Keynote Speakers
Ivan Arzhantsev
Professor, Dean, HSE University (Moscow)
Topic: "Locally nilpotent derivations and automorphisms of algebraic varieties"
The course consists of three parts. The first part is about the algebraic theory of locally nilpotent derivations. In the second part we study automorphisms of affine algebraic varieties using gradings on the algebra of regular functions and describing homogeneous locally nilpotent derivations of such algebras. In the third part we apply these results to study automorphisms of projective varieties. This approach is based on the concept of Cox rings. Among other things, following Demazure, we obtain a description of the automorphism group of a complete toric variety.
Pavel Kolesnikov
Leading Research Fellow, IM SB RAS, NSU (Novosibirsk)
Topic: "The arithmetics of binary quadratic operads"
Operads are combinatorial objects emerging in various fields of mathematics. Formally, an operad is a category with an additional structure (multi-morphisms) that has only one object. On the other hand, an operad may be considered as a governing object for an equational class of algebraic systems (a variety), with generators--operations and relations--identities. In the mini-course, we consider the class of binary quadratic operads (in the sense of V. Ginzburg and M. Kapranov) corresponding to the classes of non-associative algebras defined by quadratic identities (like associative, Lie, Poisson, Novikov algebras, etc.). We will learn how to compute Manin products of such operads (black and white product), discuss Koszul duality of operads, and consider applications of these constructions.
Maxim Vsemirnov
Director, Professor, PDMI RAS, SPbU (Saint-Petersburg),
Topic: "The (2,3)-generated and Hurwitz groups"
First, we consider the finite quotients of the famous modular group PSL(2,Z). For more than a century this problem attracted a lot of interest. Next, we focus on an important subclass of such groups, the so-called Hurwitz groups. We discuss constructive methods, which allow us to build many Hurwitz generators for matrix groups of large rank, and also explain why Hurwitz groups in low dimensions are rather rare. To deal with these obstructions we consider tools from representation theory, like Scott's formula. At the end some open problems and directions of further research will be discussed.