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 – 2021 |
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# | Code | Course Title | ECTS * | Grad / Pass |
* | Acad Year 21/22 | Sum-mer Term | Acad Year 22/23 | |||||||

Fall | Spring | Fall | Spring | ||||||||||||

Stream 1. “Science, Technology and Engineering (STE)” / 36 ECTS credits |
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1 | MA060461 | Research seminar “Cluster Integrable Systems and Supersymmetric Gauge Theories” | 6 | G | Re/Dr | 3 | 3 | X | X | ||||||

2 | MA060271 | Geometric Representation Theory | 3 | G | RE/Dr | 3 | 3 | X | X | ||||||

3 | MA060459 | Mapping Class Groups | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

4 | MA060465 | Statistical Mechanics, Percolation Theory and Conformal Invariance | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

5 | MA030458 | Symmetric Functions | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

6 | MA060462 | Affine Quantum Groups | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

7 | MA060257 | Modern dynamical systems | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

8 | MA060178 | Gauge fields and complex geometry | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

9 | MA060315 | Quantum integrable systems | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

10 | DA060259 | Vertex Operator Algebras | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

11 | DА060179 | Classical integrable systems | 6 | G | RE/Dr | X | X | 3 | 3 | ||||||

12 | MA060464 | Research seminar “Quantum mechanics” | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

13 | MA060332 | Introduction to quantum theory | 6 | G | RE/Dr | 3 | 3 | ||||||||

14 | MA060316 | Quantum field theory | 6 | G | RE/Dr | 3 | 3 | ||||||||

Stream 2. “Research Immersion” / 12 ECTS credits |
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15 | MB12006 | Research Immersion | 12 | P | C | 12 | |||||||||

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

Stream 4. “Research & MSc Thesis Project” / 48 ECTS credits |
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17 | DG120268 | 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 | ||

18 | MD060001 | Early Research Project | 6 | P | C | 3 | 3 | ||||||||

19 | MD120002 | Thesis Research project | 12 | P | C | 3 | 6 | 3 | |||||||

20 | MD090023 | Thesis Proposal, Status Review and Predefense | 9 | P | C | 3 | 3 | 3 | |||||||

21 | MD090003 | Thesis defence | 9 | SFA | C | 9 | |||||||||

Stream 5. “Options” / 18 ECTS credits |
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22 | Elective courses from Course Catalogue | 12 | E | X | X | X | X | X | X | ||||||

23 | Additional Thesis Research | P | E | X | X | X | X | X | X | ||||||

24 | Short-Term Project | 6 | P | E | X | X | X | X | X | ||||||

25 | Differential geometry of connections | 6 | G | RE/Dr | 3 | 3 | X | X | |||||||

Facultative (Extracurricular activities – maximum 20 ECTS credits overall, maximum 10 ECTS credits per year) |
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26 | MO03001 | English toolkit | 3 | P | E | 3 | |||||||||

27 | MO03002 | Academic writing essential | 3 | P | E | 3 | |||||||||

28 | MO06003 | Master your thesis in english | 3 | P | E | 3 | |||||||||

29 | MO06003 | Independent study period | 3 | P | E | 3 | 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) / DG060268 / 21-23
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 “”Cluster integrable systems and supersymmetric gauge theories” (Term 1-2) / MA060461 / 21-23
This research seminar will be devoted to the study of N=2 supersymmetric gauge theories ## Symmetric functions (Term 1-2) / MA060458 / 21-23
The theory of symmetric functions has numerous applications in various domains of mathematics and mathematical physics. At the beginning of the course, standard material will be presented, and then we will move on to more advanced topics. ## Geometric Representation Theory (Term 1-2) / DA060271 / 21-23
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. ## Mapping class groups (Term 1-2) / MA060459 / 21-23
For an oriented surface (2-manifold) S, the group Diff(S) of orientation-preserving diffeomorphisms of S is a huge infinite-dimensional topological group. By definition, the mapping class group of S is the group Mod(S) obtained from the group Diff(S) by taking the quotient by the identity component. Equivalently, Mod(S) is the group consisting of isotopy classes of orientation-preserving diffeomorphisms of S onto itself. ## Classical Integrable Systems (Term 1-2/5-6) / DA060179 / 21-23
Course description: A self-contained introduction to the theory of soliton equations with an emphasis on their algebraic-geometrical integration theory. Topics include: ## Statistical mechanics, percolation theory and conformal invariance (Term 1-2) / MA060465 / 21-23
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. ## Introduction to Schramm-Loewner Evolution (Term 1B-2) / MA060494 / 21-23
The Schramm-Loewner Evolution (SLE) was introduced in 1998 in order to describe all possible conformally invariant scaling limits that appear in many lattice models of statistical physics. Since then the subject has received a lot of attention and developed into a thriving area of research in its own right which has a lot of interesting connections with other areas of mathematics and physics. Beyond the aforementioned lattice models it is now related to many other areas including the theory of `loop soups’, the Gaussian Free Field, and Liouville Quantum Gravity. The emphasis of the course will be on the basic properties of SLE and how SLE can be used to prove the existence of a conformally invariant scaling limit for lattice models. ## Research seminar “Quantum Mechanics” (Term 1-2) / MA060464 / 21-23
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 ## Differential Geometry of Connections (Term 1-2) / MA060460 / 21-23
In this course we present the basic concepts of modern differential geometry: metric, curvature, connection, etc. The goal of our study is to develop tools for practical efficient computations (including the art of manipulation with indices) supported by a deeper understanding of the geometric meaning of all notions and theorems. We will develop an approach based on the notion of connection with all its different aspects: covariant derivative, parallel transport, collection of Christoffel symbols, matrix-valued one-form etc. ## Affine Quantum Groups (Term 1-2) / MA046062 / 21-23
The main source of the theory of quantum groups was integrable models. Affine quantum groups (better to say quantum universal enveloping of affine Lie algebras) are symmetries of the XXZ model, they appeared simultaneously with quantum group theory in the mid-80s. Affine quantum groups have many new properties (compared to quantum universal enveloping algebras of simple Lie algebras) — different realizations, different comultiplications. They have many applications, in addition to the integrable models mentioned above, we mention cluster algebras and geometric representation theory. ## Modern Dynamical Systems (Term 3-4) / MA06257 / 21-23
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) / MA060178 / 21-23
1. Self-duality equations, Bogomolny equations. ## Quantum Integrable Systems (Term 3-4) / MA060315 / 21-23
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 ## Vertex operator algebras (Term 3-4) / MA0604259/ 21-23
Infinite-dimensional Lie algebras (such as Virasoro algebra or affine Kac-Moody ## Introduction to Quantum Theory (Term 3-4) / MA060332 / 21-23
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 ## Quantum Field Theory (Term 3-4) / MA060316 / 21-23
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. |