Master program curriculum // Educational program “Mathematical and Theoretical Physics”, Field of Science and technology 03.04.01 Applied Math and Physics, Fulltime study, study period – 2 years, year of admission – 2024 

#  Course Title // Code  E C T S *  Year 1  Summer Term  Year 2  
Fall  Winterm  Spring  Fall  Winterm  Spring  
Stream 1. “Science, Technology and Engineering (STE)” / 36 ECTS credits  
> Elective Part / 36 ECTS credits  
1  Classical Groups, their Invariants and Representations // MA060690  6  3  3  X  X  
2  Geometric Representation Theory // DA060271  6  3  3  X  X  
3  Introduction in Algebraic Topology // MA060691  6  3  3  X  X  
4  Cluster Varieties and Integrable Systems // MA060597  6  3  3  X  X  
5  Cluster Integrable Systems // MA060692  6  3  3  X  X  
6  Characteristic Classes // MA060693  6  3  3  X  X  
7  Path Integral: Stochastic Processes and Basics of Quantum Mechanics // MA060542  6  3  3  X  X  
8  Introduction to TwoDimensional Conformal Field Theory // MA060694  6  3  3  X  X  
9  Integrable Systems of Classical Mechanics // MA060695  6  3  3  X  X  
10  Derived Equivalences of Satake Type // MA060696  6  3  3  X  X  
11  Hamiltonian Mechanics // MA060697  6  3  3  X  X  
12  Around the Ising Model in 20 Hours // MA060698  6  3  3  X  X  
13  Methods of Conformal Field Theory for Quantum Field Theory and String Theory // MA060699  6  3  3  X  X  
14  Modern dynamical systems // MA060257  6  3  3  X  X  
15  Integrable ManyBody Systems and Nonlinear Equations // MA060602  6  3  3  X  X  
16  Introduction to Quantum Field Theory // MA060505  6  3  3  X  X  
17  Geometry in Field Theory, First Step // MA060724  6  3  3  X  X  
Stream 2. “Research Immersion” / 12 ECTS credits  
18  Research Immersion // MB12006  12  12  
Stream 3. “Enterpreneurship and Innovation” (E&I) / 6 ECTS credits  
> Compulsory Part / 6 ECTS credits  
19  Innovation Workshop // MC06001  6  6  
20  Mathematical Modelling in Innovation // DC060021  6  6  
Stream 4. “Research & MSc Thesis Project” / 48 ECTS credits  
> Compulsory Part / 48 ECTS credits  
21  Research seminar – Modern Problems of Mathematical Physics // DG120268  12  1.5  1.5  1.5  1.5  1.5  1.5  1.5  1.5  
22  Early Research Project // MD060001  6  3  3  
23  Thesis Research project // MD120002  12  3  3  6  
24  Thesis Proposal, Status Review and Predefense // MD090023  9  3  3  X  3  
25  Thesis defence // MD090003  9  9  
Stream 5. “Options” / 18 ECTS credits  
> Compulsory Part / 6 ECTS credits  
26  ShortTerm Project // MEOX0041  6  X  X  X  X  X  
27  Additional Thesis Research // MEOX0040  6  X  X  X  X  
> Elective Part / 12 ECTS credits  
28  English Toolkit // ME030568  3  3  
29  Academic Writing Essential // MF030002  3  3  X  
30  First Steps to Thesis in English // ME030566  3  3  
31  Master Your Thesis in English // ME030567  3  3  
32  Master Your Thesis in English 2 // ME030668  3  3  
33  Elective Courses from Course Catalogue  6  X  X  X  X  X  X  
Facultative (Extracurricular activities – maximum 20 ECTS credits overall, maximum 10 ECTS credits per year)  
34  Elective Courses from Course Catalogue  
35  Independent Study Period **// MF030010  X  
Minimum overload per Year  60  60  
Maximum overload per Year  70  70  
TOTAL  120140  
*) ECTS – European Credit Transfer and Accumulation System, X – curriculum element can be chosen in specified Terms, SFA – State Final Assessment, **)Independent Study Period workload is counted in astronomical hours 
Research seminar – Modern Problems of Mathematical Physics (Term 14) / DG120268 / 2426Instructors: Andrei Marshakov, Alexey Litvinov Classical groups, their invariants and representations (Term 12) / MA060690 / 2426Instructor: Grigori Olshanski The title of the course is deliberately copied from the famous book by Hermann Weyl (1939; 1946). The material in the book forms the core of representation theory. For this reason, working through this material is useful for everyone who wants to deal with any problems in representation theory or apply its results. The purpose of the course is to introduce students to the main ideas and results of Weyl’s book, as well as to their further development. Of course, in addition to Weyl’s book, we will use other, more modern sources. Geometric representation theory (Term 12) / DA060271 / 2426Instructor: 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 LanglandsShelstad fundamental Lemma, the proof of the KazhdanLusztig 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 (DeligneLanglands conjecture). This is a course for master students knowing the basics of algebraic geometry, sheaf theory, homology and Ktheory. Introduction in algebraic topology (Term 12) / MA060691 / 2426Instructor: Maxim Kazarian Symplectic geometry is a kind of skewsymmetric analogue of Riemannian geometry. This domain of differential geometry serves as a geometric basis of calculus of variations, quantum and classical mechanics, geometric optics and thermodynamics. The language of symplectic geometry is used everywhere in modern mathematics: in the Lie group theory, the theory of differential equations, integrable systems, singularity theory, topology. The course will cover both basic concepts of the theory and some specific problems needed for particular applications. Cluster varieties and integrable systems (Term 12) / MA060597 / 2426Instructor: Andrei Marshakov We start with definition of the cluster Poisson varieties from the angle of view of Hamiltonian mechanics and geometry of Poisson manifolds. Then we turn to description of the basic features of the FockGoncharov construction for the Lie groups, being (together with the spaces of mould of flat connections) an important example of Poisson manifold, possessing cluster structure. Then we formulate main properties of integrable systems on Poisson submanifolds in Lie groups. Finally, we plan to discuss the equivalence of the class of such systems for Aseries with their combinatorial construction of Goncharov and Kenyon, and possibly their deautonomization and relation with supersymmetric gauge theories. Cluster integrable systems (Term 12) / MA060692 / 2426Instructor: Mikhail Bershtein Characteristic classes (Term 12) / MA060693 / 2426Instructor: Alexander Gaifullin The course will include introduction to theory of characteristic classes, namely, the StiefelWhitney, Chern and Pontryagin classes of vector bundles and the MillerMoritaMumford classes of fibre bundles with fibre an oriented surface. The exposition will be based on the splitting principle and on the computation of the cohomology rings of the Grassmann manifolds. We will also discuss the connection with theory of cohomological operations (Steenrod squares), including the formulae due to Wu and Thom. Path integral: stochastic processes and basics of quantum mechanics (Term 12) / MA060542 / 2426Instructor: Andrei Semenov 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. Introduction to twodimensional conformal field theory (Term 12) / MA060694 / 2426Instructor: Alexey Litvinov Conformal field theories are relatively simple quantum field theories that serve as starting points for perturbation theory for more generic quantum field theories with the mass gap. In two dimensions, unlike higher dimensions, the algebra of conformal transformations is infinitedimensional. As a result, it has been possible to exactly solve certain nontrivial twodimensional conformal field theories. This course provides introduction to basic concepts of twodimensional conformal field theory. We will review basic ideas of the bootstrap approach to quantum field theory and describe the mathematical structures that appear in conformal field theory: representation theory of the Virasoro algebra, differential equations of correlation functions, conformal blocks etc. Integrable systems of classical mechanics (Term 12) / MA060695 / 2426Instructor: Vadim Prokofev The course is devoted to the study of integrable systems, their general properties and methods applied to their study them. We will use the example of integrable systems of classical mechanics. Using the example of such models as the CalogeroMoser, RuijenaarsSchneider, Toda systems, we will talk about the Lax representation, rmatrix, Bäcklund transformations – methods used for integrable systems in general. In addition, attention will be paid to the relationship of integrable systems with each other and with other branches of mathematical physics. Derived equivalences of Satake type (Term 12) / MA060696 / 2426Instructor: Artem Prikhodko Hamiltonian mechanics (Term 12) / MA060697 / 2426Instructor: Vladimir Poberezhny Hamiltonian mechanics is one of basic disciplines in mathematical physics. The scope of our course is to introduce students to modern views on the fundamentals of the theory of integrable systems and mathematical physics. Mastering its program makes it possible the further study of advanced courses of mathematical physics. Around the Ising model in 20 hours (Term 34) / MMA060698 / 2426Instructor: Semen Shlosman Methods of conformal field theory for quantum field theory and string theory (Term 34) / MA060699 / 2426Instructor: Alexey Litvinov The course assumes a basic knowledge of twodimensional conformal field theory. The course will emphasize various methods and approaches of twodimensional conformal field theory with application in quantum field theory and string theory. In particular, such issues as: conformal bootstrap and classification of partition functions in CFT, conformal field theory with the boundary, counting the number of physical states in string theory, Dbranes in string theory, renormalization group and Zamolodchikov’s ctheorem, integrable perturbations of twodimensional conformal theory will be considered. Modern dynamical systems (Term 34) / MA060257 / 2426Instructors: 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. Integrable manybody systems and nonlinear equations (Term 34) / MA060602 / 2426Instructor: Anton Zabrodin This course is devoted to manybody integrable systems of classical mechanics such as CalogeroMoser, RuijsenaarsSchneider and their spin generalizations. These systems play a significant Geometry in field theory, first step (Term 34) / MA060724 / 2426Instructor: Alexei Rosly This is a continuation of the subject started in “Some Uses of Twistors in Field Theory”, 2024.
