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I study geometry of varieties with a "good" action of an algebraic
group (main examples are toric varieties, flag varieties and more
generally spherical varieties). My purpose is to extend some important
result known for toric varieties to other spherical varieties. There is
a beautiful and rich theory, known as the theory of Newton polytopes,
that connects various geometric and topological invariants of a toric
variety with the combinatorial invariants of a certain polytope
associated with the variety. E.g. the Euler characteristic of complete
intersections in toric varieties can be found explicitly, the
cohomology ring can be described in terms of polytopes, etc. I would
like to extend the theory of Newton polytopes to a more general class
of varieties. So far I was able to extend formulas for the Euler
characteristic of complete intersections to arbitrary reductive groups
and to their regular compactifications (see
[4],
[5]).
My current work is on a relation between Schubert
calculus on flag varieties and combinatorics of the Gelfand-Zetlin polytopes
(see [1],
[3]).
I am also interested in a newly developed theory of algebraic cobordism. This theory
is in many aspects similar to complex (topological) cobordism, and one of the main challenges is to
find algebro-geometric replacement for the topological methods of complex cobordism.
Together with Jens Hornbostel we have computed algebraic cobordism rings of comlete flag varieties and
established Schubert calculus for Bott-Samelson resolutions of Schubert varieties by purely algebro-geometric
methods (see [2]).
My past research concerned constructible sheaves
on reductive groups (see
[6],
[7]) and generalized
hypergeometric functions (see [8]).
Hausdorff Center for Mathematics, University of Bonn
Department of Mathematics, Stony Brook University
Department of Mathematics, University of Toronto
Independent University of Moscow
Homepage of Vladlen Timorin
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