curriculum 21/23

Research seminar “Modern Problems of Mathematical Physics” (Term 1-8) / DG060268 / 21-23

Instructor: 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. 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, stochastic integrability, quantum/classical duality, supersymmetric gauge theories, etc.
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Research seminar “”Cluster integrable systems and supersymmetric gauge theories” (Term 1-2) / MA060461 / 21-23

Instructors: Pavlo Gavrylenko, Andrei Marshakov

This research seminar will be devoted to the study of N=2 supersymmetric gauge theories
and related topics. It turns out that comparing to the N=1 theories, N=2 supersymmetry allows to compute much more quantities. In particular, low-energy effective action can be
described in terms of single function, prepotential. Seiberg-Witten solution of the N=2 theory gives explicit description of the prepotential in terms of periods of some meromorphic differential on algebraic curve. It turns out that this description is deeply related to classical integrable systems.
This will be the working seminar where we are going to discuss some topics related to Seiberg-Witten theory in 4D and 5D: partition functions, relation to the integrable systems and their deautonomizations (isomonodromic deformations). On the integrable systems side we will consider cluster integrable systems (like relativistic Toda chains), which come from the dimer models or from double Bruhat cells in Poisson-Lie groups. We are going to discuss their Lax representation, discrete and continuous flows, relation to the dimer models, etc.
We expect some talks given by participant
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Symmetric functions (Term 1-2) / MA060458 / 21-23

Instructor: Grigori Olshanski

The theory of symmetric functions has numerous applications in various domains of mathematics and mathematical physics. At the beginning of the course, standard material will be presented, and then we will move on to more advanced topics.
Tentative program:
The algebra Sym of symmetric functions. Generators of Sym. The scalar product, involution map, and Hopf algebra structure. Schur functions, skew Schur functions. Combinatorial formula. Cauchy identity and dual Cauchy identity. Jacobi-Trudi formula and its dual version.
Frobenius coordinates. Giambelli formula. Symmetric group characters. Murnaghan-Nakayama rule. Polynomial functions on Young diagrams. The Gessel-Viennot method. Supersymmetric functions. Interpolation symmetric polynomials. Multidimensional symmetric orthogonal polynomials. Generalized Schur polynomials and Macdonald’s “9th variation”. Beyond Schur polynomials (if time permits): Hall-Littlewood polynomials and other generalizations.
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Geometric Representation Theory (Term 1-2) / DA060271 / 21-23

Instructors: 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.
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Mapping class groups (Term 1-2) / MA060459 / 21-23

Instructor: Alexander Gaifullin

For an oriented surface (2-manifold) S, the group Diff(S) of orientation-preserving diffeomorphisms of S is a huge infinite-dimensional topological group. By definition, the mapping class group of S is the group Mod(S) obtained from the group Diff(S) by taking the quotient by the identity component. Equivalently, Mod(S) is the group consisting of isotopy classes of orientation-preserving diffeomorphisms of S onto itself.
Theory of the mapping class groups of surfaces lies on the crossroad of algebraic and hyperbolic geometry, three-dimensional topology, geometric, homological and combinatorial group theory. More precisely, it is related to:
– Moduli spaces of complex curves (equivalently, of hyperbolic surfaces) via interpretation of the mapping class group as an orbifold fundamental group of the moduli space;
– Topology of three-manifolds via interpretation of a Heegaard splitting as gluing along an element of the mapping class group;
– Braids and hence knots; in fact, usual braid groups are the mapping class groups of a 2-disc with punctures.
– Outer automorphism groups of free groups; this relationship is caused by the fact that the mapping class group acts by outer automorphisms of the fundamental group of the surface, and the fundamental group of the surface is not so far from being free (as it is given by 2g generators and only 1 relation).
– Arithmetic groups such as SL(n,Z) and Sp(2g,Z) via the action of the mapping class group on the homology of the surface.
The course will start from basic facts on surfaces and their mapping class groups. After this introductory part, we will discuss various methods in theory of mapping class groups arising from relationships listed above
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Classical Integrable Systems (Term 1-2/5-6) / DA060179 / 21-23

Instructor: Igor Krichever

Course description: A self-contained introduction to the theory of soliton equations with an emphasis on their algebraic-geometrical integration theory. Topics include:
1. General features of the soliton systems.
2. Algebraic-geometrical integration theory.
3. Hamiltonian theory of soliton equations.
4. Perturbation theory of soliton equations and its applications to Topological Quantum
Field Theories and Sieberg-Witten solutions of N=2 Supersymmetric Gauge Theiorie
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Statistical mechanics, percolation theory and conformal invariance (Term 1-2) / MA060465 / 21-23

Instructor: 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:
Crossing probabilities as a characteristic of sub-, super- and at- criticality.
Critical percolation and its power-law behavior.
The Russo-Seymour-Welsh theory of crossing probabilities – a cornerstone of critical percolation
Cardy’s formula for crossing probabilities
Parafermionic observables and S. Smirnov theory
Conformal invariance of two-dimensional percolation a la Khristoforov.
Conformal invariance of two-dimensional Ising model
O(N)-symmetric models
Continuous symmetry in 2D systems: The Mermin–Wagner Theorem and the absence of Goldstone bosons.
The Berezinskii–Kosterlitz–Thouless transition
Reflection Positivity and the chessboard estimates in statistical mechanics
Infrared bounds and breaking of continuous symmetry in 3D
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Introduction to Schramm-Loewner Evolution (Term 1B-2) / MA060494 / 21-23

Instructor: Dmitry Belyaev

The Schramm-Loewner Evolution (SLE) was introduced in 1998 in order to describe all possible conformally invariant scaling limits that appear in many lattice models of statistical physics. Since then the subject has received a lot of attention and developed into a thriving area of research in its own right which has a lot of interesting connections with other areas of mathematics and physics. Beyond the aforementioned lattice models it is now related to many other areas including the theory of `loop soups’, the Gaussian Free Field, and Liouville Quantum Gravity. The emphasis of the course will be on the basic properties of SLE and how SLE can be used to prove the existence of a conformally invariant scaling limit for lattice models.
Topics will include:
Introduction to the theory of conformal maps, their boundary behaviour, half-plane capacity, Beurling estimates;
Loewner differential equation and its basic properties;
Introduction to stochastic analysis, Ito calculus, optional stopping theorem.
Schramm-Loewner Evolution, conformal invariance, domain Markov property, motivation, Schramm’s principle;
Basic properties of SLE, phase transition, locality of SLE(6), restriction property of SLE(8/3);
Dimension of SLE curves;
Convergence to SLE, general approach, Cardy’s formula.
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Research seminar “Quantum Mechanics” (Term 1-2) / MA060464 / 21-23

Instructor: Vladimir Losyakov

Advanced course in quantum mechanics, in which the basic principles quantum theory is supplemented and applied to the study of specific physical systems. Modern methods of research of quantum systems are proposed – the construction of integrable potentials, the integral along trajectories, and the concepts of density matrix and effective action are introduced. The course involves a transition to the consideration of free field theories, their canonical quantization, and discussion of differences
quantum mechanics from quantum field theory.
The purpose of the course is to consolidate the basic principles and methods of quantum theory, study the transition from quantum mechanics to quantum field theory. The course introduces the basic concepts necessary for studying the courses of the program “Mathematical physics”.
The course is designed as a solution to specific problems in quantum theory (see the course content).
The course involves significant independent work on solving problems.
I would like the results of the course to coincide with the goals.
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Differential Geometry of Connections (Term 1-2) / MA060460 / 21-23

Instructor: Maxim Kazarian

In this course we present the basic concepts of modern differential geometry: metric, curvature, connection, etc. The goal of our study is to develop tools for practical efficient computations (including the art of manipulation with indices) supported by a deeper understanding of the geometric meaning of all notions and theorems. We will develop an approach based on the notion of connection with all its different aspects: covariant derivative, parallel transport, collection of Christoffel symbols, matrix-valued one-form etc.
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Affine Quantum Groups (Term 1-2) / MA046062 / 21-23

Instructor: Mikhail Bershtein

The main source of the theory of quantum groups was integrable models. Affine quantum groups (better to say quantum universal enveloping of affine Lie algebras) are symmetries of the XXZ model, they appeared simultaneously with quantum group theory in the mid-80s. Affine quantum groups have many new properties (compared to quantum universal enveloping algebras of simple Lie algebras) — different realizations, different comultiplications. They have many applications, in addition to the integrable models mentioned above, we mention cluster algebras and geometric representation theory.
In the course, we will discuss the basic constructions associated with quantum affine algebras and touch on their applications. Some basic familiarity with quantum groups and affine Lie algebras is assumed, and familiarity with the Bethe ansatz is also desirable.
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Modern Dynamical Systems (Term 3-4) / MA06257 / 21-23

Instructors: 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.
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.
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Gauge Fields and Complex Geometry (Term 3-4) / MA060178 / 21-23

Instructor: Alexei Rosly

1. Self-duality equations, Bogomolny equations.
2. Relation to holomorphic bundles.
3. Relation to holomorphic bundles on twistor space.
4. Conformal symmetry and complex geometry in twistor space.
5. Elements of superfield formulation of SUSY field theories.
6. Chirality type constraints and complex geometry.
7. Some examples of superfield theories which require complex geometry.
8. BPS conditions in SUSY theories and complex geometry.
9. Elements of Hitchin’s integrable systems and related complex geometry
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Quantum Integrable Systems (Term 3-4) / MA060315 / 21-23

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, calculation 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
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Vertex operator algebras (Term 3-4) / MA0604259/ 21-23

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 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.
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Introduction to Quantum Theory (Term 3-4) / MA060332 / 21-23

Instructors: Pavlo Gavrylenko, Vladimir Losyakov

One of the most striking breakthrough of the XX century is the creation of the entirely new area of physics named quantum physics. It emerged that the whole world around us obeys the laws of quantum mechanics, while the laws of classical physics that we are familiar with (such as, for example, Newton’s equations) describe only macroscopic objects and can be obtained in limiting case. After that a lot of phenomena in different areas of physics found their explanation. Also quantum mechanics had a very
significant impact on the development of mathematics and mathematical physics. Today quantum mechanics is one of the keystone parts of theoretical and mathematical physics.
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Quantum Field Theory (Term 3-4) / MA060316 / 21-23

Instructor: Andrei Semenov

At present time Quantum Field Theory (QFT) is the main theoretical tool used for the description of the phenomena occurring in the microworld. Examples include interactions between elementary particles, hadron structure and so on. At the same time, QFT methods are widely used in all areas of modern theoretical physics such as condensed matter physics, statistical mechanics, turbulence theory and others. Moreover, the creation of QFT has stimulated the development of many modern areas of mathematics.
The course is aimed at the study of the basic ideas and methods of QFT, as well as the discussion of its applications in various areas of modern theoretical and mathematical physics. Topics include quantization of scalar and gauge theories, path integral approach, perturbative expansions and Feynman diagrams, (1+1) dimensional exactly soluble models and some other ideas of modern science.
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