Master program curriculum // 03.04.01 Applied Math and Physics, “Mathematical and Theoretical Physics”, full-time, onsite form of study, study period – 2 years, year of admission – 2017 |
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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 | MA12268m | Modern problems of mathematical physics | 12 | G | С | A | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | |

1 | MA12268p | Modern problems of theoretical physics | 12 | G | С | B | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | |

2 | MA06173 | Lie Groups and Lie algebras and their representation | 6 | G | RE | A | 3 | 3 | |||||||

3 | MA06174 | Hamiltonian mechanics | 6 | G | RE | A | 3 | 3 | |||||||

4 | MA06175 | Differential and symplectic geometry | 6 | G | RE | A | 3 | 3 | |||||||

5 | MA06176 | Strings and cluster varieties | 12 | G | RE | A | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | |

6 | MA06256 | Geometric representation theory | 6 | G | RE | A | 3 | 3 | |||||||

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

8 | MA06261 | Advanced quantum mechanics | 6 | G | RE | B | 6 | ||||||||

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

10 | MA06179 | Integrable systems 1 | 6 | G | RE | A | 3 | 3 | |||||||

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

12 | MA06259 | Vertex operator algebras | 6 | G | RE | A | 3 | 3 | |||||||

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

14 | MA06260 | String theory and conformal theory | 6 | G | RE | A | 3 | 3 | |||||||

15 | MA06180 | Statistical physics | 6 | G | RE | A | 3 | 3 | |||||||

16 | ME06010 | Integrable systems 2 | 6 | G | RE | A | 3 | 3 | |||||||

In collaboration with Higher School of Economics (HSE) |
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17 | ME06013 | Applied methods of analysis | 6 | G | RE | A | 3 | 3 | |||||||

18 | ME06014 | Random processes | 6 | G | RE | A | 3 | 3 | |||||||

19 | MA06177 | Quantum mechanics | 6 | G | RE | A | 3 | 3 | |||||||

20 | ME06011 | Quantum field theory | 6 | G | RE | A | 3 | 3 | |||||||

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

22 | MA06263 | Introduction to the theory of disordered systems | 6 | G | RE | B | 3 | 3 | |||||||

23 | MA06264 | Introduction to the quantum field theory | 6 | G | RE | B | 3 | 3 | |||||||

24 | MA06265 | Asymptotic methods in complex analysis | 6 | G | RE | B | 3 | 3 | |||||||

25 | MA06266 | Quantum mesoscopics. Quantum Hall effect | 6 | G | RE | B | 3 | 3 | |||||||

26 | MA06267 | One-dimensional quantum systems | 6 | G | RE | B | 3 | 3 | |||||||

Stream 2. “Research” / 6 ECTS credits |
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27 | MB06006 | Research Immersion | 6 | P | C | A B |
6 | ||||||||

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

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

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

31 | MD03006 | Thesis defence | 6 | FSA | C | A B |
6 | ||||||||

Stream 5. “Options” / 24 ECTS credits |
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32 | Elective courses from Course catalogue | P | E | A B |
X | X | X | X | X | X | X | ||||

33 | ME06008 | Research project | P | E | A B |
X | X | X | X | X | X | ||||

Learning activities outside of Curriculum / maximum 30 ECTS credits overall, maximum 15 ECTS credits per year |
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Сompulsory elements | 1 | 2 | 7 | 8 | 6 | 13 | 8 | 7 | 14 | ||||||

Recommended electives | 31 | 38 | 31 | 32 | 10 | 11 | 1 | 2 | |||||||

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

Total by year (without facultative) | 75 | 75 | |||||||||||||

TOTAL | 120-150 | ||||||||||||||

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

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

Examples of Learning activities outside of Curriculum: English | Academic writing | Individual study period | Colloquium | Scientific seminar |

## Lie Groups and Lie Algebras, and their Representations / MA06173 / 17-18
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 / MA06271 / 17-18
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: ## Geometric Representation Theory / MA06256 / 17-18
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 / Term 1 17-18
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 / 17-18
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. ## Integrable Systems, 2 / ME06010 / 17-18
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). ## Vertex Operator Algebras / MA06259 / 17-18
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. ## String Theory and Conformal Theory / MA06260 / 17-18
Conformal field theory is a quantum field theory that is invariant under conformal transformations. The course is devoted to a two-dimensional theory, there is an infinite-dimensional algebra of local conformal transformations. In the course, we will discuss aspects of the conformal theory, basic, but not included in the usual introductory courses. A small preliminary acquaintance with string theory and conformal field theory is assumed. We will mainly focus on the mathematical aspects of the theory, the relations with the representation theory, geometry, combinatorics, special functions ## Statistical Physics / MA06180 / 17-18
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. |