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 – 2022 |
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# | Code | Course Title | ECTS * | Grad / Pass |
* | Acad Year 22/23 | Sum-mer Term | Acad Year 23/24 | |||||||

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

Stream 1. “Science, Technology and Engineering (STE)” / 36 ECTS credits |
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> Elective Part / 36 ECTS credits | |||||||||||||||

1 | MA060513 | Universal enveloping algebras and Yangians | 6 | G | Re/Dr | 3 | 3 | X | X | ||||||

2 | DA060271 | Geometric Representation Theory | 3 | G | RE/Dr | 3 | 3 | X | X | ||||||

3 | MA060427 | Phase transitions: introduction to statistical physics and percolation | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

4 | MA060424 | Critical points of functions | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

5 | MA060509 | Introduction to cluster algebras and varieties | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

6 | MA060543 | Cohomology of groups and classifying spaces | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

7 | MA060425 | Quiver representations and quiver varieties | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

8 | MA060542 | Path integral: stochastic processes and basics of quantum mechanics | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

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

10 | MA060178 | Hitchin systems and complex geometry | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

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

12 | MA060505 | Introduction to quantum field theory | 6 | G | RE/Dr | 3 | 3 | X | X | ||||||

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

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

> Compulsory Part / 6 ECTS credits | |||||||||||||||

Stream 4. “Research & MSc Thesis Project” / 48 ECTS credits |
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> Compulsory Part / 48 ECTS credits | |||||||||||||||

15 | 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 | ||

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

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

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

19 | MD090003 | Thesis defence | 9 | SFA | C | 9 | |||||||||

Stream 5. “Options” / 18 ECTS credits |
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> Elective Part / 18 ECTS credits | |||||||||||||||

20 | Elective courses from Course Catalogue | 12 | X | X | X | X | X | X | |||||||

21 | MEOX0040 | Additional Thesis Research | 6 | P | E | X | X | X | X | X | X | ||||

22 | MEOX0041 | 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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23 | MF030001 | English toolkit | 3 | P | E | 3 | |||||||||

24 | MF030002 | Academic writing essential | 3 | P | E | 3 | |||||||||

25 | MF030003 | Master your thesis in english | 3 | P | E | 3 | |||||||||

26 | MF030010 | 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 / 22-24
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: ## Universal Enveloping Algebras and Yangians (Term 1-2) / MA060513 / 22-24
Tentative program of the course: ## Geometric Representation Theory (Term 1-2) / DA060271 / 22-24
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. ## Cohomology of Groups and Classifying Spaces (Term 1-2) / MA060543 / 22-24
The course will include an introduction to theory of cohomology of groups from geometric viewpoint. The concept of a classifying space of a group will be central in this course. We will start with general algebraic definitions and theorems concerning homology and cohomology of (discrete) groups. Then we proceed with various constructions of classifying spaces of groups and methods for computing their cohomology. An important part of the course will be devoted to examples such as certain finite groups, braid groups, Coxeter groups, mapping class groups, etc. ## Phase Transitions: introduction to Statistical Physics and Percolation (Term 1-2) / MA060427 / 22-24
TThis 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. ## Introduction to quantum field theory (Term 3-4) / MA060505 / 22-24
As you know, the modern theory of fundamental physics (the “standard model of elementary particle physics”) is a quantum field theory (QFT). In addition to this central role in modern physics, quantum field theory also has many applications in pure mathematics (for example, from it came the so-called quantum knot invariants and Gromov-Witten invariants of symplectic manifolds). Within the framework of this course, the basic concepts of QFT will be introduced “from scratch”. The Fock space and the formalism of operators on it, as well as the formalism of the “continuum integral” will be defined. The main example under consideration will be the quantum scalar field theory. A scalar field in physical terminology is a field that, at the classical level, is defined by one number at each point (i.e., in fact, its state at a given time is just a numerical function on space), unlike a vector field (an example of which, in particular, is an electromagnetic field). However, considering the quantum theory of a scalar field (even separately, and simpler than for the Higgs field) is in any case very useful, since it allows you to get acquainted with the apparatus and phenomena of QFT on a simpler example than vector and spinor fields. The course will consider the “perturbation theory” (that is, in fact, a method for calculating the first orders of smallness in a small parameter expansion) for a scalar field and describe ways to calculate various probabilities of events with particles. ## Critical points of functions (Term 1-2) / MA060460 / 22-24
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. ## Introduction to cluster algebras and varieties (Term 1-2) / MA060509 / 22-24
Cluster algebras and cluster varieties appeared almost simultaneously in the early 2000s; the algebras were introduced in the works of Fomin and Zelevinsky, and varieties in the works of Fock and Goncharov. These notions rather quickly found numerous applications and became popular, for the last 15 years this popularity has not decreased in any way. ## Modern Dynamical Systems (Term 3-4) / MA06257 / 22-24
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. ## Hitchin systems and complex geometry (Term 3-4) / MA060508 / 22-24
The subject of this course will be mainly the complex geometry. The choice of topics, however, is determined by their uses in Field Theory and Theory of Integrable Systems.
1. The self-duality equations (SDE) in the Yang-Mills theory as the 0-curvature equation with a spectral parameter (Belavin-Zakharov). A construction by Ward. ## Quantum Integrable Systems (Term 3-4) / MA060315 / 22-24
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 1-2) / MA060425/ 22-24
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. ## Path integral: stochastic processes and basics of quantum mechanics (Term 1-2) / MA060542 / 22-24
One of the most powerful methods of modern theoretical physics is the method of functional integration or path integration. The foundations of this approach were developed by N. Wiener at the beginning of the 20th century, but it spread widely after R. Feynman, who applied this approach in quantum mechanics. At present, the functional integral has found its application in the theory of random processes, polymer physics, quantum and statistical mechanics, and even in financial mathematics. Despite the fact that in some cases its applicability has not yet been mathematically rigorous proven, this method makes it possible to obtain exact and approximate solutions of various interesting problems with surprising elegance. The course is devoted to the basics of this approach and its applications to the theory of random processes and quantum mechanics. In the first part of the course, using the example of stochastic differential equations, the main ideas of this approach will be described, as well as various methods for exact and approximate calculation of functional integrals. Further, within the framework of the course, the main ideas of quantum mechanics will be considered, and both the operator approach and the approach using functional integration will be considered. It will be demonstrated that, from the point of view of formalism, the description of random processes and the description of quantum mechanical systems are very similar. This will make it possible to make a number of interesting observations, such as, for example, the analogy between supersymmetric quantum mechanics and the diffusion of a particle in an external potential. In the final part of the course, depending on the interests of the audience, various applications of the functional integration method will be discussed, such as polymer physics, financial mathematics, etc. |