Wednesday, February 27, 17:00, room 210

This week, instead of a regular talk, we invite you to the defense of the habilitation thesis (note the change of time and place):

17:00, room 210

Valentina Kiritchenko 

Geometry of spherical varieties and Newton-Okounkov polytopes
https://www.hse.ru/sci/diss/226346766

After the famous Kushnirenko and Bernstein theorems were proved in the 70-s, Askold Khovanskii asked how to extend these results to the setting where a complex torus is replaced by an arbitrary connected reductive group. In particular, he advertised widely the problem of finding the right analogs of Newton polytopes for spherical varieties. The latter are natural generalizations of toric varieties and include classical examples such as Grassmannians, flag varieties and complete conics.
Kushnirenko’s theorem was extended to spherical varieties by Michel Brion and Boris Kazarnovskii in the 80-s while the notion of Newton polytopes was generalized by Andrei Okounkov in the 90-s. I will talk about these and more recent results focusing on explicit description of geometric and topological invariants of spherical varieties in terms of geometric and combinatorial invariants of their Newton-Okounkov polytopes.
As this talk is a part of the thesis defense, it will be followed by comments of the committee members including Brion and Okounkov. Some of them will give talks on Thursday:

Michel Brion (Université Grenoble Alpes, Institut Fourier)

Automorphism groups of almost homogeneous varieties

February 28, Thursday, 17:00, room 208    (the seminar of the Laboratory of Algebraic Geometry)

The automorphism group of a projective algebraic variety X is known to be a “locally algebraic group”, extension of a discrete group (the group of components) by a connected algebraic group. The group of components of Aut(X) is quite mysterious: recently, Lesieutre constructed examples for which this group is not  finitely generated. In this talk, we will discuss the structure of Aut(X) when X has an action of an algebraic group with an open dense orbit. In particular, we will see that the group of components is arithmetic (and hence finitely presented) under this assumption.

Гаянэ Панина (доклад по совместной работе с И. Некрасовым)

Компактификации M_{0,n}, связанные с самодвойственными по Александеру комплексами: кольца Чжоу, psi-классы и числа пересечения

28 февраля, четверг, 17:00, ауд. 210 (совместное заседание семинаров “Комбинаторика инвариантов Васильева” и “Характеристические классы и теория пересечений”)

Каждый  симплициальный комплекс, двойственный самому себе  по Александеру, задает  компактификацию пространства M_{0,n}    (ASD компактификацию), которая является гладким алгебраическим многообразием. ASD компактификации включают в себя (но не исчерпываются ими)  конфигурационные пространства  шарнирных многоугольников.
Мы дадим явное описание кольца Чжоу ASD компактификации, изучим  аналог тавтологических расслоений Концевича, найдем их классы Черна,  старшие пересечения классов Черна,  и выведем рекурсию для чисел пересечения.
Для понимания доклада достаточно самых общих начальных представлений о теории когомологий, классе Эйлера, раздутиях, действиях групп и пр.

Wednesday, February 20, 18:30, room 306

Anastasia Sukachyova

Littlewood–Richardson rule and Knutson–Tao puzzles

The purpose of the talk is to define Littlewood-Richardson coefficients which are essentially structure constants of cohomology rings of Grassmannians, flag varieties, Schur and Schubert polynomials. We will discuss two different ways to calculate them: the first is based on Littlewood–Richardson tableaux, while the other relies on Knutson and Tao’s puzzles.

These coefficients appear in enumerative geometry, so we will review a couple of simple examples to understand how they can be used in practice.
All necessary definitions will be given during the talk.

Wednesday, February 6 and 13, 18:30, room 306

Anatoly Kouchnirenko 

Arnold filtrations, Stanley–Reisner rings, and simplicial Newton polytopes (in Russian)

Два ключевых раздела алгебраической комбинаторики, связанных с геометрией многогранников – это теория Стэнли-Рейснера и теория многогранников Ньютона. Первая позволяет исследовать комбинаторику граней выпуклого многогранника средствами коммутативной алгебры. Вторая связывает объем и другие геометрические характеристики целочисленного многогранника с инвариантами построенных по нему алгебраических многообразий.
Доклады познакомят слушателей с обоими сюжетами, а также (во второй день) с их взаимосвязью, которую докладчику недавно удалось применить в интересной конкретной задаче. Более подробный анонс доступен по ссылке. Лекции будут доступны студентам второго курса.

Wednesday, January 23 and 30, 18:30, room 306

Alexander Esterov

Geometry and algebra of lattice polytopes

We shall overview geometry of lattice polytopes and its applications, and discuss the seminar topics for this term. At the same time, I will recall some facts and notions that will be used in the subsequent talks by Anatoly Kouchnirenko (see below).
This includes: classification and statistics of lattice polytopes, Ehrhart polynomials of lattice polytopes and Hilbert series of graded algebras, multiplicities of roots and length of local algebras, shellable complexes and f-vectors of polytopes

Upcoming talks: February 6&13: Anatoly Kouchnirenko Arnold filtrations, Stanley–Reisner rings, and simplicial Newton polytopes (in Russian)

Wednesday, January 16, 18:30, room 306

Alexander Esterov

Lattice polytopes and systems of polynomial equations

The fundamental theorem of algebra states that the number of roots of the univariate polynomial (counted with multiplicities) equals the degree of the polynomial. We shall discuss several generalizations of this fact to systems of polynomial equations of several variables.
The most geometric generalization (the Kouchnirenko theorem) states that, for generic systems of equations, the number of solutions of the system equals the volume of a certain lattice polytope, associated to this system. Motivated by this application, we shall overview geometry of lattice polytopes and briefly discuss the seminar topics for this term.

December 19, Wednesday, 18:30, room 306

Fedor Selyanin

Tropical Grassmannian (continued)

Russian version: Тропический грассманиан  (продолжение, см. ниже анонс на русском)

We will identify the tropical Grassmannian with the space of phylogenetic trees. The latter is an interesting combinatorial object, which was first defined and studied in applications of mathematics to biology.

Второй доклад будет посвящён собственно тропикализации грассманового многообразия G(2,n), которая окажется пространством филогенетических деревьев, красивым комбинаторным объектом.

December 12, Wednesday, 18:30, room 306

Fedor Selyanin

Tropical Grassmannian

Russian version: Тропический грассманиан (см. ниже анонс на русском)

The first part of the talk is an introduction to tropical geometry. We explain basic concepts of tropical geometry using concrete examples. The second part of the talk will be devoted to tropicalization of the Grassmannian G(2,n). We identify the tropical Grassmannian with the space of phylogenetic trees. The latter is an interesting combinatorial object, which was first defined and studied in applications of mathematics to biology.

В начале лекции будет введение в тропическую геометрию, на наглядных примерах объясню её базовые понятия. Вторая часть лекции будет посвящена собственно тропикализации грассманового многообразия G(2,n), которая окажется пространством филогенетических деревьев, красивым комбинаторным объектом.

December 5, Wednesday, 18:30, room 306

Yury Rudko

Gelfand-Zetlin bases for representations of SL(n)

We construct Gelfand-Zetlin bases in irreducible representations of SL(n) using the chain of subgroups SL(n)>SL(n-1)>…>SL(2). The key step is reduction of an irreducible representation of SL(n) to SL(n-1). Using reduction we show that basis vectors are labeled by lattice points in a convex polytope (called Gelfand-Zetlin polytope) defined by simpe inequalities. We work out concrete examples such as the adjoint representation of SL(3).

November 28, Wednesday, 18:30, room 306

Valentina Kiritchenko

Newton-Okounkov polytopes of flag varieties for classical groups

For classical groups SL(n), SO(n) and Sp(2n), we define uniformly geometric valuations on the corresponding complete flag varieties. The valuation in every type comes from a natural coordinate system on the open Schubert cell, and is combinatorially related to the Gelfand-Zetlin pattern in the same type. For SL(n) and Sp(2n), we identify the corresponding Newton-Okounkov polytopes with the Feigin-Fourier-Littelmann-Vinberg polytopes. For SO(n), we compute low-dimensional examples and formulate open problems. All necessary definitions will be given in the talk.

November 21, Wednesday, 18:30, room 306

Ilya Dumanskiy

PBW filtration and FFLV basis in type A (continued)

Abstract: Last time we constructed a generating set in an irreducible representation of SL(n). This set is compatible with the PBW filtration and its elements are parameterized by lattice points in a convex polytope (called Feigin-Fourier-Littelmann-Vinberg polytope). By counting the number of lattice points in FFLV polytopes we prove that this generating set is in fact a basis. We remind all definitions and constructions from the previous talk.