Fellow

Vladimir Leonidovich Popov

Division of Mathematics and Information Sciences

- Mathematical logic, algebra, and number theory
- popovvl@mi-ras.ru

Mathematician; Principal Research Fellow of the Russian Academy of Sciences; Corresponding Member of the Russian Academy of Sciences; Professor at Department of Applied Mathematics of MIEM-HSE

**Information**

Membership Number: FCA2304

Membership Type: Fellowship

Division: Mathematics and Information Sciences

Corresponding Email: popovvl@mi-ras.ru

Homepage(s): https://www.mathnet.ru/php/person.phtml?&personid=8935&option_lang=eng

**Present and Previous Positions**

Chair of Algebra and Mathematical Logic at Moscow State University MIEM (1995–2012; half-time since 2002)

Professor at Department of Applied Mathematics of MIEM-HSE (part time)

Principal Research Fellow at the Steklov Mathematical Institute, Russian Academy of Sciences (since 2017main place of work).

**Fields of Scholarship and Research Interests**

Algebraic transformation groups; invariant theory; algebraic groups, Lie groups, Lie algebras and their representations; algebraic geometry; automorphism groups of algebraic varieties; discrete reflection groups

The results obtained include the following:

● A criterion for closedness of orbits in general position, one of the basic facts of modern Invariant theory (1970–72).

● Pioneering results of modern theory of embeddings (compactifications) of homogeneous algebraic varieties (in particular, toric and spherical varieties), which determined its rapid modern development (1972–73).

● Computing the Picard group of any homogeneous algebraic variety of any linear algebraic group (1972–74).

● Creation of a new direction in Invariant theory—classifying linear actions with certain exceptional properties, e.g., with a free algebra of invariants (jointly with V. G. Kac and E. B. Vinberg), with a free module of covariants, with an equidimensional quotient, and the others. Developing the appropriate methods and obtaining the classifications themselves. Finiteness theorems for the actions with a fixed length of the chain of syzygies (1976–83). The ideology of exceptional properties has then became wide spreaded.

● Solution to the generalized Hilbert’s 14th problem (1979).

● The estimates of the degrees of basic invariants of connected semisimple linear groups first obtained 100 years after the attempt by Hilbert to obtain them (1981–82). They gave rise to modern constructive Invariant theory .

● A theory of contractions of any actions to horospherical ones, which has become an indispensable tool for the modern theory of algebraic transformation groups (1986).

● Pioneering results on the description of algebraic subgroups of the affine Cremona groups that led to a surge of activity in this area in recent decades are obtained (1986–2011).

● The characterization of affine algebraic groups as automorphism groups of simple finite-dimensional (not necessarily associative) algebras (2003, jointly with N. L. Gordeev). In particular, the extension to any finite group of the famous characterization of the largest simple sporadic finite group (the Fischer–Griess Monster). The result is published in Annals of Mathematics and recognized as one of the best in the Steklov Mathematical Institute in 2002.

● A theory of the phenomenon discovered in 1846 by Cayley (2005, jointly with N. Lemire, Z. Reichstein): classification of algebraic groups admitting a birational equivariant map on its Lie algebra. Solution to the old (1975) problem of classifying Caley unimodular groups. The result is published in Journal of the American Mathematical Society and recognized as one of the best in the Russian Academy of Sciences in 2005.

● Proving the algorithmic solvability of the belonging problem of a point of an algebraic variety to the orbit closure of another its point with respect to the action of an algebraic group on this variety and, in particular, proving the algorithmic solvability of the coincidence problem of the orbits of these points (2009).

● Classification of simple Lie algebras whose fields of rational functions are purely transcendental over the subfields of adjoint invariants (2010, jointly with J.-L. Colliot-Thélène, B. Kunyavskiĭ, Z. Reichstein). This result is at the heart of counter-examples to the famous Gelfand–Kirillov conjecture of 1966 on the fields of fractions of the universal enveloping algebras of simple Lie algebras. It is published in Compositio Mathematica and recognized as one of the best in the Steklov Mathematical Institute in 2010.

● Answers to the old (1969) questions of Grothendieck to Serre on the cross-sections and quotients for the actions of semisimple algebraic groups on themselves by conjugation. Constructing the minimal system of generators of the algebras of class functions and that of the representations of rings of such groups (2011).

● Defining the general notion of Jordan group and initiating exploration (carried out since then by many specialists) of the Jordan property of automorphism groups of varieties and manifolds, in particular, groups of birational self-maps and biregular automorphisms of algebraic varieties. Obtaining classification of algebraic surfaces and curves whose groups of birational self-maps are Jordan (2011).

● Solving the problem, posed in 1965 by A. Borel: obtaining the classification of infinite discrete groups generated by complex affine unitary reflections; exploring their remarkable connections with number theory, combinatorics, coding theory, algebraic geometry and singularity theory (1967, 1980–82, 2005, 2023).

**Honors, Awards and Other Membership**

Corresponding Member of the Russian Academy of Sciences

**Selected Publications**

https://scholar.google.com/citations?user=Qcve-A0AAAAJ&hl=en

https://www.researchgate.net/profile/Vladimir-Popov-17