Curriculum 17/18

Mathematical and Theoretical Physics MSc Program Structure

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
# 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
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
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)
17 ME06013 Applied methods of analysis 6 G RE A 3 3
18 ME06014 Random Matrices, Random Processes and Integrable Systems 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)
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
27 MB06006 Research Immersion 6 P C A
Stream 3. “Enterpreneurship and Innovation” (E&I) / 6 ECTS credits
28 MC06001 Innovation workshop 6 P C A
Stream 4. “Research & MSc Thesis Project” / 36 ECTS credits
29 MD06001 Early Research Project 6 P C A
3 3
30 MD24002 Thesis Research project 24 P C A
6 6 6 6
31 MD03006 Thesis defence 6 FSA C A
Stream 5. “Options” / 24 ECTS credits
32 Elective courses from Course catalogue P E A
33 ME06008 Research project P E A
Learning activities outside of Curriculum / maximum 30 ECTS credits overall, maximum 15 ECTS credits per year
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
Maximum overload per Year 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

Instructor: Grigori Olshanski

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:
— linear Lie groups and their Lie algebras;
— universal enveloping algebras;
— Haar measure on a linear Lie group;
— general facts about representations of compact groups and their characters;
— radial part of Haar measure;
— Weyl’s formula for characters of the unitary groups;
— Weyl’s unitary trick;
— classification and realization of representations;
— symmetric functions

Hamiltonian Mechanics / MA06271 / 17-18

Instructor: Andrei Marshakov

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 equations and dynamical systems,
== Lie groups and algebras,
== differential geometry on manifolds.
Many modern directions in mathematics (e.g. symplectic geometry) got their origin from the problems of classical mechanics.This course is recommended to all students, interested in mathematical physics, and it does not imply any special preliminary education in physics.
The preliminary program of the course includes:
1. Lagrangian formalism: minimal action principle, Euler-Lagrange equations, symmetries and integrals of motion, Noether theorem.
2. Simplest examples: dynamics for a single degree of freedom, Kepler’s problem etc.
3. Basis of the Hamiltonian formalism: phase space, Legebdre transform, Hamilton equations, the Poisson and symplectic structures, Darboux theorem.
4. The Hamilton-Jacobi equation, canonical transform, Liouville theorem.
5. Integrable systems: separation of variables, Liouville integrability. Systems with Lax representation.
6. Examples of integrable systems: Toda and Calogero problems, integrable systems on Lie groups, geometry of spectral curves etc.

Geometric Representation Theory / MA06256 / 17-18

Instructors: Mikhail Finkelberg, Alexander Braverman, assistant: Alexei Litvinov

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.

Dynamical Systems and Ergodic Theory / MA06257 / Term 1 17-18

Instructors: Aleksandra Skripchenko, Anton Zorich

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.
In accordance with this strategy, the course comprises several blocks closely related to each other. The first three of them (including very short introduction) are mainly mandatory. The decision, which of the topics listed below these three blocks would depend on the background and interests of the audience.

Gauge Theory and Gravitation / MA06178 / 17-18

Instructor: Alexei Rosly

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.
A traditional approach to Classical Field Theory has a perfect base in the 2nd volume of Landau-Lifshitz’ course. However, since that prominent book was written, new elements came forward, which required more knowledge of differential geometry and topology. In our lecture course, we shall get familiar with most important basic facts from these branches of mathematics with application to field theory. For example, understanding instantons (even at a classical level) requires good knowledge of a number of notions from modern math courses, such as vector bundles, connections, homotopy groups. Therefore our course has to go beyond the reach of Landau-Lifshitz’ volume 2.

Integrable Systems, 2 / ME06010 / 17-18

Instructor: Anton Zabrodin

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, caclulation 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 / MA06259 / 17-18

Instructor: Evgeny Feigin

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.
Our goal is to give an introduction to the theory of vertex operator algebras from the modern mathematical point of view. We describe the main definitions, constructions and applications of the theory. The course is aimed at PhD students and master students.

String Theory and Conformal Theory / MA06260 / 17-18

Instructor: Mikhail Bershtein

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

Instructor: Semen Shlosman

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.
The topics will include:
== Grand canonical, canonical and microcanonical ensembles,
== DLR equation,
== Thermodynamic limit,
== Gibbs distributions and phase transitions,
== One-dimensional models,
== Correlation inequalities ( GKS, GHS, FKG),
== Spontaneous symmetry breaking at low temperatures,
== Uniqueness at high temperatures and in non-zero magnetic field,
== Non-translation-invariant Gibbs states and interfaces,
== Dobrushin Uniqueness Theorem
== Pirogov–Sinai Theory
== O(N)-symmetric models
== the Mermin–Wagner Theorem
== Reflection Positivity and the chessboard estimate infrared bounds