curriculum 22/23

Research seminar “Modern Problems of Mathematical Physics” (Term 1-8) / DG060268 / 22-24

Instructor: Andrei Marshakov

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 theta-functions,
quantum integrable models (quantum-mechanical and field-theoretical),
models of statistical physics, stochastic integrability,
quantum/classical duality, supersymmetric gauge theories, cluster algebras etc.
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Universal Enveloping Algebras and Yangians (Term 1-2) / MA060513 / 22-24

Instructor: Grigori Olshanski

Tentative program of the course:
1. Reminder: the classification of irreducible finite-dimensional representations of a reductive Lie algebra $\mathfrak g$; the universal enveloping algebra $U(\matfrak g)$; the PBW theorem.
2. The structure of the center $Z(U(\mathfrak g))$; the Harish-Chandra map.
3. The symmetrization map $S(\mathfrak g)\to U(\mathfrak g)$ and related combinatorics
4. The Perelomov–Popov formula (mainly for $\mathfrak g=\mathfrak{gl}(N,\mathbb C)$; for other classical Lie algebras — without proof)
5. The Capelli identity
6. Quantum immanants (after Okounkov)
7. The Yangian for $\mathfrak g=\mathfrak{gl}(N,\mathbb C)$ and the $R$-matrix formalism
8. The twisted Yangians
9. Works of Gelfand–Kirillov
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Geometric Representation Theory (Term 1-2) / DA060271 / 22-24

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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Cohomology of Groups and Classifying Spaces (Term 1-2) / MA060543 / 22-24

Instructor: Alexander Gaifullin

The course will include an introduction to theory of cohomology of groups from geometric viewpoint. The concept of a classifying space of a group will be central in this course. We will start with general algebraic definitions and theorems concerning homology and cohomology of (discrete) groups. Then we proceed with various constructions of classifying spaces of groups and methods for computing their cohomology. An important part of the course will be devoted to examples such as certain finite groups, braid groups, Coxeter groups, mapping class groups, etc.
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Phase Transitions: introduction to Statistical Physics and Percolation (Term 1-2) / MA060427 / 22-24

Instructor: Semen Shlosman

TThis is a course on rigorous results in statistical mechanics, random fields, and percolation theory. We start with percolation, which is the simplest system, exhibiting singular behavior, and undergoing phase transitions. We then go to more realistic models of interacting particles, like the Ising model and XY-model, and study phase transitions, occurring there.
The topics will include:
Percolation models, infinite clusters.
Crossing probabilities for rectangles
Critical percolation
The Russo-Seymour-Welsh theory
Cardy’s formula in Carleson form and the Smirnov theorem.
Gibbs distribution
Dobrushin-Lanford-Ruelle equation
Ising model
Spontaneous symmetry breaking at low temperatures
O(N)-symmetric models
The Mermin–Wagner Theorem
The Berezinskii–Kosterlitz–Thouless transition
Reflection Positivity and the chessboard estimates
Infrared bounds
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Introduction to quantum field theory (Term 3-4) / MA060505 / 22-24

Instructors: Vladimir Losyakov, Petr Dunin-Barkowski

As you know, the modern theory of fundamental physics (the “standard model of elementary particle physics”) is a quantum field theory (QFT). In addition to this central role in modern physics, quantum field theory also has many applications in pure mathematics (for example, from it came the so-called quantum knot invariants and Gromov-Witten invariants of symplectic manifolds).
“Ordinary” quantum mechanics deals with systems with a fixed number of particles. In QFP, the objects of study are fields (not in the sense of a “field of complex numbers”, but in the sense of an “electromagnetic field”), whose elementary perturbations are analogs of quantum mechanical particles, but can appear and disappear (“born” and “die”); at the same time, the number of degrees of freedom turns out to be infinite.

Within the framework of this course, the basic concepts of QFT will be introduced “from scratch”. The Fock space and the formalism of operators on it, as well as the formalism of the “continuum integral” will be defined. The main example under consideration will be the quantum scalar field theory. A scalar field in physical terminology is a field that, at the classical level, is defined by one number at each point (i.e., in fact, its state at a given time is just a numerical function on space), unlike a vector field (an example of which, in particular, is an electromagnetic field). However, considering the quantum theory of a scalar field (even separately, and simpler than for the Higgs field) is in any case very useful, since it allows you to get acquainted with the apparatus and phenomena of QFT on a simpler example than vector and spinor fields. The course will consider the “perturbation theory” (that is, in fact, a method for calculating the first orders of smallness in a small parameter expansion) for a scalar field and describe ways to calculate various probabilities of events with particles.
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Critical points of functions (Term 1-2) / MA060460 / 22-24

Instructor: Maxim Kazarian

The theory of critical points of functions is of the main subjects of Singularity theory studying local geometry of singularities of differentiable maps as well as its relationship with global topological invariants of manifolds. In the course we will discuss classification of critical points, its relationship with the ADE-series of simple Lie algebras and the corresponding reflection groups, their deformations and adjacencies. The study of a local topological structure of singularities will include description of Milnor fiber and vanishing cycles. We will discuss also application of the theory critical points to the study of caustics and wave fronts in geometric optics and classical mechanics, as well as enumeration of contact singularities in complex projective geometry.
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Introduction to cluster algebras and varieties (Term 1-2) / MA060509 / 22-24

Instructor: Mikhail Bershtein

Cluster algebras and cluster varieties appeared almost simultaneously in the early 2000s; the algebras were introduced in the works of Fomin and Zelevinsky, and varieties in the works of Fock and Goncharov. These notions rather quickly found numerous applications and became popular, for the last 15 years this popularity has not decreased in any way.
The course will be devoted to an introduction to these concepts. We will mainly concentrate on connections with Lie groups, Poisson structures, and integrable systems.
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Modern Dynamical Systems (Term 3-4) / MA06257 / 22-24

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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Hitchin systems and complex geometry (Term 3-4) / MA060508 / 22-24

Instructor: Alexei Rosly

The subject of this course will be mainly the complex geometry. The choice of topics, however, is determined by their uses in Field Theory and Theory of Integrable Systems.

Plan // 1 item of the plan ≠ 1 lecture

1. The self-duality equations (SDE) in the Yang-Mills theory as the 0-curvature equation with a spectral parameter (Belavin-Zakharov). A construction by Ward.
2. Holomorphic Hermitian bundles over complex manifolds.
3. The main example of a twistor transform: a 1:1 correspondence between solutions to the SDE on M and certain holomorphic vector bundles on P. (On both sides of this correspondence, equivalence classes in a natural sense are understood.)
4. The geometry of CPⁿ, line bundles and homogenous functions, cohomology v CPⁿ with coefficients in line bundles.
5. The Penrose twistor transform for solutions to the massless wave equations.
6. The SDE in gravity, their twistor description (Penrose et al.). (First, in less detail, more could come later if we had time.)
7. What other non-linear equations can be reformulated in terms of complex geometry on twistor space.
8. A twistor formulation of the self-dual Yang-Mills theory. The latter is such a toy theory, where the SDE form a part of the field equations.
9. Hyperkähler and quaternionic manifolds are also such Riemann manifolds which allow for a twistor space.
10. A construction of so-called MHV tree amplitudes with help of twistors.
11. It is not only the world sheet which can have a twistor space, but the target space as well can have it. Some examples of twistor formulation for supersymmetric theories written in superspace.
12. The ADHM (Atiyah-Drinfeld-Hitchin-Manin) description of instantons (solutions to the SDE in Yang-Mills theory). This will probably be a not quite complete presentation.
Below is what for what we will possibly have no time…
13. Green’s function in the background of an instanton constructed with help of twistors (Atiyah).
14. Sometimes, CP³ can be replaced with CP² (Donaldson).
15. The ALE metrics (asymptotically locally euclidean metrics = gravitational instantons).
16. The Yang-Mills instantons over an ALE space.
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Quantum Integrable Systems (Term 3-4) / MA060315 / 22-24

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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Quiver representations and quiver varieties (Term 1-2) / MA060425/ 22-24

Instructor: Evgeny Feigin

The theory of quivers is one of the central topics in various fields of modern mathematics and mathematical physics, such as algebraic geometry, representation theory, combinatorics, quantum field theory, integrable systems. The theory has lots of beautiful and deep theorems and is very popular due to a huge number of applications, including McKay correspondence, instantons and ADHM construction, geometric realization of the Kac-Moody Lie algebras. Many of the recent results and applications of the theory of quivers are based on the quiver verieties, introduced by Hiraku Nakajima 20 years ago. The course will cover the basic material on the structure theory of quivers and their representations, such as path algebras, Gabriel’s theorem, Hall algebras, preprojective algebras and Auslander-Reiten quivers. Based on the general theory of quiver representations we will discuss the definition of the Nakajima quiver varieties and several explicit examples and applications. The course is aimed at the graduate students or advanced bachelor students. The basic knowledge of algebraic geometry, differential geometry, and the theory of Lie groups and Lie algebras is expected.
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Path integral: stochastic processes and basics of quantum mechanics (Term 1-2) / MA060542 / 22-24

Instructor: Andrei Semenov

One of the most powerful methods of modern theoretical physics is the method of functional integration or path integration. The foundations of this approach were developed by N. Wiener at the beginning of the 20th century, but it spread widely after R. Feynman, who applied this approach in quantum mechanics. At present, the functional integral has found its application in the theory of random processes, polymer physics, quantum and statistical mechanics, and even in financial mathematics. Despite the fact that in some cases its applicability has not yet been mathematically rigorous proven, this method makes it possible to obtain exact and approximate solutions of various interesting problems with surprising elegance. The course is devoted to the basics of this approach and its applications to the theory of random processes and quantum mechanics. In the first part of the course, using the example of stochastic differential equations, the main ideas of this approach will be described, as well as various methods for exact and approximate calculation of functional integrals. Further, within the framework of the course, the main ideas of quantum mechanics will be considered, and both the operator approach and the approach using functional integration will be considered. It will be demonstrated that, from the point of view of formalism, the description of random processes and the description of quantum mechanical systems are very similar. This will make it possible to make a number of interesting observations, such as, for example, the analogy between supersymmetric quantum mechanics and the diffusion of a particle in an external potential. In the final part of the course, depending on the interests of the audience, various applications of the functional integration method will be discussed, such as polymer physics, financial mathematics, etc.
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