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 – 2019 |
|||||||||||||||
# | Code | Curriculum element | ECTS | Grad / Pass |
* | ** | Acad Year 19/20 | Acad Year 20/21 | |||||||
Fall | Spring | Fall | Spring | ||||||||||||
Stream 1. “Science, Technology and Engineering (STE)” / Coursework / 48 ECTS credits | |||||||||||||||
Track A – “Mathematical Physics” | |||||||||||||||
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 | |
2 | MA06173 | Lie Groups and Lie algebras and their representations | 6 | G | RE/Dr | A | 3 | 3 | X | X | |||||
3 | MA06271 | Geometric representation theory | 6 | G | RE/Dr | A | 3 | 3 | X | X | |||||
4 | MA06175 | Differential geometry of connections | 6 | G | RE/Dr | A | 3 | 3 | X | X | |||||
5 | MA06381 | Supersymmetric gauge theories | 6 | G | RE/Dr | A | 3 | 3 | X | X | |||||
6 | MA06382 | Supersymmetric gauge theories and integrable systems | 6 | G | RE/Dr | A | 3 | 3 | |||||||
7 | MA06257 | Modern dynamical systems | 6 | G | RE/Dr | A | 3 | 3 | |||||||
8 | MA06178 | Gauge fields and complex geometry | 6 | G | RE/Dr | A | 3 | 3 | |||||||
9 | MA06315 | Quantum integrable systems | 6 | G | RE/Dr | A | 3 | 3 | |||||||
10 | MA06258 | Differential topology | 6 | G | RE/Dr | A | 3 | 3 | |||||||
11 | MA06259 | Vertex operator algebras | 6 | G | RE/Dr | A | 3 | 3 | |||||||
12 | MA06321 | Virasoro algebra and conformal field theory | 6 | G | RE/Dr | A | 3 | 3 | X | X | |||||
13 | ME06179 | Classical integrable systems | 6 | G | RE/Dr | A | 3 | 3 | |||||||
14 | MA06383 | Statistical mechanics, percolation theory and conformal invariance | 6 | G | RE/Dr | A | 3 | 3 | X | X | |||||
15 | MA06387 | Statistical mechanics | 6 | G | RE/Dr | A | 3 | 3 | |||||||
16 | MA06174 | Hamiltonian mechanics | 6 | G | RE/Dr | A | 3 | 3 | X | X | |||||
17 | MA06317 | Mathematical methods of science | 6 | G | RE/Dr | A | 3 | 3 | X | X | |||||
18 | MA06332 | Introduction to quantum theory | 6 | G | RE/Dr | A | 3 | 3 | |||||||
19 | ME06384 | Quantum theory | 6 | G | RE/Dr | A | 3 | 3 | X | X | |||||
20 | MA06316 | Quantum field theory | 6 | G | RE/Dr | A | 3 | 3 | |||||||
Track B – “Theoretical Physics” | |||||||||||||||
21 | 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 | |
22 | MA06207 | Advanced quantum mechanics | 6 | G | RE | B | 6 | ||||||||
23 | MA06262 | Functional methods in the theory of disordered systems | 6 | G | RE | B | 3 | 3 | |||||||
24 | MA06138 | Theory of phase transition | G | экз | RE | B | 3 | 3 | |||||||
25 | MA06274 | Introduction to the theory of disordered systems | 6 | G | RE | B | 3 | 3 | |||||||
26 | MA06273 | Introduction to the quantum field theory | 6 | G | RE | B | 3 | 3 | |||||||
27 | MA06275 | Asymptotic methods in complex analysis | 6 | G | RE | B | 3 | 3 | |||||||
28 | MA03385 | Numerical simulations of quantum many-body systems | 6 | G | RE | B | 3 | ||||||||
29 | MA06278 | Quantum mesoscopics. Quantum Hall effect | 6 | G | RE | B | 3 | 3 | |||||||
30 | MA06276 | One-dimensional quantum systems | 6 | G | RE | B | 3 | 3 | |||||||
Stream 2. “Research” / 12 ECTS credits | |||||||||||||||
31 | MB12006 | Research Immersion | 12 | P | C | A B |
12 | ||||||||
Stream 3. “Enterpreneurship and Innovation” (E&I) / 6 ECTS credits | |||||||||||||||
32 | MC06001 | Innovation workshop | 6 | P | C | A B |
6 | ||||||||
Stream 4. “Research & MSc Thesis Project” / 36 ECTS credits | |||||||||||||||
33 | MD06001 | Early Research Project | 6 | P | C | A B |
3 | 3 | |||||||
34 | MD24002 | Thesis Research project | 21 | P | C | A B |
6 | 6 | 6 | 3 | |||||
35 | MD06003 | Thesis defence | 9 | FSA | C | A B |
9 | ||||||||
Stream 5. “Options” / 18 ECTS credits | |||||||||||||||
36 | Elective courses from Course Catalogue | E | A B |
X | X | X | X | X | X | X | |||||
37 | Research project | P | E | A B |
X | X | X | X | X | X | |||||
Facultative (Extracurricular activities – maximum 20 ECTS credits overall, maximum 10 ECTS credits per year) | |||||||||||||||
38 | MO03001 | English toolkit | 3 | P | E | A B |
3 | ||||||||
39 | MO03002 | Academic writing essential | 3 | P | E/Dr | A B |
3 | ||||||||
40 | MO06003 | Master your thesis in english 1 | 6 | P | RE/Dr | A B |
2 | 2 | 2 | ||||||
41 | MO06003 | Master your thesis in english 2 | 6 | P | RE/Dr | A B |
2 | 2 | 2 | ||||||
42 | 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 / 19-21Instructor: Pavlo Gavrylenko 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. Lie Groups and Lie Algebras, and their Representations (Term 1-2) / MA06173 / 19-21Instructor: 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). Geometric Representation Theory (Term 1-2) / MA06271 / 19-21Instructors: 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 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. Differential Geometry of Connections (Term 1-2) / MA06175 / 19-21Instructor: Maxim Kazaryan Supersymmetric Gauge Theories (Term 1-2) / MA06381 / 19-21Instructors: Pavlo Gavrylenko, Andrei Marshakov, Alexei Yung This course is an introduction to the supersymmetry, which is one of the main basic principles, being used both in field theory and in string theory. The main emphasis will be made on the consideration of dynamics of supersymmetric gauge field theories: in particular, at the non-perturbative level. Course starts from the introduction of the main concepts: supersymmetric algebra, its representations, chiral and vector multiplets. One will consider Wess-Zumino model and introduce superpotential. Then one will consider supersymmetric quantum electrodynamics, discuss supersymmetric Higgs mechanism and Higgs branches. One will consider in detail supersymmetric quantum chromodynamics. Its vacuum structure at the classical level will be discussed. Then one will introduce instantons and explain how they change vacuum structure by generating Affleck-Dine-Seiberg superpotential. Finally we will consider Seiberg duality Supersymmetric Gauge Theories and Integrable Systems (Term 3-4) / MA06382 / 19-21Instructors: Pavlo Gavrylenko, Andrei Marshakov Modern Dynamical Systems (Term 3-4) / MA06257 / 19-21Instructors: 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. Gauge Fields and Complex Geometry (Term 3-4) / MA06178 / 19-21Instructor: Alexei Rosly Quantum Integrable Systems (Term 3-4) / MA06315 / 19-21Instructor: 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.
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 Differential Topology (Term 3-4) / MA06258 / 19-21Instructor: Alexander Gaifullin Vertex Operator Algebras (Term 3-4) / MA06259 / 19-21Instructor: 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 dimensional 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. Virasoro Algebra and Conformal Field Theory (Term 1-2) / MA06321 / 19-21Instructor: 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, which possess an infinite dimensional algebra of local conformal transformation, which includes Virasoro Lie algebra. Classical Integrable Systems (Term 5-6) / MA06179 / 19-21Instructor: Igor Krichever 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. Statistical Mechanics, Percolation Theory and Conformal Invariance (Term 1-2) / MA06383 / 19-21Instructor: Semen Shlosman This is a course on rigorous results in statistical mechanics, random fields and percolation theory. Some of it will be dedicated to the theory of phase transitions, uniqueness or non-uniqueness of the lattice Gibbs fields. We will also study the models at the criticality, where one hopes to find (in dimension 2) the onset of conformal invariance. We will see that it is indeed the case for the percolation and the Ising model. The topics will include: Statistical Mechanics (Term 5-6) / MA06387 / 19-21Instructor: Semen Shlosman Hamiltonian Mechanics (Term 1-2) / MA06174 / 19-21Instructor: Igor Krichever Mathematical Methods of Science (Term 1-2) / MA06317 / 19-21Instructor: Sergei Khoroshkin 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 a minor 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. In the endpoint the students are assumed to acquire the use of Green functions, distributions, Laplace and Fourier transforms, asymptotic evaluations in mathematical physics Introduction to Quantum Theory (Term 3-4) / MA06332 / 19-21Instructor: Vladimir Losyakov Quantum Theory (Term 1-2) / MA06384 / 19-21Instructor: Vladimir Losyakov The course is addressed to students who know the basics and basic principles of quantum theory. Physical applications of quantum theory on examples of specific problems will be considered. It is planned to study the following issues: supersymmetric quantum mechanics, electromagnetic field interacting with an external source, the Caldeira-Leggett model, the band structure of onedimensional systems, the basics of quantum Informatics. As can be seen from the list, the purpose of the course is to prepare the student for the study of quantum Quantum Field Theory (Term 3-4) / MA06316 / 19-21Instructor: Andrei Semenov |