[2] Constructions using compass and ruler and Galois theory,
preprint [In Russian]
pdf,ps
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 knowlege of high school mathematics.
[3] Chern classes of reductive groups and an adjunction formula,
Annales de l'Institut Fourier, 56 no. 3 (2006), 1225-1256
pdf,ps
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.
[4] A Gauss-Bonnet theorem for constructible sheaves on
reductive groups, Mathematical Research Letters 9 no. 5-6 (2002), 791-800
pdf,ps
Abstract: 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.
[5] The monodromy group of generalized hypergeometric equations,
Diploma, Independent University of Moscow, Moscow 2001 [In Russian]
pdf,ps
Abstract: 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.