Time Schedule / MSc / Fall 2017

9:30-11:10 (Year 1)
Geometric representation theory /
A.Litvinov (assistant)
starts on 2 Oct

(Year 1)
Dynamical Systems
Ergodic Theory /
starts on 5 Sep
11:30-13:30 (Years 1+2)
Modern problems of mathematical and theoretical physics /
Research seminar /
starts on 4 Sep
12:00-13:20 (Year 1)
Hamiltonian mechanics / Lecture /
starts on 6 Sep
Hamiltonian mechanics / Seminar / V.Poberezhny
12:00-13:30 (Year 1)
String theory and conformal theory /
starts on 14 Sep
Center for Advanced Studies Seminar
14:30-16:00 (Year 2)
Integrable systems 2 /
starts on 19 Sep
  14:00-15:40 (Years 1+2)
Strings and cluster varieties / Research seminar /
starts on 7 Sep
15:00- or 17:00- (Year 2)
Statistical physics /
starts on 29 Sep
14:00-15:20 (Year 1)
Applied methods of analysis / Lecture /
starts on 7 Sep
Applied methods of analysis / Seminar /
  16:30-17:10 (Year 1)
Differential and symplectic geometry /
starts on 19 Sep
18:00-21:00 (Years 1+2)
Joint seminar of Independent University of Moscow and Skoltech Center for Advanced Studies /
“P-adic Hodge theory and Topological Hochschild Homology” / Research seminar
seminar info /
starts on 13 Sep
Skoltech / IUM

17:00-18:20 (Year 1)
Lie groups and Lie algebras and their representation / Lecture / G.Olshanski
starts on 21 Sep
18:30-19:50 /
Lie groups and Lie algebras and their representation / Seminar /
Joint seminar on Mathematical Physics
of National Research University HSE and Skoltech Center for Advanced Studies

Hamiltonian Mechanics / MA06271 / Term 1, 2017-2018

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.

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.

Geometric Representation Theory / MA06256 / Term 1, 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.

Integrable Systems, 2 / ME06010 / Term 5, 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.

Statistical Physics / MA06180 / Term 1, 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