01.01.06 (Mathematical logic, algebra, and number theory)
Phone:
+7 (495) 396 63 98
Fax:
+7 (495) 396 63 98
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Keywords:
algebraic transformation groups; invariant theory; algebraic groups and their representation theory; homogeneous spaces; Lie groups and algebras; algebro-geometric aspects of algebraic transformation groups theory; affine algebraic geometry; automorphism groups of algebraic varieties; discrete reflection groups.
A criterion of stability of action of algebraic group on algebraic variety is obtained. A theory of affine quasihomogeneous horospherical varieties is developed (jointly with E. Vinberg). The Picard group of any homogeneous space is calculated. A theory of normal affine embeddings of homogeneous spaces of SL(2) is developed. The methods of classifying representations of reductive groups with free algebra of invariants (jointly with V. Kac and E. Vinberg) and with free module of covariants are found. The explicit classifications are obtained for irreducible representations of connected simple groups. Spinors of dimension 14 are classified. It is given a solution to the generalized 14th Hilbert problem. It is given a solution to the main problem of constructive invariant theory for connected semisimple groups. Discrete complex reflection groups are classified. Finiteness theorems for representations of a given group with a fixed length of the chain of syzygies of the algebra of invariants are proved. A method of contractions of algebraic group actions is elaborated. Rationality of singularities of normal affine spherical varieties is proved. Nontriangulable actions of additive group on affine space of arbitrary dimension are constructed. Normal affine orbit closures with nonrational singularities are constructed. Theorems on linearization of actions of connected reductive groups on affine spaces of small (< 5) dimension are proved (part of these results jointly with H. Kraft). Canonical presentation by generators and relations of affine coordinate algebras of connected semisimple groups is obtained. A method of constructing a number of self-dual projective varieties is found. The classifications of self-dual nilpotent orbit closures in semisimple Lie algebras and (jointly with E. Tevelev) in isotropy modules of symmetric spaces are obtained. It is proved that any linear algebraic group is realizable as the full automorphism group of a finite dimensional simple algebra (jointly with N. Gordeev).
Biography
Graduated from faculty of Mathematics and Mechanics of M. V. Lomonosov Moscow State University (MSU) in 1969 (department of high algebra). Ph.D. thesis was defended in 1972. D.Sci. thesis (Habilitation) was defended in 1984. The list of my publications contains more than 75 titles (including 2 monographs and 1 textbook). In 1973–1992 (resp. 1984–2000) has run, together with E. Vinberg (resp. E. Vinberg and A. Onishchik) the research seminar at MSU on invariant theory (resp. Lie groups and algebras, and invariant theory). As senior fellow of The Erwin Schroedinger Institute (Vienna), in August–December of 2000 I organized (together with B. Kostant (USA) and F. Pauer (Austria)) the international program of this institute on algebraic transformation groups and applications, and in October 2001, the international conference "Interesting algebraic varieties arising in algebraic transformation groups theory". I am organizer and scientific editor of the series "Algebraic transformation groups and Invariant theory" published by Springer-Verlag. From 1989 to 1999 I was member of the Editorial Board of the journal "Geometria Dedicata" (published by Kluwer). From 1996 to present I am Executive Managing Editor of the journal "Transformation Groups" (published by Birkhauser, Boston).
In 1986 I was invited speaker (45 minutes address) at the International Congress of Mathematicians, Berkeley (USA). In 1989 I was invited speaker at Colloque en l'honneur de J. Dixmier (Paris). In 1992 I was invited speaker at the International Conference in commemoration of the 150th anniversary of the birth of Sophus Lie (Oslo). In 1995 and 1998 I was invited speaker at Special Sessions of the Annual American Mathematical Society meetings resp. in Chicago and Louisville (USA). In 2000 I was invited speaker at the International Colloquium "Algebra, Arithmetic and Geometry" (Tata Institute, Bombay). In 2001 I was invited plenary speaker (60 minutes invited address) at 15 Kongress, Oesterreichische Mathematische Gesellschaft, (Vienna). In 1996–1998 I was Principal Investigator of the fSU-USA cooperative CRDF project "Algebraic Transformation Groups and Applications". In 1998–2000 I was fSU Team Leader of the joint Swiss-Franco-sFU INTAS project "Algebraic Transformation Groups with Application in Representation Theory and Algebraic Geometry". My results on invariant theory were the subject of J. Dixmier's talk at Seminar Bourbaki (exp. 659).
Main publications
Popov V. L., “Modern developments in invariant theory”, Proceedings of the International Congress of Mathematicians, Vol. 1 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 394–406
Popov V. L., Discrete complex reflection groups, Commun. Math. Inst. Rijksuniv. Utrecht, 15, Rijksuniversiteit Utrecht, Mathematical Institute, Utrecht, 1982, 89 pp.
Popov V. L., Groups, generators, syzygies, and orbits in invariant theory, Transl. Math. Monogr., 100, AMS, 1992, 245 pp.
Popov V. L., Vinberg E. B., “Invariant theory”, Algebraic geometry. IV, Encyclopaedia Math. Sci., 55, Springer-Verlag, Berlin, 1994, 123–284
Popov V. L., “Self-dual algebraic varieties and nilpotent orbits”, Proc. Intern. Colloq. “Algebra, Arithmetic and Geometry” (Mumbai, 2000), Tata Inst. Fund. Res. Stud. Math., 16, Tata Inst. Fund. Research, Bombay, 2001, 496–520
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