Representation Theory
Lecturer：Evgeny Smirnov - Higher School of Economics
Abstract: Representation Theory studies abstract algebraic structures, such as groups or rings, by representing their elements as linear transformations of vector spaces. It is a fundamental tool to study groups, algebras and their actions using linear algebra.
Representation Theory plays an important role in many recent developments of mathematics and theoretical physics. In our course we introduce basic concepts and results of the classical theories of complex representations of finite groups, Lie groups and Lie algebras.
The course consists of three parts. The first part is devoted to representation theory of finite groups; we will use this example to get acquainted with the fundamental concepts of the theory, to be generalized later. In the second part, we will discuss the representation theory of algebras in general and then study in detail the so-called path algebras of quivers and apply this to problems of linear algebra. In the final part, we study representations of Lie groups and Lie algebras.
Time：Mondays 15:10-17:00
Thursdays 18:40-20:30
Join Zoom Meeting：
https://us02web.zoom.us/j/6687697417?pwd=aCszVjJ6YlBDRmJBdE5SVTJydWF1UT09
Meeting ID：668 769 7417
Password：123451
Contact informtion of teaching assistants will be notified later.
Geometry, Quantization and Semi-Classical Asymptotics
Lecturer：Andrei Shafarevich - Lomonosov Moscow State University
Abstract: The main goal of the course is to present geometric and analytical ideas related to semi-classical theory, the theory of quantization of isotropic manifolds, as well as with description of singularities for PDE’s.
Contents.
Week 1. Symplectic manifolds and classical mechanics. Hamiltonian systems.
Week 2. Integrable systems.
Week 3. Quantization of classical systems. Dirac quantization.
Week 4. Pseudo-differential operators. Weyl symbolds.
Week 5. Sobolev spaces and pseudo-differential operators.
Week 6. Geometric quantization.
Week 7. Semi-classical asymptotics. Quantization of Lagrangian manifolds.
Week 8. Maslov canonical operator. Maslov class and Maslov index.
Week 9. Quantization of complex vector bundles over isotropic manifolds.
Week 10. Semi-classical quantization and Morse theory.
Week 11. Propagation of singularities for hyperbolic systems.
Week 12. Lagrangian manifolds and Fourier integral operators. Wave fronts.
Time：Mondays 18:40-20:30
Tuesdays 18:40-20:30
Join Zoom Meeting：
https://us02web.zoom.us/j/6997780703?pwd=eU9abGY3TytSZDlwd09lZzlLRVY0UT09
Meeting ID：699 778 0703
Password：8242931
Contact informtion of teaching assistants will be notified later.
Fundamentals of Calculus of Variations and Optimal Control
Lecturer：Eugene Stepanov - Higher School of Economics
Abstract: The course is an introduction to the modern calculus of variations and its basic tools with an emphasis on concrete problems of variational nature, both classical and modern ones. Among the problems treated, there will be some shape optimization problems, classical mechanics, isoperimetric problems, existence and uniqueness of solutions to some PDEs of variational nature and optimal control problems.
Time：Tuesdays 13:00-14:50
Fridays 13:00-14:50
Join Zoom Meeting：
https://us02web.zoom.us/j/6687697417?pwd=aCszVjJ6YlBDRmJBdE5SVTJydWF1UT09
Meeting ID：668 769 7417
Password：123451
Contact informtion of teaching assistants will be notified later.
Geometry and Topology of nilmanifolds
Lecturer：Dmitry Millionshchikov - Lomonosov Moscow State University
Abstract: Homogeneous spaces play an important role in the apparatus of modern fundamental and applied mathematics and theoretical physics. The concept of symmetry brings together geometry, topology, algebra and analysis. This course is devoted to such an important subclass of homogeneous spaces as nilmanifolds, i.e. homogeneous spaces of nilpotent groups. The study and application of nilmanifolds and invariant geometric structures (Riemannian and pseudo-Riemannian metrics, complex, hypercomplex, symplectic and affine structures) has experienced rapid development in recent decades. This course is intended to be an introduction to this rapidly developing field, located at the intersection of different mathematical sciences. On the other hand, using the example of nilmanifolds, we will touch upon such important sections of topology as the theory of invariant de Rham cohomology, study Massey products and the concept of formality, and establish its connection with symplectic and Kähler structures. The second part of the course will be devoted to the study of left-invariant geometric structures on nilmanifolds, in particular, we will discuss invariant nilsoliton metrics. We will try to skip the technically complicated proofs, focusing on the most important examples, ideas and basic geometric constructions.
Time：Wednesdays 18:40-20:30
Fridays 18:40-20:30
Join Zoom Meeting：
https://us02web.zoom.us/j/6997780703?pwd=eU9abGY3TytSZDlwd09lZzlLRVY0UT09
Meeting ID：699 778 0703
Password：8242931
Contact informtion of teaching assistants will be notified later.
