Personal webpage of Valentina Kiritchenko

Faculty of Mathematics

June 6, Wednesday, 15:30, room 211

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Ivan Arjantsev

Automorphisms of varieties, Cox rings and locally nilpotent derivations 

We plan to consider automorphism groups of complex algebraic varieties. Groups which arise in this way are of quite different nature; compare, for example, the automorphsim groups of projective spaces, affine spaces and algebraic tori. We are going to discuss some approaches to the study of automorphsim groups.

Any connected linear algebraic group is generated by its maximal torus and one-parameter root subgroups. This observation was used by Demazure (1970) to describe the automorphism group of a compact toric variety. For affine varieties, torus actions correspond to gradings on the algebra of regular functions, while root subgroups are in bijection with homogeneous locally nilpotent derivations. The theory of Cox rings allows to reduce the study of automorphisms of a wide class of varieties to the affine case.

In this talk we assume only knowledge of basic algebra, all other objects will be defined and illustrated by examples.

May 30, Wednesday, 15:30, room 211

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Ivan Kochkin

Minimal free resolutions and syzygies

A minimal free resolution is an invariant associated with a graded module over a ring. We focus on graded modules over the ring of polynomials in several variables. We calculate minimal free resolutions by syzygies (generalization of relations on relations) in special cases and discuss connections with the Hilbert function and projective geometry. All necessary definitions will be given in the talk.

May 23, Wednesday, 15:30, room 211

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Evgeny Krasilnikov

Kadomtsev-Petviashvili hierarchy

The Kadomtsev-Petviashvili (KP) hierarchy is an infinite system of nonlinear partial differential equations for a function F(p1,p2,…) of infinitely many variables. We will consider some interesting solution series of KP connected with simple graphs. Then we will describe an algorithm of generating all possible solutions of KP.

May 16, Wednesday, 15:30, room 211

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Andrei Konovalov

Quadric hypersurfaces, Grassmannians and triality

The set of k-dimensional linear subspaces of a smooth complex quadric hypersurface of dimension 2k is a subvariety of Gr(k+1,2k+2). Like in the classical case of two-dimensional quadric, this variety splits into two irreducible components. In my talk, I will use Schubert calculus to prove the “triality” property of six-dimensional quadric: variety of 3-planes in this quadric is a disjoint union of two components and each of them is isomorphic to the original quadric.