Preprints and publications

This is an extended abstract of my talk on the Oberwolfach meeting ``New developments in Newton-Okounkov bodies'' in August, 2011. We define analogs of Demazure operators acting on convex polytopes. This leads to a simple inductive construction of Gelfand-Zetlin polytopes and their generalizations.

• (joint with Evgeny Smirnov and Vladlen Timorin) *
Gelfand-Zetlin polytopes and Demazure characters*, Proceedings of the International Conference "50 years of IITP", ISBN 978-5-901158-15-9, 5 pages

pdf

This is a short self-contained exposition of our formula for the Demazure characters of Schubert varieties in terms of the exponential sums over faces of Gelfand-Zetlin polytopes. All necessary definitions are given in the text.

• (joint with Amalendu Krishna) *
Equivariant Cobordism of Flag Varieties and of Symmetric Varieties*, preprint

pdf

We give a Borel-type presentation for the equivariant algebraic as well as complex cobordism of varieties of complete flags for arbitrary reductive groups. We also compute the equivariant algebraic cobordism ring for the symmetric varieties of minimal rank.

• (joint with Evgeny Smirnov and Vladlen Timorin) *
Schubert calculus and Gelfand-Zetlin polytopes*, preprint

pdf

We describe a new approach to the Schubert calculus on complete flag varieties
using the volume polynomial associated with Gelfand-Zetlin polytopes.
This approach allows us to compute the intersection products of Schubert cycles
by intersecting faces of a polytope. We also prove a formula for all Demazure characters of a given representation of GL_n(C) via exponential sums over integral points in faces of the Gelfand-Zetlin polytope associated with the representation.

• *
From moment polytopes to string bodies*, Oberwolfach reports, 19/2010, 30-33

pdf

This is an extended abstract of (the second part) of my talk on the Oberwolfach meeting ``Algebraic groups''
in April, 2010. String bodies
(recently constructed by Kaveh and Khovanskii) for varieties with a reductive group
action are natural extensions of moment (or Newton) polytopes for toric varieties.
I discuss a particular application of string bodies to geometry of non-toric varieties
based on our recent results with Evgeny Smirnov and Vladlen Timorin. Namely, we develop
a new approach to Schubert calculus on the varieties of complete flags using the volume
polynomial on Gelfand-Zetlin polytopes (=string bodies of flag varieties).

• (joint with Jens Hornbostel) *
Schubert calculus for algebraic cobordism*, Journal fur die reine und angewandte Mathematik (Crelles Journal), Volume 2011, no. 656, pp 59–85

pdf

We establish a Schubert calculus for Bott-Samelson resolutions
in the algebraic cobordism ring of a complete flag variety G/B extending the results of Bressler-Evens
to the algebro-geometric setting. All our methods are purely algebraic or algebro-geometric.
In particular, we prove an algebraic Chevalley-Pieri formula for the product of a
Bott-Samelson resolution with a divisor in cobordisms and use this formula to give an algorithm for
multiplying two Bott-Samelson resolutions.

• *
Gelfand-Zetlin polytopes and flag varieties*,
International Mathematics Research Notices, Volume 2010. no. 13. pp 2512—2531

pdf

I construct a correspondence between the Schubert cycles on the flag variety for the group
GL_n(C) and some faces of the
Gelfand-Zetlin polytope of the irreducible representation of GL_n(C) with a strictly dominant
highest weight. The construction is inspired by a geometric presentation of Schubert cells by Bernstein,
I.Gelfand and S.Gelfand.
The correspondence between the Schubert cycles and faces is then used to relate Schubert calculus on
flag varieties to combinatorics of the Gelfand-Zetlin polytopes. The main result in this direction is
the Chevalley formula for the intersection product of a Schubert cycle with a divisor stated in terms of
the Gelfand-Zetlin polytope. The whole picture resembles the picture for toric varieties and their
polytopes.

• *
Flag varieties and Gelfand-Zetlin polytopes*, Oberwolfach Reports, workshop ``Toric geometry'',
Report 01/2009, 16-19

pdf

This is an extended abstract of my talk on the Oberwolfach meeting ``Toric geometry'' in January, 2009.
I discuss some results and open problems for regular compactifications of reductive groups motivated by analogous results for toric varieties.

•
*
On intersection indices of subvarieties in reductive groups*,
Moscow Mathematical Journal (special volume in honor of Askold Khovanskii), **7** no.3 (2007), 489-505

pdf

This paper is a continuation of the one below. It contains an explicit formula for the intersection indices
of the Chern classes of an arbitrary reductive group with hypersurfaces.
This formula has the following applications.
First, it allows to extend to arbitrary reductive groups the explicit answer given by D.Bernstein,
Khovanskii and Kouchnirenko for the Euler characteristic of
all complete intersections in the complex torus (C^*)^n.
Second, for any regular compactification of a reductive group, it computes the intersection indices of
the Chern classes of the compactification with hypersurfaces.
The formula is similar to the Brion-Kazarnovskii formula
for the intersection indices of hypersurfaces in reductive groups. The proof uses an algorithm of
De Concini and Procesi for computing such intersection indices. In particular, it is shown that this
algorithm produces the Brion-Kazarnovskii formula.

• *Chern classes of reductive groups and an adjunction formula*,
Annales de l'Institut Fourier, **56** no. 4 (2006), 1225-1256

pdf

In this paper, I construct noncompact analogs of the Chern
classes of equivariant vector bundles over complex reductive groups.
For the tangent bundle, these Chern classes yield an adjunction
formula for the Euler characteristic of complete
intersections in reductive groups. In the case when a complete intersection is a curve,
this formula gives an explicit answer for the Euler characteristic and the genus of the curve.
This is the first step towards extension to the reductive case of
the explicit answer given by D.Bernstein, Khovanskii and Kouchnirenko for the Euler characteristic of
all complete intersections in the complex torus (C^*)^n.

• *A Gauss-Bonnet theorem, Chern classes and an adjunction formula for reductive groups*, PhD Thesis, University of Toronto, 2004

pdf (in English), pdf (in Russian)

A major part of the thesis was rewritten in two of my papers.
Here is a brief summary of those results in the thesis that are not contained
anywhere else. Let G be a connected complex reductive group. In the first part of the thesis (concerning a Gauss-Bonnet theorem) there are two more corollaries: a formula for the Euler characteristic of an adjoint invariant (possibly singular) subvariety X of G in terms of invariants of the strata in a Whithney stratification of X and the vanishing of the Euler characteristic of adjoint invariant sheaves on G that are supported on a set of non-semisimple elements of G (e.g. character sheaves have this property). In the second part (concerning an adjunction formula), there is an elementary proof of an adjunction formula for one hyperplane section in G similar to the proof by Bernstein in the torus case.

• *A Gauss-Bonnet theorem for constructible sheaves on
reductive groups*, Mathematical Research Letters **9** no. 5-6 (2002), 791-800

pdf

In this paper, I prove an analog of the Gauss-Bonnet formula for constructible sheaves on reductive groups. This formula holds for all constructible sheaves equivariant under the adjoint action and expresses the Euler characteristic of a sheaf via the Gaussian degrees of the components of its characteristic cycle. As a corollary from this formula I get that if a perverse sheaf on a reductive group is equivariant under the adjoint action, then its Euler characteristic is nonnegative.

• *The monodromy group of generalized hypergeometric equations*,
Diploma, Independent University of Moscow, 2001 [In Russian]

pdf

A generalized hypergeometric equation of order n is a Fuchsian differential equation in
complex domain with 3 singular points 0, 1, \infty, which has n-1 linearly independent solutions holomorphic at
the singular point 1. It depends on 2n complex parameters. I study in detail the representation of its solutions
via generalized hypergeometric series and give a simple description of its monodromy group for almost all sets of
parameters. My method is based on an interesting linear algebraic problem connected with the Deligne-Simpson problem.

Course notes and and popularizing articles

This is an article written for ``Kvant'' journal for high school students. Schubert's method (Schubert calculus) of solving problems of enumerative geometry is explained on the elementary level and illustrated by two classical problems. First, we compute the number of lines passing through 4 given lines in a 3-space. Second, we compute the number of conics tangent to 5 given conics (famous Steiner's problem solved correctly by Chasles).

• *Geometry of spherical varieties*, course notes [In Russian]

pdf

These are notes to my course on geometry of spherical varieties taught at the Independent University of
Moscow in fall 2009. The notes cover geometry of flag varieties, toric varieties, complete conics and
regular compactifications of reductive groups.

• Intersection Theory, course notesThis is a preliminary version of the notes on intersection theory. These notes have grown out of the graduate mini-course that I taught in Jacobs University Bremen in 2008. The following topics are covered: intersection indices of subvarieties, degree of affine and projective varieties, divisors, self-intersection index of a divisor; Bezout theorem for projective spaces and for toric varieties; intersection product and Chow ring of a variety; Schubert calculus on Grassmannians and enumerative geometry. The notes can serve as an introduction to intersection theory for graduate students. Only basic knowledge of algebraic geometry is assumed.
• |

• *Constructions using compass and ruler and Galois theory*,
course notes [In Russian]

pdf

These notes are based on the course which I taught during the summer school ``Contemporary mathematics''
in Dubna in 2005. The notes contain an elementary introduction to Galois theory. In particular, the problem
of constructubility of regular n-gons is discussed in great detail. More generally, Galois theory for
cyclotomic extensions is constructed following Gauss'
``Disquisitiones Arithmeticae''. To understand these notes
it is enough to have a good knowledge of high school mathematics.