curriculum 20/22

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 study, study period – 2 years, year of admission – 2020
# Code Course Title ECTS * Grad
/ Pass
* Acad Year 20/21 Sum-mer Term Acad Year 21/22
Fall Spring Fall Spring
Stream 1. “Science, Technology and Engineering (STE)” / 36 ECTS credits
Track – “Mathematical Physics”
1 MA060382 Research seminar “Supersymmetric gauge theories and integrable systems” 6 G Re/Dr 3 3 X X
2 MA030423 Representations of classical groups and related topics 3 G RE/Dr 1.5 1.5 X X
3 MA060271 Geometric representation theory 6 G RE/Dr 3 3 X X
4 MA060258 Elliptic operators in topology of manifolds 6 G RE/Dr 3 3 X X
5 ME060179 Classical integrable systems 6 G RE/Dr X X 3 3
6 MA060427 Phase transitions: Introduction to statistical physics and percolation 6 G RE/Dr 3 3
7 ME060428 Quantum mechanics 6 G RE/Dr 3 3 X X
8 MA060317 Mathematical methods of science 6 G RE/Dr 3 3 X X
9 MA060424 Critical points of functions 6 G RE/Dr 3 3 X X
10 MA060257 Introduction to quantum groups 6 G RE/Dr 3 3 X X
11 MA060257 Modern dynamical systems 6 G RE/Dr 3 3
12 MA06178 Gauge fields and complex geometry 6 G RE/Dr 3 3
13 MA060315 Quantum integrable systems 6 G RE/Dr 3 3
14 MA060425 Quiver representations and quiver varieties 6 G RE/Dr 3 3
15 MA060332 Introduction to quantum theory 6 G RE/Dr 3 3
16 MA060316 Quantum field theory 6 G RE/Dr 3 3
Track “Theoretical Physics”
17 MA060207 Advanced quantum mechanics 6 G Re/Dr 3 3 X X
18 MA060262 Functional methods in the theory of disordered systems 6 G Re/Dr 3 3
19 MA060138 Theory of phase transition 6 G Re/Dr 3 3
20 MA060274 Introduction to the theory of disordered systems 6 G Re/Dr 3 3
21 MA060273 Introduction to the quantum field theory 6 G Re/Dr 3 3
22 MA030385 Numerical simulations of quantum many-body systems 6 G Re/Dr 3
24 MA060278 Quantum mesoscopics. Quantum Hall effect 6 G Re/Dr 3 3
24 MA060276 One-dimensional quantum systems 6 G Re/Dr 3 3
Stream 2. “Research Immersion” / 12 ECTS credits
25 MB12006 Research Immersion 12 P C 12
Stream 3. “Enterpreneurship and Innovation” (E&I) / 6 ECTS credits
26 MC06001 Innovation workshop 6 P C 6
Stream 4. “Research & MSc Thesis Project” / 48 ECTS credits
27 MA12268 Research seminar – Modern problems of mathematical physics 12 G С/Dr 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
28 MA12319 Research seminar – Modern problems of theoretical physics 12 G С/Dr 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
29 MD06001 Early Research Project 6 P C 3 3
30 MD24002 Thesis Research project 12 P C 3 3 6
31 MD090023 Thesis Proposal, Status Review and Predefense 9 P C 3 3 3
32 MD090003 Thesis defence 9 SFA C 9
Stream 5. “Options” / 18 ECTS credits
33 Elective courses from Course Catalogue E X X X X X X
34 Additional Thesis Research P E X X X X X
35 Short-Term Project P E X X X X X
Facultative (Extracurricular activities – maximum 20 ECTS credits overall, maximum 10 ECTS credits per year)
36 MO03001 English toolkit 3 P E 3
37 MO03002 Academic writing essential 3 P E X 3
38 MO06003 Master your thesis in english 3 P E 3 X
39 MO06003 Independent study period 3 P E X
Minimum overload per Year 60 60
Maximum overload per Year 70 70
TOTAL 120-140
*) ECTS – European Credit Transfer and Accumulation System, G – Graded course, P – Pass/Fail course, X – curriculum element can be chosen in specified Terms, SFA – State Final Assessment, C – compulsory curriculum element, RE – recommended elective, E – elective, Dr – suitable for PhD

Research seminar “Modern Problems of Mathematical Physics” (Term 1-8) / MA060382 / 20-22

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. 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, stochastic integrability, quantum/classical duality, supersymmetric gauge theories, etc.
Full Syllabus

Research seminar “Supersymmetric Gauge Theories and Integrable Systems” (Term 1-2) / DA060382 / 20-22

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

Representations of Classical Groups and Related Topics (Term 1-2) / MA030423 / 20-22

Instructor: Grigori Olshanski

The course is focused on fundamental results of the representation theory of classical matrix groups, which find numerous applications in various domains of mathematics. Particular attention will be paid to links with algebraic combinatorics.
Tentative program:
– Characters of classical groups (general linear, orthogonal, and symplectic).
– Second Weyl character formulas
– Classical invariant theory and applications
– Representations in traceless tensors
– Brauer duality
– Highest weight representations and Littlewood formulas
– Center of universal enveloping algebra
– Perelomov-Popov theorem
– Capelli identity
– Multidimensional interpolation polynomials
– Binomial formula for characters
– Okounkov’s quantum immanants
– Applications to asymptotic representation theory
Full Syllabus

Geometric Representation Theory (Term 1-2) / MA060271 / 20-22

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

Elliptic operators in topology of manifolds (Term 1-2) / MA060258 / 20-22

Instructor: Alexander Gaifullin

The course will be devoted to applications of elliptic differential operators in topology of manifolds. We start with basics of Hodge theory including a detailed proof of the existence of the harmonic representative in a de Rham cohomology class. This proof is based on the usage of Sobolev spaces, and we will pay attention to this technique. Further, we proceed with the notion of the index of an elliptic differential operator towards the Atiyah-Singer theorem and its applications. Examples of several most important operators will be studied in details
Full Syllabus

Classical Integrable Systems (Term 1-2/5-6) / MA060179 / 20-22

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
Full Syllabus

Phase Transitions: Introduction to Statistical Physics and Percolation (Term 1-2) / MA060427 / 20-22

Instructor: Semen Shlosman

This is a course on rigorous results in statistical mechanics, random fields, and percolation theory. We start with percolation, which is the simplest system, exhibiting singular behavior, and undergoing phase transitions. We then go to more realistic models of interacting particles, like the Ising model and XY-model, and study phase transitions, occurring there.
The topics will include:
= = Percolation models, infinite clusters.
= = Crossing probabilities for rectangles
= = Critical percolation
= = The Russo-Seymour-Welsh theory
= = Cardy’s formula in Carleson form and the Smirnov theorem.
= = Gibbs distribution
= = Dobrushin-Lanford-Ruelle equation
= = Ising model
= = Spontaneous symmetry breaking at low temperatures
= = O(N)-symmetric models
= = The Mermin–Wagner Theorem
= = The Berezinskii–Kosterlitz–Thouless transition
= = Reflection Positivity and the chessboard estimates
= = Infrared bounds
Full Syllabus

Quantum Mechanics (Term 1-2) / MA060428 / 20-22

Instructor: Vladimir Losyakov

Advanced course in quantum mechanics, in which the basic principles quantum theory is supplemented and applied to the study of specific physical systems. Modern methods of research of quantum systems are proposed – the construction of integrable potentials, the integral along trajectories, and the concepts of density matrix and effective action are introduced. The course involves a transition to the consideration of free field theories, their canonical quantization, and discussion of differences quantum mechanics from quantum field theory.
The purpose of the course is to consolidate the basic principles and methods of quantum theory, study the transition from quantum mechanics to quantum field theory. The course introduces the basic concepts necessary for studying the courses of the program “Mathematical physics”.
The course is designed as a solution to specific problems in quantum theory (see the course content).
The course involves significant independent work on solving problems.
I would like the results of the course to coincide with the goals.
Full Syllabus

Mathematical Methods of Science (Term 1-2) / MA060317 / 20-22

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 basic notions of classical mechanics and field 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
Full Syllabus

Critical Points of Functions (Term 1-2) / MA060424 / 20-22

Instructor: Maxim Kazarian

This course is an introduction to the supersymmetry, which is one of the main basic The theory of critical points of functions is of the main subjects of Singularity theory studying local geometry of singularities of differentiable maps as well as its relationship with global topological invariants of manifolds.
In the course we will discuss classification of critical points, its relationship with the ADE-series of simple Lie algebras and the corresponding reflection groups, their deformations and adjacencies. The study of a local topological structure of singularities will include description of Milnor fiber and vanishing cycles. We will discuss also application of the theory critical points to the study of caustics and wave fronts in geometric optics and classical mechanics, as well as enumeration of contact singularities in complex projective geometry.
Full Syllabus

Introduction to quantum groups (Term 1-2) / 20-22

Instructor: Mikhail Bershtein

Quantum groups were introduced in the mid-80’s and very quickly became one of the most important topics in mathematics and mathematical physics. They are still actively studied, and their knowledge is necessary for work in many areas.
The purpose of the course is an introduction to quantum groups. The content will be based on classic works of the 80’s and early 90’s, we will not get to the latest results. Initial knowledge about quantum groups is not assumed, but acquaintance with Lie algebras and Groups, Poisson brackets, and the first notions of category theory is assumed.

Modern Dynamical Systems (Term 3-4) / MA06257 / 20-22

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 / 20-22

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 / 20-22

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

Quiver Representations and Quiver Varieties (Term 3-4) / MA060425 / 20-22

Instructor: Evgeny Feigin

The theory of quivers is one of the central topics in various fields of modern mathematics and mathematical physics, such as algebraic geometry, representation theory, combinatorics, quantum field theory, integrable systems. The theory has lots of beautiful and deep theorems and is very popular due to a huge number of applications, including McKay correspondence, instantons and ADHM construction, geometric realization of the Kac-Moody Lie algebras. Many of the recent results and applications of the theory of quivers are based on the quiver verieties, introduced by Hiraku Nakajima 20 years ago. The course will cover the basic material on the structure theory of quivers and their representations, such as path algebras, Gabriel’s theorem, Hall algebras, preprojective algebras and Auslander-Reiten quivers. Based on the general theory of quiver representations we will discuss the definition of the Nakajima quiver varieties and several explicit examples and applications. The course is aimed at the graduate students or advanced bachelor students. The basic knowledge of algebraic geometry, differential geometry, and the theory of Lie groups and Lie algebras is expected.
Full Syllabus

Introduction to Quantum Theory (Term 3-4) / MA060332 / 20-22

Instructor: Vladimir Losyakov

One of the most striking breakthrough of the XX century is the creation of the entirely new area of physics named quantum physics. It emerged that the whole world around us obeys the laws of quantum mechanics, while the laws of classical physics that we are familiar with (such as, for example, Newton’s equations) describe only macroscopic objects and can be obtained in limiting case. After that a lot of phenomena in different areas of physics found their explanation. Also quantum mechanics had a very significant impact on the development of mathematics and mathematical physics. Today quantum mechanics is one of the keystone parts of theoretical and mathematical physics.
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Quantum Field Theory (Term 3-4) / MA060316 / 20-22

Instructor: Andrei Semenov

At present time Quantum Field Theory (QFT) is the main theoretical tool used for the description of the phenomena occurring in the microworld. Examples include interactions between elementary particles, hadron structure and so on. At the same time, QFT methods are widely used in all areas of modern theoretical physics such as condensed matter physics, statistical mechanics, turbulence theory and others. Moreover, the creation of QFT has stimulated the development of many modern areas of mathematics.
The course is aimed at the study of the basic ideas and methods of QFT, as well as the discussion of its applications in various areas of modern theoretical and mathematical physics. Topics include quantization of scalar and gauge theories, path integral approach, perturbative expansions and Feynman diagrams, (1+1) dimensional exactly soluble models and some other ideas of modern science
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