-matrices, transfer matrices, Yang-Baxter equation.
== Functional Bethe ansatz and the method of Baxter’s Q-operators, functional relations for transfer matrices, transfer matrices as classical tau-functions.
The knowledge of quantum mechanics and statistical physics for understanding of the course is highly desirable but not absolutely necessary. Out of the physical context ansatz Bethe in its finite-dimensional version is simply a method for diagonalization of big matrices of a special form. In this sense it does not require anything except the basic notions of linear algebra.

Differential Topology / MA06258 / 18-19

Instructor: Alexander Gaifulin

We plan to discuss two topics, which are central in topology of smooth manifolds, the h-cobordism theorem and theory of characteristic classes. 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). Characteristic classes, in particular, Pontryagin classes are very natural invariants of smooth manifolds. Computation of characteristic classes can help one to distinguish between non-diffeomorphic manifolds. We plan to finish the course with the theorem by J. Milnor on non-trivial smooth structures on the 7-dimensional sphere. This theorem is based both on methods of Morse theory and theory of characteristic classes

Representations of Affine Kac-Moody Algebras / MA06320 / 18-19

Instructor: Evgeny Feigin

Infinite-dimensional Lie algebras (such as Virasoro algebra or affine Kac-Moody algebras) turn out to be very important in various areas of modern mathematics and mathematical physics. In particular, they are very useful in the description of some field theories. In this context one arranges infinite number of the Lie algebra elements into a single object called field. This idea generalizes to the general theory of vertex operator algebras. VOAs capture the main properties of the infinite diemensional Lie algebras and have rich additional structure. Vertex operator algebras proved to be very useful in many situations; the classical example is the KP integrable hierarchy. They are also extensively used in modern algebraic geometry.
Our goal is to give an introduction to the theory of vertex operator algebras from the modern mathematical point of view. We describe the main definitions, constructions and applications of the theory. The course is aimed at PhD students and master students.

Classical Integrable Systems / MA06179 / 18-19

Instructor: Igor Krichever

Course description: A self-contained introduction to the theory of soliton equations with an emphasis on their algebraic-geometrical integration theory. Topics include:
1. General features of the soliton systems.
2. Algebraic-geometrical integration theory.
3. Hamiltonian theory of soliton equations.
4. Perturbation theory of soliton equations and its applications to Topological Quantum
Field Theories and Sieberg-Witten solutions of N=2 Supersymmetric Gauge Theiories

Affine Lie Algebras and Conformal Field Theory / MA06321 / 18-19

Instructor: Mikhail Bershtein

The course will be devoted to conformal theories (more precisely, their chiral parts – vertex algebras) with the symmetry of the affine Lie algebra and the theories that are obtained from them by basic constructions. The basic examples are Wess-Zumino-Witten theory, coset theories, Drinfeld-Sokolov reductions. In general, we will discuss the mathematical aspects associated with the theory of representations and geometries. The course is supposed to be relatively advanced, see Course Prerequisites below.

Statistical Physics / MA06180 / 18-19

Instructor: Semen Shlosman

This is a course on rigorous results in statistical physics and random fields. Most of it will be dedicated to the theory of phase transitions, uniqueness or non-uniqueness of the lattice Gibbs fields.
The topics will include:
== Grand canonical, canonical and microcanonical ensembles,
== DLR equation,
== Thermodynamic limit,
== Gibbs distributions and phase transitions,
== One-dimensional models,
== Correlation inequalities ( GKS, GHS, FKG),
== Spontaneous symmetry breaking at low temperatures,
== Uniqueness at high temperatures and in non-zero magnetic field,
== Non-translation-invariant Gibbs states and interfaces,
== Dobrushin Uniqueness Theorem
== Pirogov–Sinai Theory
== O(N)-symmetric models
== the Mermin–Wagner Theorem
== Reflection Positivity and the chessboard estimate infrared bounds

Applied Methods of Analysis / MA06317 / 18-19

Instructor: Sergei Khoroshkin

The course is addressed to undergraduates of the first year and contains applications of various mathematical methods for solving problems of mathematical physics. The course assumes familiarity with various sections of theoretical physics (classical mechanics, field theory, quantum mechanics, statistical physics, hydrodynamics, elasticity theory) on the example of solving specific problems. The main purpose of the course is to encourage undergraduates to independent research work. For this reason, the main element of the course is an independent solution to the problem, requiring the study of additional material.

Quantum Field Theory / MA06316 / 18-19

Instructor: Vladimir Losyakov

The course is addressed to undergraduates of the first year and contains applications of various mathematical methods for solving problems of mathematical physics. The course assumes familiarity with various sections of theoretical physics (classical mechanics, field theory, quantum mechanics, statistical physics, hydrodynamics, elasticity theory) on the example of solving specific problems. The main purpose of the course is to encourage undergraduates to independent research work. For this reason, the main element of the course is an independent solution to the problem, requiring the study of additional material.

Quantum Mechanics / MA06322 / 18-19

Instructor: Andrei Semenov

Random Matrices, Random Processes and Integrable Systems / MA06318 / 18-19

Instructor: Alexander Povolotsky

In recent years, researchers have found remarkable connections between, at first glance, completely different problems of mathematics and theoretical physics. Mathematically, these are combinatorial and probabilistic problems about systems with a large number of degrees of freedom. Among them, the problem of describing the eigenvalues ​​of matrices with random elements, the problem of statistics of random Young diagrams, the problem of tiling various regions of the plane by dominoes or lozenges, the problem of enumerating nonitersecting paths on lattices. On the physical side, these are the problems of interface growth, traffic flows, polymers in random media, and so on.
It turns out that all these problems have a general mathematical structure, which can be encoded in the term “integrability”. This structure is behind a lot of beautiful exact mathematical results. Moreover, these results have remarkable universality property, which play the same role as the law of large numbers and the central limit theorem in the probability theory. Looking at our random systems from afar, we find that they posess completely nonrandom limiting forms. Random fluctuations around these forms are described by a small number of universal probability distributions that are completely independent of the details of original system. The course attendees are going to get acquainted with the range of issues described and to learn about the latest achievements in the field.