Master program curriculum // Educational program “Mathematical and Theoretical Physics”,Field of Science and technology 03.04.01 Applied Math and Physics, full-time, onsite form of study, study period – 2 years, year of admission – 2018 |
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# | Code | Curriculum element | ECTS | Grad / Pass |
* | ** | Year 1 | Year 2 | |||||||

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

Stream 1. “Science, Technology and Engineering (STE)” / Coursework / 48 ECTS credits |
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1.1 | MA12268 | Research seminar – Modern problems of mathematical physics | 12 | G | С/Dr | A | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | |

1.2 | MA12319 | Research seminar – Modern problems of theoretical physics | 12 | G | С/Dr | B | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | |

2 | MA06176 | Research seminar – Strings and cluster varieties | 12 | G | RE/Dr | A | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | |

3 | MA06173 | Lie Groups and Lie algebras and their representations | 6 | G | RE/Dr | A | 3 | 3 | |||||||

4 | MA06174 | Hamiltonian mechanics | 6 | G | RE/Dr | A | 3 | 3 | |||||||

5 | MA06175 | Differential and symplectic geometry | 6 | G | RE/Dr | A | 3 | 3 | |||||||

6 | MA06271 | Geometric representation theory | 6 | G | RE/Dr | A | 3 | 3 | |||||||

7 | MA06257 | Dynamical systems and ergodic theory | 6 | G | RE/Dr | A | 3 | 3 | |||||||

8 | MA06207 | Advanced quantum mechanics | 6 | G | RE | B | 6 | ||||||||

9 | MA06178 | Gauge theory and gravitation | 6 | G | RE/Dr | A | 3 | 3 | |||||||

10 | MA06315 | Quantum integrable systems | 6 | G | RE/Dr | A | 3 | 3 | |||||||

11 | MA06258 | Differential topology | 6 | G | RE/Dr | A | 3 | 3 | |||||||

12 | MA06320 | Representations of affine Kac-Moody algebras | 6 | G | RE/Dr | A | 3 | 3 | |||||||

13 | MA06262 | Functional methods in the theory of disordered systems | 6 | G | RE | B | 3 | 3 | |||||||

14 | ME06179 | Classical integrable systems | 6 | G | RE/Dr | A | 3 | 3 | |||||||

15 | MA06260 | Affine Lie algebras and conformal field theory | 6 | G | RE/Dr | A | 3 | 3 | |||||||

16 | MA06180 | Statistical physics | 6 | G | RE/Dr | A | 3 | 3 | |||||||

In collaboration with Higher School of Economics (HSE) |
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17 | ME06317 | Applied methods of analysis | 6 | G | RE/Dr | A | 3 | 3 | |||||||

18 | ME06318 | Random matrices, random processes and integrable systems | 6 | G | RE/Dr | A | 3 | 3 | |||||||

19 | MA06322 | Quantum mechanics | 6 | G | RE/Dr | A | 3 | 3 | |||||||

20 | ME06316 | Quantum field theory | 6 | G | RE/Dr | A | 3 | 3 | |||||||

21 | Elective courses from HSE course catalog | E | A | x | x | x | x | x | x | x | |||||

In collaboration with Moscow Institute of Physics and Technology (MIPT) |
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22 | MA06138 | Theory of phase transition | G | экз | RE | B | 3 | 3 | |||||||

23 | MA06274 | Introduction to the theory of disordered systems | 6 | G | RE | B | 3 | 3 | |||||||

24 | MA06273 | Introduction to the quantum field theory | 6 | G | RE | B | 3 | 3 | |||||||

25 | MA06275 | Asymptotic methods in complex analysis | 6 | G | RE | B | 3 | 3 | |||||||

26 | MA06278 | Quantum mesoscopics. Quantum Hall effect | 6 | G | RE | B | 3 | 3 | |||||||

27 | MA06276 | One-dimensional quantum systems | 6 | G | RE | B | 3 | 3 | |||||||

28 | Elective courses from MIPT course catalog | E | B | x | x | x | x | x | x | x | |||||

Stream 2. “Research” / 12 ECTS credits |
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29 | MB12006 | Research Immersion | 12 | P | C | A B |
12 | ||||||||

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

Stream 4. “Research & MSc Thesis Project” / 36 ECTS credits |
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31 | MD06001 | Early Research Project | 6 | P | C | A B |
3 | 3 | |||||||

32 | MD24002 | Thesis Research project | 24 | P | C | A B |
6 | 6 | 6 | 6 | |||||

33 | MD06003 | Thesis defence | 6 | FSA | C | A B |
6 | ||||||||

Stream 5. “Options” / 18 ECTS credits |
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34 | Elective courses from Course Catalogue | E | A B |
X | X | X | X | X | X | X | |||||

35 | Research project | P | E | A B |
X | X | X | X | X | X | |||||

Learning activities outside of Curriculum / maximum 20 ECTS credits overall, maximum 10 ECTS credits per year |
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36 | Academic writing | P | E | A B |
x | x | x | x | x | x | x | ||||

37 | MO03001 | English toolkit | 3 | P | E | A B |
3 | ||||||||

38 | MO03002 | Academic writing essential | 3 | P | E/Dr | A B |
3 | ||||||||

39 | MO06003 | Master your english for thesis | 6 | P | RE/Dr | A B |
2 | 2 | 2 | ||||||

40 | MO06003 | Independent study period | A B |
x | |||||||||||

Minimum overload per Year | 60 | 60 | |||||||||||||

Maximum overload per Year | 70 | 70 | |||||||||||||

TOTAL | 120-140 | ||||||||||||||

*) C – сompulsory curriculum element, RE – recommended elective, E – elective, Dr – suitable for PhD | |||||||||||||||

**) Track A – “Mathematical Physics”, Track В – “Theoretical Physics” |

## Research seminar “Modern Problems of Mathematical Physics” (Term 1-8) / MA12268 / 18-19
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: now it is devoted to the study of N=2 supersymmetric gauge theory and its links with random matrix models, ABJM theory, localization, complex curves, and integrable systems. Other 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 ## Research seminar “Strings and Cluster Varieties” (Term 1-8) / MA12176 / 18-19
The course is directed to substantive work of the master and PhD students in order to understand recently found relations among supersymmetric gauge theories, refined topological strings, cluster varieties and integrable systems. The plan of wrk on the course consists of several introductive lectures on the various consistuents of the subject as well as student talks on recent original papers and results of their own investigation. The core topics include relation between cluster varieties and Painleve equations and approaches to the Seiberg-Witten theories with fundamental matter based on Toda systems and spin chains ## Lie Groups and Lie Algebras, and their Representations / MA06173 / 18-19
We shall begin with the basics of the theory of Lie groups and Lie algebras. Then we shall provide an accessible introduction to the theory of finite-dimensional representations of classical groups on the example of the unitary groups U(N). Tentative plan: ## Hamiltonian Mechanics / MA06174 / 18-19
This is the first among the base courses in the theoretical physics, aimed for the master students. Matematical methods of modern theory of Hamiltonian systems are based on the concepts, arosen in different fields of mathematics: ## Differential and Symplectic Geometry / MA06175 / 18-19
Symplectic geometry is the mathematical language of classical mechanics a hamiltonian dynamic. The course will cover the base of symplectic and contact geometry: symplectic forms, symplectomorphisms, Hamilton vector fields, as well as contact counterparts of these objects. Some applications in differential geometry will include geometry of caustics and wave fronts and their singularities. Among topological applications we will discuss different aspects of Arnold’s conjecture on fixed points of symplectomorphisms and its variations ## Geometric Representation Theory / MA06271 / 18-19
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. ## Dynamical Systems and Ergodic Theory / MA06257 / 18-19
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. ## Gauge Theory and Gravitation / MA06178 / 18-19
The present course could be also entitled ‘Classical Field Theory’, which menas it deals with all basic material needed in a study of fields preceeding to a study of their quantum properties. This requires in particular understanding such tools as Lagrangian, action functional, field equations (Euler-Lagrange equations). We shall also learn what are the most important symmetry principles which put certain constraints on a field theory. With this are related conservation laws. Typical important symmetries to mention are Lorentz and Poincare symmetry, conformal symmetry, gauge symmetry, general coordinate covariance. ## Quantum Integrable Systems / MA06315 / 18-19
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). ## Differential Topology / MA06258 / 18-19
We plan to discuss two topics, which are central in topology of smooth manifolds, the h-cobordism theorem and theory of characteristic classes. The h-cobordism theorem proved by S. Smale in 1962 is the main (and almost the only) tool for proving that two smooth manifolds (of dimension greater than or equal to 5) are diffeomorphic. In particular, this theorem implies the high-dimensional Poincare conjecture (for manifolds of dimensions 5 and higher). Characteristic classes, in particular, Pontryagin classes are very natural invariants of smooth manifolds. Computation of characteristic classes can help one to distinguish between non-diffeomorphic manifolds. We plan to finish the course with the theorem by J. Milnor on non-trivial smooth structures on the 7-dimensional sphere. This theorem is based both on methods of Morse theory and theory of characteristic classes ## Representations of Affine Kac-Moody Algebras / MA06320 / 18-19
Infinite-dimensional Lie algebras (such as Virasoro algebra or affine Kac-Moody algebras) turn out to be very important in various areas of modern mathematics and mathematical physics. In particular, they are very useful in the description of some field theories. In this context one arranges infinite number of the Lie algebra elements into a single object called field. This idea generalizes to the general theory of vertex operator algebras. VOAs capture the main properties of the infinite diemensional Lie algebras and have rich additional structure. Vertex operator algebras proved to be very useful in many situations; the classical example is the KP integrable hierarchy. They are also extensively used in modern algebraic geometry. ## Classical Integrable Systems / MA06179 / 18-19
Course description: A self-contained introduction to the theory of soliton equations with an emphasis on their algebraic-geometrical integration theory. Topics include: ## Affine Lie Algebras and Conformal Field Theory / MA06321 / 18-19
The course will be devoted to conformal theories (more precisely, their chiral parts – vertex algebras) with the symmetry of the affine Lie algebra and the theories that are obtained from them by basic constructions. The basic examples are Wess-Zumino-Witten theory, coset theories, Drinfeld-Sokolov reductions. In general, we will discuss the mathematical aspects associated with the theory of representations and geometries. The course is supposed to be relatively advanced, see Course Prerequisites below. ## Statistical Physics / MA06180 / 18-19
This is a course on rigorous results in statistical physics and random fields. Most of it will be dedicated to the theory of phase transitions, uniqueness or non-uniqueness of the lattice Gibbs fields. ## Applied Methods of Analysis / MA06317 / 18-19
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 familiarity with various sections of theoretical physics (classical mechanics, field theory, quantum mechanics, statistical physics, hydrodynamics, elasticity 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. ## Quantum Field Theory / MA06316 / 18-19
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 familiarity with various sections of theoretical physics (classical mechanics, field theory, quantum mechanics, statistical physics, hydrodynamics, elasticity 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. ## Quantum Mechanics / MA06322 / 18-19
## Random Matrices, Random Processes and Integrable Systems / MA06318 / 18-19
In recent years, researchers have found remarkable connections between, at first glance, completely different problems of mathematics and theoretical physics. Mathematically, these are combinatorial and probabilistic problems about systems with a large number of degrees of freedom. Among them, the problem of describing the eigenvalues of matrices with random elements, the problem of statistics of random Young diagrams, the problem of tiling various regions of the plane by dominoes or lozenges, the problem of enumerating nonitersecting paths on lattices. On the physical side, these are the problems of interface growth, traffic flows, polymers in random media, and so on. |