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

Graduated from Mathematics and Mechanics Faculty of Moscow State University Lomonosov (MSU) (Department of High Algebra) in 1969. PhD (Candidate of Physics and Mathematics) (1972). Habilitation (Doctor of Physics and Mathematics) (1984). Full Professor (1986). Chair of Algebra and Mathematical Logic at Moscow State University MIEM (1995–2012; half-time since 2002). Since 2012 Professor at Department of Applied Mathematics of MIEM-HSE (part time). Since January 2002 Leading Research Fellow, and since May 2017 Principal Research Fellow at the Steklov Mathematical Institute, Russian Academy of Sciences (main place of work).

Invited speaker at the International Congress of Mathematicians, Berkeley, USA (1986). The results of 1982–1983 are the subject of J. Dixmiers talk at Séminaire N. Bourbaki (J. Dixmier, Quelques résults de finitude en théorie des invariants (daprès V. L. Popov), Séminaire Bourbaki, 38ème année 1985–86, no. 659, pp. 163–175).

Core member of the panel for Section 2, "Algebra" of the Program Committee for the 2010 International Congress of Mathematicians (2008–2010).

Fellow of The American Mathematical Society, the inaugural class (elected in November 2012 ``in recognition of distinguished contributions to the profession’’), see http://www.ams.org/profession/fellows-list-institution

Corresponding Member of the Russian Academy of Sciences (elected in October 2016).

Fellow of The Core Academy (Hong Kong) (elected in October 2023), see https://www.coreacad.org/Member.aspx?ProId=43

Invited plenary speaker at the XVth Austrian–German Mathematical Congress (Ősterreichische Mathematische Gesellschaft–XV Kongress, Jahrestagung der Deutschen Mathematiker-vereinigung), Vienna, 2001.

Honorable International John-von-Neumann Professur awarded by Technische Universität München, Germany (2008). Invited Noted Scholar, Heidelberg University, Germany (1998–1999). Invited Noted Scholar, the University of British Columbia, Vancouver, Canada (1996).

Invited speaker at the international colloquia and conferences in Russia, France, UK, Italy, Germany, USA, Canada, Japan, Switzerland, Israel, Netherlands, Belgium, Spain, Norway, Sweden, India, Australia, Singapore, Hungary, Poland, Argentina, Uruguay, in particular, at Colloque en lhonneur de J. Dixmier (Paris, 1989), at the International Conference commemorating 150th birthday of Sophus Lie (Oslo, 1992), at Special Sessions of the Annual American Mathematical Society meetings in Chicago (1995) and Louisville, USA (1998), at the International Colloquium "Algebra, Arithmetic and Geometry" (Tata Institute, Bombay, 2000), at the International Conference commemorating 80th birthday of B. Kostant" (Vancouver, 2008).

Honorable Colligwood Lecture at Durham University, UK (2007).

Delivered courses "Invariant Theory", "Discrete Groups Generated by Complex Reflections", "Algebraic Transformation Groups and Singularities of Algebraic Varieties", "Algebraic Groups", "Algebraic Geometry" at the invitation of several leading mathematical centers in Germany (Heidelberg University, TUM), Switzerland (ETH Zürich), Netherlands (University of Utrecht), USA (University of Michigan), Canada (UBC), Austria (The Erwin Schrödinger Institute, Innsbruck University), Australia (Sydney University), Sweden (Lund University), Russia (Steklov Mathematical Institute, Moscow). For many years conducted seminars at the Mechanics and Mathematics Department of the Moscow State University: from 1970 to 1986 on invariant theory (jointly with E. B. Vinberg), from 1986 to 2000 on Lie groups and invariant theory (jointly with E. B. Vinberg and A. L. Onishchik). They formed the national school of the algebraic transformation group theory.

In 1995, together with E. B. Vinberg, founded the journal "Transformation Groups" published by Birkhäuser Boston. Editor-in-Chief of this journal from 2020 to present and Executive Managing Editor from 1996 to 2020, see https://www.springer.com/journal/31? Member of the Editorial Boards of the journals: "Izvestiya: Mathematics" (2006–present) and "Mathematical Notes" (2003–present) of the Russian Academy of Sciences, "European Mathematical Society Newsletter" (2015–2022) of the EMS, "Transactions of the Moscow Mathematical Society" of the MMS, MCIME, and AMS, "Geometriae Dedicata" of Kluwer (1989–1999), "Journal of Mathematical Sciences" of Springer (2001–2000). Founder and Title Editor of the series "Invariant Theory and Algebraic Transformation Groups" of Encyclopaedia of Mathematical Sciences published by Springer (1998–present).

Member, Board of Moscow Mathematical Society (1998–2000).

More than 190 publications, among them 4 monographs, 1 textbook and the papers published in Annals of Mathematics, Journal of the American Mathematical Society, Compositio Mathematica, Transformation Groups, Izvestiya: Mathematics, Sbornik: Mathematics, Journal fur die reine und angewandte Mathematik, Commentarii Mathematici Helvetici, Contemporary Mathematics, Journal of Algebra, Functional Analysis and Its Applications, Comptes Rendus de lAcademie des Sciences Paris, Transactions of the Moscow Mathematical Society, Indagationes Mathematicae, Mathematical Notes, Russian Mathematical Surveys, Journal of the Ramanujan Mathematical Society, Documenta Mathematica, Pacific Journal of Mathematics, European Journal of Mathematics. The results are included in many monographs and textbooks (D. Mumford, J. Fogarty, Geometric Invariant Theory; H. Kraft, Geometrische Methoden in der Invariantentheorie; H. Derksen, G. Kemper, Computational Invariant Theory; F. Grosshans, Algebraic Homogeneous Spaces and Invariant Theory; H. Kraft, P. Slodowy, T. A. Springer, Algebraic Transformation Groups and Invariant Theory; W. F. Santos, A. Rittatore, Actions and Invariants of Algebraic Groups; B. Sturmfels, Algorithms in Invariant Theory; G. Freudenburg, Algebraic Theory of Locally Nilpotent Derivations; M. Lorenz, Multiplicative Invariant Theory; E. A. Tevelev, Projective Duality and Homogeneous Spaces and the others).

Organizer of several international conferences, in particular, "Semester on Algebraic Transformation Groups" at The Erwin Schrödinger Institute, Vienna (joint with B. Kostant, 2000), and the conference "Interesting Algebraic Varieties Arising in Algebraic Transformation Groups Theory" at The Erwin Schrödinger Institute, Vienna (2001).

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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).

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On the results obtained (citations):

● From Introduction to the book J. Olver, Classical Invariant Theory, London Math. Soc. Student Texts 44 Cambridge Univ. Press, 1999:

``[…] a vigorous, new Russian school of invariant theorists, led by Popov [181] and Vinberg [226] who have pushed the theory into fertile new areas. […]"

● On the book Popov, V. L. Groups, Generators, Syzygies, and Orbits in Invariant Theory. Transl. of Math. Monographs, 100. Amer. Math. Soc., Providence, RI, 1992. vi+245 pp.:

– From the review by G. Schwarz (Bull of Amer. Math. Soc., 29 (1993), no. 2, 299–304):

``[…] Popov is a leader in Invariant theory, and the articles in this book were important to that field’s development. […]’’

``[…] There has been an explosion of activity in this area over the last ten years. Popovs work was seminal. […]’’

– From the review by M. Brion (Math. Reviews 92g:14054:

``[… ] The author’s results have been the starting point for research trends in invariant theory: for example, classification of representations of semisimple groups with ``good " properties, and also embedding theory of homogeneous spaces. […]’’

● On the work V. L. Popov, E. B. Vinberg, Invariant Theory, Encycl. Math. Sci., Vol. 55, Springer-Verlag, Berlin, 1994, pp. 123–284:

– From the review by N. Andruskiewitsch (Zentralblatt Math. 735.14010):

``[…] The paper under review, written by two of the main contributors in this last period, […] should be considered as a book, which is probably the format it would have if translated. […]"

– From the review by P. E. Newstead (Math. Reviews 92d:14010) :

``This article is […] by two of today’s leading experts in the field and will undoubtedly serve as a major source of information on the subject. […]"

— From the paper S. Fomin, P. Pylyavskyy, Tensor diagrams and cluster algebras, Adv. Math. 300 (2016), 717--787:

``Our main sources of inspiration outside cluster theory included the timeless texts by H. Weyl [57] and V. Popov--E. Vinberg [48]" (here [48] is the reference to the paper V. L. Popov, E. B. Vinberg, Invariant theory, in: Algebraic geometry. IV, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer-Verlag, Berlin, 1994, pp. 123–284).

● From the paper Y. André, Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence, Ann. Sci. Ec. Norm. Sup. (4) 47 (2014), no. 2, 449--467:

``After pioneering work by Grosshans, Luna, Popov, Vinberg and others in the seventies, the study of quasi-homogeneous G-varieties, i.e., algebraic G-varieties with a dense G-orbit, has now become a rich and deep theory.’’

● From the paper D. Luna et Th. Vust, Plongements d’espaces homogènes, Comment. Math. Helvetici 58 (1983), 186–245:

``Nous devons notre point de départ bien évidemment à la théorie des plongements toriques ([5], [6]), mais aussi à article [10] de V. L. Popov, dans lequel est donnée la classification des espaces Presque-homogènes affines normaux sous SL(2)’’ (here [10] stands for V. L. Popov, “Quasihomogeneous affine algebraic varieties of the group SL(2)”, Math. USSR-Izv., 7:4 (1973), 793–831).

● From the Introduction to Chap. III of the book H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspekte der Mathematik, Bd. D1, Vieweg, Braunschweig, 1985:

``[…] Zum Abschluss geben wir – sozusagen als Krönung der hier entwickelten Methoden – die vollständige Klassifikation der sogenannten SL(2)-Einbettungen, d.h. derjenigen affinen SL(2)-Varietäten, welche einen dichten Orbit enthalten. Dieses schöne Resultat geht auf V. L. Popov zurück [P1] (here [Po1] stands for V. L. Popov, “Quasihomogeneous affine algebraic varieties of the group SL(2)”, Math. USSR-Izv., 7:4 (1973), 793–831).

● From the book Algebraic Transformation Groups and Invariant Theory, DMV Seminar, Band 13, Birkhäuser, 1989, p. 72:

``In this paragraph we explain some classical results about the Picard group Pic G ([…]; [Po 74]; […])" (here [Po 74] stands for V. L. Popov, Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles, Math. USSR Izv. 8 (1974), 301–327).

● From the paper H. Derksen, H. Kraft, Constructive Invariant theory, in: Algèbre Non Commutative, Groupes Quantiques et Invariants (Reims, 1995), Sémin. Congr., Vol. 36, Soc. Math. France, Paris, 1997, pp. 221–244:

``It took almost a century until Vladimir Popov determined a general bound for β(V ) for any semi-simple group G ([Pop 81/82])" (here [Pop 81/82] stands for V. Popov, Constructive Invariant theory, Ast_erisque 87{88 (1981), 303–334, and V. L. Popov, The constructive theory of invariants, Math. USSR Izv. 19 (1982), 359–376.

● From the paper K. D. Mulmuley, Geometric Complexity Theory V: Equivalence between blackbox derandomization of polynomial identity testing and derandomization of Noethers Normalization Lemma, in: 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, New Brunswick, New Jersey, 2012, pp. 629--638:

``$\ldots$ But Hilbert could only show that his algorithm for constructing finitely many generators for $K[V]^G$ worked in finite time. He could not prove any explicit upper bound on its running time. Such a bound was proved in $[\rm P]$ a century later $\ldots$" (here $[\rm P]$ is the reference to the paper V. Popov, The constructive theory of invariants, Math. USSR Izv. 10 (1982), 359--376).

● From the paper J. Elmer, M. Kohls, Zero-separating invariants for finite groups, J. Algebra 411 (2014), 92–113:

``One of the most celebrated results of 20th century invariant theory is the theorem of Nagata [12] and Popov [13] which states that $k[X]^G$ is finitely generated for all affine G-varieties X if and only if G is reductive. (here [13] stands for V. L. Popov, Hilberts theorem on invariants, Soviet Math. Dokl., 20:6 (1979), 1318–1322).

● From the book (p. 161) D. Mumford, J. Fogarty, Geometric Invariant Theory, 2nd ed., Ergebnisse der Math. Und ihrer Grenzgebiete, Bd. 34, Springer-Verlag, Berlin, 1982:

``[…] The striking result due to Kac, Popov, Vinberg ([…], [166], […]) is the following Theorem […]‘’ (here [166] stands for V. G. Kac, V. L. Popov, E. B. Vinberg, “Sur les groupes linéaires algébriques dont lalgèbre des invariants est libre”, C. R. Acad. Sci. Paris Sér. A-B, 283:12 (1976), A875–A878).

● From the paper H. Flenner, M. Zaidenberg, Locally nilpotent derivations on affine surfaces with a C-action, Osaka J. Math. 42 (2005), no. 4, 931–974:

``By classical results […] of Popov [Po], […]" (here [Po] stands for V. L. Popov, Classification of affine algebraic surfaces that are quasihomogeneous with respect to an algebraic group, Math. USSR Izv. 7 (1974), 1039–1055 (1975)).

● From the paper L. E. Renner, Orbits and invariants of visible group actions, Transform. Groups 17 (2012), no. 4, 1191–1208:

``We now introduce the following definition (Definition 1.10 below). It is one of the key notions in the study of invariants.[...] The notion of a stable action was first introduced in [7] by V. L. Popov. There he establishes a criterion of stability for semisimple groups (Theorem 1 of [7])‘’ (here Definition 1.10 is the definition of stable action and [7] is the reference to paper V. Popov, On the stability of the action of an algebraic group on an algebraic variety, Math. USSR Izv. 6 (1973), 367–379).

● From the paper N. Perrin, On the geometry of spherical varieties, Transform. Groups 19 (2014), no. 1, 171–223:

``It is a classical problem to ask which product of projective rational homogeneous spaces $\prod_i G/P_i$ has a dense G-orbit. This is solved in [141] if all the parabolic subgroups agree‘’ (here [141] is the reference to the paper V. L. Popov, Generically multiple transitive algebraic group actions, in: Proceedings of the International Colloquium on Algebraic Groups and Homogeneous Spaces (Mumbai, 2004), Tata Institute of Fundamental Research, Vol. 19, Narosa, internat. distrib. by AMS, New Delhi, 2007, pp. 481–523).

● From the paper A. Guld, Boundedness properties of automorphism groups of forms of flag varieties, arXiv:1806.05400v1 [math.AG] 14 Jun 2018:

`` Recently there have been great interest in investigating the finite subgroups of biregular and birational automorphism groups of algebraic varieties. The Jordan property lies in the center of attention. <…> Research about investigating Jordan properties for birational and biregular automorphism groups of varieties was initiated by V. L. Popov in [Po11]” (here [Po11] is the reference to the paper V. L. Popov. On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, Proceedings of the conference on Affine Algebraic Geometry held in Professor Russell’s honour, 1–5 June 2009, McGill Univ., Montreal., Centre de Recherches Mathématiques CRM Proc. and Lect. Notes, Vol. 54, 289–311, 2011).

● From the paper Bharat Adsul, Milind Sohoni, K. V. Subrahmanyam, Geometric Complexity Theory --- Lie Algebraic Methods for Projective Limits of Stable Points, arXiv:2201.00135v1 [math.RT] 1 Jan 2022:

``A fundamental problem of invariant theory is that of obtaining a good description of the G orbit closures of points y ∈ V and deciding whether a point x belongs to the orbit closure of y. These problems go back to Hilbert and are of fundamental importance in the construction of moduli spaces […] The ubiquitous appearance of this problem in many areas of mathematics is also surveyed in the introduction of [Pop09]. […] In Popov [Pop09] the author gives a constructive algorithm to determine if x is in the orbit closure of y. He does this in both, the affine as well as the projective setting. This establishes the decidability of the orbit closure membership problem in the sense of computability theory.’’ (Here [Pop09] is the reference to the paper Vladimir L Popov. Two orbits: When is one in the closure of the other? , Proc. Steklov Inst. of Math., 264(1):146–158, 2009).

1. V. L. Popov, “Group varieties and group structures”, Izv. Math., 86:5 (2022), 903–924              
2. Vladimir L. Popov, “On the equations defining affine algebraic groups”, Pacific J. Math., 279:1-2, Special issue. In memoriam: Robert Steinberg (2015), 423–446 http://msp.org/pjm/2015/279-1/p19.xhtml, arXiv: 1508.02860            
3. J.-L. Colliot-Thélène, B. Kunyavskiĭ, V. L. Popov, Z. Reichstein, “Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?”, Compos. Math., 147:2 (2011), 428–466          
4. V. L. Popov, “Cross-sections, quotients, and representation rings of semisimple algebraic groups”, Transform. Groups, 16:3, special issue dedicated to Tonny Springer on the occasion of his 85th birthday (2011), 827–856            
5. N. Lemire, V. L. Popov, Z. Reichstein, “Cayley groups”, J. Amer. Math. Soc., 19:4 (2006), 921–967              
6. N. L. Gordeev, V. L. Popov, “Automorphism groups of finite dimensional simple algebras”, Annals of Math. (2), 158:3 (2003), 1041–1065            
7. V. L. Popov, Groups, Generators, Syzygies, and Orbits in Invariant Theory, Translations of Mathematical Monographs, 100, Amer. Math. Soc., Providence, RI, 1992 , vi+245 pp.    
8. V. L. Popov, “Modern developments in invariant theory”, Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), v. 1, Amer. Math. Soc., Providence, RI, 1987, 394–406 www.researchgate.net/publication/261556502_Modern_developments_in_invariant_theory  
9. V. L. Popov, Discrete Somplex Reflection Groups, Lectures delivered at the Math. Institute Rijksuniversiteit Utrecht in October 1980, Commun. Math. Inst. Rijksuniv. Utrecht, 15, Rijksuniversiteit Utrecht Mathematical Institute, Utrecht, 1982 , 89 pp. www.researchgate.net/publication/261552178_Discrete_complex_reflection_groups . Second enlarged edition published in Communications in Mathematics, vol. 30 (2022), no. 3 (published August 22, 2023), 303–375, cm.episciences.org/11725    
10. V. L. Popov, “Hilbert's theorem on invariants”, Soviet Math. Dokl., 20:6 (1979), 1318–1322        

1. Algebra and Arithmetic, Algebraic, and Complex Geometry, Collected papers. In memory of Academician Alexey Nikolaevich Parshin, Trudy Mat. Inst. Steklova, 320, ed. V. L. Popov, S. O. Gorchinskiy, A. B. Zheglov, D. V. Osipov, 2023, 324 с.
   http://mi.mathnet.ru/book1934
2. Algebra, number theory, and algebraic geometry, Collected papers. Dedicated to the memory of Academician Igor Rostislavovich Shafarevich, Trudy Mat. Inst. Steklova, 307, ed. A. N. Parshin, V. L. Popov, S. O. Gorchinskiy, Vik. S. Kulikov, 2019, 328 с.
   http://mi.mathnet.ru/book1773

• Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
• Steklov International Mathematical Center
• National Research University Higher School of Economics, Moscow
• Moscow Institute of Electronics and Mathematics — Higher School of Economics
• Moscow Mathematical Society