Curriculum 19/21

Mathematical and Theoretical Physics MSc Program Structure

Master program curriculum //
Educational program “Mathematical and Theoretical Physics”,
Field of Science and technology 03.04.01 Applied Math and Physics,

full-time, onsite form of study, study period – 2 years, year of admission – 2019
# Code Curriculum element ECTS Grad
/ Pass
* ** Acad Year 19/20 Acad Year 20/21
Fall Spring Fall Spring
Stream 1. “Science, Technology and Engineering (STE)” / Coursework / 48 ECTS credits
Track A – “Mathematical Physics”
1 MA12268 Research seminar – Modern problems of mathematical physics 12 G С/Dr A 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
2 MA06173 Lie Groups and Lie algebras and their representations 6 G RE/Dr A 3 3 X X
3 MA06271 Geometric representation theory 6 G RE/Dr A 3 3 X X
4 MA06175 Differential geometry of connections 6 G RE/Dr A 3 3 X X
5 MA06381 Supersymmetric gauge theories 6 G RE/Dr A 3 3 X X
6 MA06382 Supersymmetric gauge theories and integrable systems 6 G RE/Dr A 3 3
7 MA06257 Modern dynamical systems 6 G RE/Dr A 3 3
8 MA06178 Gauge fields and complex geometry 6 G RE/Dr A 3 3
9 MA06315 Quantum integrable systems 6 G RE/Dr A 3 3
10 MA06258 Differential topology 6 G RE/Dr A 3 3
11 MA06259 Vertex operator algebras 6 G RE/Dr A 3 3
12 MA06321 Virasoro algebra and conformal field theory 6 G RE/Dr A 3 3 X X
13 ME06179 Classical integrable systems 6 G RE/Dr A 3 3
14 MA06383 Statistical mechanics, percolation theory and conformal invariance 6 G RE/Dr A 3 3 X X
15 MA06387 Statistical mechanics 6 G RE/Dr A 3 3
16 MA06174 Hamiltonian mechanics 6 G RE/Dr A 3 3 X X
17 MA06317 Mathematical methods of science 6 G RE/Dr A 3 3 X X
18 MA06332 Introduction to quantum theory 6 G RE/Dr A 3 3
19 ME06384 Quantum theory 6 G RE/Dr A 3 3 X X
20 MA06316 Quantum field theory 6 G RE/Dr A 3 3
Track B – “Theoretical Physics”
21 MA12319 Research seminar – Modern problems of theoretical physics 12 G С/Dr B 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
22 MA06207 Advanced quantum mechanics 6 G RE B 6
23 MA06262 Functional methods in the theory of disordered systems 6 G RE B 3 3
24 MA06138 Theory of phase transition G экз RE B 3 3
25 MA06274 Introduction to the theory of disordered systems 6 G RE B 3 3
26 MA06273 Introduction to the quantum field theory 6 G RE B 3 3
27 MA06275 Asymptotic methods in complex analysis 6 G RE B 3 3
28 MA03385 Numerical simulations of quantum many-body systems 6 G RE B 3
29 MA06278 Quantum mesoscopics. Quantum Hall effect 6 G RE B 3 3
30 MA06276 One-dimensional quantum systems 6 G RE B 3 3
Stream 2. “Research” / 12 ECTS credits
31 MB12006 Research Immersion 12 P C A
Stream 3. “Enterpreneurship and Innovation” (E&I) / 6 ECTS credits
32 MC06001 Innovation workshop 6 P C A
Stream 4. “Research & MSc Thesis Project” / 36 ECTS credits
33 MD06001 Early Research Project 6 P C A
3 3
34 MD24002 Thesis Research project 21 P C A
6 6 6 3
35 MD06003 Thesis defence 9 FSA C A
Stream 5. “Options” / 18 ECTS credits
36 Elective courses from Course Catalogue E A
37 Research project P E A
Facultative (Extracurricular activities – maximum 20 ECTS credits overall, maximum 10 ECTS credits per year)
38 MO03001 English toolkit 3 P E A
39 MO03002 Academic writing essential 3 P E/Dr A
40 MO06003 Master your thesis in english 1 6 P RE/Dr A
2 2 2
41 MO06003 Master your thesis in english 2 6 P RE/Dr A
2 2 2
42 MO06003 Independent study period A
Minimum overload per Year 60 60
Maximum overload per Year 70 70
TOTAL 120-140
*) C – сompulsory curriculum element, RE – recommended elective, E – elective, Dr – suitable for PhD
**) Track A – “Mathematical Physics”, Track В – “Theoretical Physics”

Research seminar “Modern Problems of Mathematical Physics” (Term 1-8) / MA12268 / 19-21

Instructor: Pavlo Gavrylenko

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.
Full Syllabus

Lie Groups and Lie Algebras, and their Representations (Term 1-2) / MA06173 / 19-21

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
Full Syllabus

Geometric Representation Theory (Term 1-2) / MA06271 / 19-21

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 Geometry of Connections (Term 1-2) / MA06175 / 19-21

Instructor: Maxim Kazaryan

Supersymmetric Gauge Theories (Term 1-2) / MA06381 / 19-21

Instructors: Pavlo Gavrylenko, Andrei Marshakov, Alexei Yung

This course is an introduction to the supersymmetry, which is one of the main basic principles, being used both in field theory and in string theory. The main emphasis will be made on the consideration of dynamics of supersymmetric gauge field theories: in particular, at the non-perturbative level.

Course starts from the introduction of the main concepts: supersymmetric algebra, its representations, chiral and vector multiplets. One will consider Wess-Zumino model and introduce superpotential. Then one will consider supersymmetric quantum electrodynamics, discuss supersymmetric Higgs mechanism and Higgs branches. One will consider in detail supersymmetric quantum chromodynamics. Its vacuum structure at the classical level will be discussed. Then one will introduce instantons and explain how they change vacuum structure by generating Affleck-Dine-Seiberg superpotential. Finally we will consider Seiberg duality
Full Syllabus

Supersymmetric Gauge Theories and Integrable Systems (Term 3-4) / MA06382 / 19-21

Instructors: Pavlo Gavrylenko, Andrei Marshakov

The course will be devoted to the study of N=2 supersymmetric gauge theories and related topics. It turns out that comparing to the N=1 theories, N=2 allows to compute much more quantities. In particular, low-energy effective action can be described in terms of single function, prepotential. Seiberg-Witten solution of the N=2 theory gives explicit description of the prepotential in terms of periods of some meromorphic differentials on algebraic curves. It turns out that this description is deeply related to classical integrable systems.
During the course we will learn basics of the N=2 theories, classical solutions, holomorhy arguments, and so on, study Seiberg-Witten exact solution, and then its underlying integrable systems. We are also going to learn some modern developments of this topic, like Nekrasov instanton computations and AGT relation.
Full Syllabus<

Modern Dynamical Systems (Term 3-4) / MA06257 / 19-21

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

Gauge Fields and Complex Geometry (Term 3-4) / MA06178 / 19-21

Instructor: Alexei Rosly

1. Self-duality equations, Bogomolny equations.
2. Relation to holomorphic bundles.
3. Relation to holomorphic bundles on twistor space.
4. Conformal symmetry and complex geometry in twistor space.
5. Elements of superfield formulation of SUSY field theories.
6. Chirality type constraints and complex geometry.
7. Some examples of superfield theories which require complex geometry.
8. BPS conditions in SUSY theories and complex geometry.
9. Elements of Hitchin’s integrable systems and related complex geometry.
Full Syllabus

Quantum Integrable Systems (Term 3-4) / MA06315 / 19-21

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, calculation 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
Full Syllabus

Differential Topology (Term 3-4) / MA06258 / 19-21

Instructor: Alexander Gaifullin

Vertex Operator Algebras (Term 3-4) / MA06259 / 19-21

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 dimensional 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.
Full Syllabus

Virasoro Algebra and Conformal Field Theory (Term 1-2) / MA06321 / 19-21

Instructor: Mikhail Bershtein

Conformal field theory is a quantum field theory that is invariant under conformal transformations.The course is devoted to a two-dimensional theory, which possess an infinite dimensional algebra of local conformal transformation, which includes Virasoro Lie algebra.
We will mainly focus on the mathematical aspects of the theory based on the relations with the representation theory of Virasoro algebra. A small preliminary acquaintance with string theory and conformal field theory is assumed.
Full Syllabus

Classical Integrable Systems (Term 5-6) / MA06179 / 19-21

Instructor: Igor Krichever

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.

Statistical Mechanics, Percolation Theory and Conformal Invariance (Term 1-2) / MA06383 / 19-21

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.

The topics will include:
Crossing probabilities
Critical percolation
The Russo-Seymour-Welsh theory
Cardy’s formula
Parafermionic observables
Conformal invariance of two-dimensional percolation
Conformal invariance of two-dimensional Ising model
O(N)-symmetric models
The Mermin-Wagner Theorem
The Berezinskii-Kosterlitz-Thouless transition
Reflection Positivity and the chessboard estimates
Infrared bounds
Full Syllabus

Statistical Mechanics (Term 5-6) / MA06387 / 19-21

Instructor: Semen Shlosman

Hamiltonian Mechanics (Term 1-2) / MA06174 / 19-21

Instructor: Igor Krichever

Mathematical Methods of Science (Term 1-2) / MA06317 / 19-21

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 a minor 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. In the endpoint the students are assumed to acquire the use of Green functions, distributions, Laplace and Fourier transforms, asymptotic evaluations in mathematical physics

Introduction to Quantum Theory (Term 3-4) / MA06332 / 19-21

Instructor: Vladimir Losyakov

Quantum Theory (Term 1-2) / MA06384 / 19-21

Instructor: Vladimir Losyakov

The course is addressed to students who know the basics and basic principles of quantum theory. Physical applications of quantum theory on examples of specific problems will be considered. It is planned to study the following issues: supersymmetric quantum mechanics, electromagnetic field interacting with an external source, the Caldeira-Leggett model, the band structure of onedimensional systems, the basics of quantum Informatics. As can be seen from the list, the purpose of the course is to prepare the student for the study of quantum
field theory. Students will be asked to choose their own topics to be discussed
Full Syllabus

Quantum Field Theory (Term 3-4) / MA06316 / 19-21

Instructor: Andrei Semenov