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 |
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# | 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 |
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Track – “Mathematical Physics” |
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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” |
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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 |
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25 | MB12006 | Research Immersion | 12 | P | C | 12 | |||||||||

Stream 3. “Enterpreneurship and Innovation” (E&I) / 6 ECTS credits |
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26 | MC06001 | Innovation workshop | 6 | P | C | 6 | |||||||||

Stream 4. “Research & MSc Thesis Project” / 48 ECTS credits |
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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 |
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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) |
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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
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. ## Research seminar “Supersymmetric Gauge Theories and Integrable Systems” (Term 1-2) / DA060382 / 20-22
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. ## Representations of Classical Groups and Related Topics (Term 1-2) / MA030423 / 20-22
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. ## Geometric Representation Theory (Term 1-2) / MA060271 / 20-22
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. ## Elliptic operators in topology of manifolds (Term 1-2) / MA060258 / 20-22
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 ## Classical Integrable Systems (Term 1-2/5-6) / MA060179 / 20-22
Course description: A self-contained introduction to the theory of soliton equations with an emphasis on their algebraic-geometrical integration theory. Topics include: ## Phase Transitions: Introduction to Statistical Physics and Percolation (Term 1-2) / MA060427 / 20-22
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. ## Quantum Mechanics (Term 1-2) / MA060428 / 20-22
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. ## Mathematical Methods of Science (Term 1-2) / MA060317 / 20-22
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 ## Critical Points of Functions (Term 1-2) / MA060424 / 20-22
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. ## Introduction to quantum groups (Term 1-2) / 20-22
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. ## Modern Dynamical Systems (Term 3-4) / MA06257 / 20-22
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. ## Gauge Fields and Complex Geometry (Term 3-4) / MA06178 / 20-22
1. Self-duality equations, Bogomolny equations. ## Quantum Integrable Systems (Term 3-4) / MA06315 / 20-22
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 ## Quiver Representations and Quiver Varieties (Term 3-4) / MA060425 / 20-22
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. ## Introduction to Quantum Theory (Term 3-4) / MA060332 / 20-22
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. ## Quantum Field Theory (Term 3-4) / MA060316 / 20-22
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. |