Valerii Nikolaevich Berestovskii was born on March 4, 1949. He graduated from the mechanics-mathematical department of Novosibirsk State University in 1971. His thesis advisor and chair of Geometry and Topology (1972-75) was Professor Ju. F. Borisov, a geometer in the Alexandrov school. Berestovskii received his Ph.D. in 1979 (Kazan St.ate University) and his D.Sc. in 1990 (Mathematical Institute at Novosibirsk). The latter dissertation was marked among the three best of eight in geometry in 1990 in the former Soviet Union. His positions have been senior lecturer (1975-79), associate professor (1979-83, 1985-90), senior researcher (1983-85), and full professor (1990-present)--at Omsk State University. He has given lectures and practical works to undergraduates in differential and analytical geometry, topology, analysis, and differential equations. He presently has one graduate student and several students.
Berestovskii has about fifty mathematical papers. His basic scientific interests are concentrated on I) spaces of bounded curvature a la Alexandrov and II) locally compact homogeneous spaces with intrinsic metric (particularly, homogeneous Riemannian and Finslerian and perhaps nonholonomic manifolds). The following is a summary of his main results:
I. 1) Every connected locally compact space with inner metric of bilaterally bounded curvature and locally prolongable shortest arcs is isometric to a Riemannian manifold with continuous metric tensor (1975).
2) A simple characterization of spaces of bounded curvature in terms of distance geometry (1981, 1986).
II. 1) Every locally compact homogeneous space with inner metric is isometric to a (perhaps nonholomorphic) homogeneous finslerian manifold if the space is locally contractible. 2) A simple set of axioms for homogeneous Riemannian and Finslerian manifolds and their nonholomorphic generalizations, and known classes of these spaces. 3) The search for shortest arcs in these manifolds is reduced to special invariant problems of optimal control on homogeneous spaces.
III. 1)Every topological n-sphere, n > 4, admits Alexandrov's
inner metric with curvature \< 1 such that in some points the space
of di- rections is not homeomorphic to (n-1)-sphere.
2) The same statement for curvature >/ 1 in dimensions n = 4k+1, k =
1,2,...
3) Every topological euclidean manifold R(n), n > 4, admits complete
Alexandrov's inner metric with curvature =< -1 such that it's Gromov's
ideal boundary is not manifold.
IV. For simply connected quotient M = G/H of compact Lie group G by it's
closed subgroup H the following statements are equivalent:
1) Every G-invariant distribution on M is integrable;
2) M has normal type in sense of L.-B.Bergery, i.e. every G-invariant Riemannian
metric on M is normal;
3) Every G-invariant Riemannian metric on M has positive scalar curvature;
4) Every G-invariant Riemannian metric on M has positive Ricci curvature;
5) Homogeneous space G/H is isomorphic to direct product of compact simply
connected isotropy irreducible homogeneous spaces (which was classified
earlier by O.V.Manturov, J.A.Wolf).
At present Berestovskii has an advanced program in the
above mentioned directions. Particularly he has built a well-argumented
list of unsol- ved problems.Some part of this list is contained in his
paper [23].All of his investigations are at the center of recent differential
geometry, primarily in it's synthetic aspects.
He was visiting professor in 1994-95 at the University of Tennessee, Knoxville,
where he reads a lectures on generalized Riemannian spaces.
