Speciality:
01.01.06 (Mathematical logic, algebra, and number theory)
E-mail: Website: http://halgebra.math.msu.su/wiki/doku.php/staff:timashev Keywords: invariant theory,
reductive group,
representation,
Lie algebra,
embeddings of homogeneous spaces,
spherical varieties.
Subject:
Main interests lie in the theory of algebraic transformation groups and in Invariant Theory. Spherical homogeneous spaces of reductive algebraic groups were studied, and the theory of equivariant embeddings of arbitrary homogeneous spaces and varieties acted on by a reductive group was developed. A classification of $B$–orbits on a spherical homogeneous space $G/TU'$ was obtained, where $B$ is a Borel subgroup in a connected reductive group $G$, $T$ is a maximal torus in $B$, and $U$ is the maximal unipotent subgroup. The Hasse graph of these orbits was described. The general Luna–Vust theory of equivariant embeddings of homogeneous spaces was reworked and extended to arbitrary $G$–varieties. This work resulted in the classification of $G$–varieties of complexity 1 (i.e. such that a generic $B$–orbit has codimension 1) in terms of combinatorial geometry, which generalizes the classification of toric and spherical varieties. Divisors on normal $G$–varieties were studied. In particular, criteria for a Cartier divisor to be globally generated and ample were given. An integral formula for intersection numbers of divisors on a variety of complexity 1 generalizing Brion's formula for spherical varieties was obtained and applied to computing the degree of an arbitrary 3–dimensional orbit in any $SL(2)$–module. Affine homogeneous spaces of reductive groups such that all their equivariant affine embeddings have finitely many orbits were classified (a joint work with I. V. Arzhantsev). A classification of 2-step nilpotent Lie algebras with the dimensions of the quotients of the lower central series less or equal than $(5,5)$ or $(6,3)$ was obtained (a joint work with L. Yu. Galitski) with the help of invariant-theoretic methods, in particular, Vinberg"s theory of $\theta$–groups.
Main publications:
Galitski L. Yu., Timashev D. A. On classification of metabelian Lie algebras // Journal of Lie Theory, 1999, 9 (1), 125–156.
Timashev D. A. Cartier divisors and geometry of normal G–varieties // Transformation Groups, 2000, 5 (2), 181–204.
Arzhantsev I. V., Timashev D. A. Affine embeddings with a finite number of orbits // Transformation Groups, 2001, 6(2), 101–110.
