Personal webpage of Valentina Kiritchenko

Faculty of Mathematics

November 14, Wednesday, 18:30, room 306

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Alexandr Popkovich

PBW filtration and FFLV basis in type A

Russian version: ПБВ фильтрация и ФФЛВ базис для типа A (см. ниже анонс на русском)

Abstract: We describe the PBW filtration on the irreducible representations  of SL(n). This filtration is induced by the filtration on the universal enveloping algebra of the Lie algebra sl_n. Using this filtration Feigin, Fourier and Littelmann constructed a basis conjectured by Vinberg in an irreducible representation of SL(n). The basis is parameterized by lattice points in a convex polytope (called Feigin-Fourier-Littelmann-Vinberg polytope). We will formulate the main results of their paper and prove some of them if time permits. All necessary definitions will be given in the talk.

Russian version: Фильтрация на универсальной обертывающей алгебре U(sl_n) индуцирует градуировку на каждом неприводимом представлении алгебры Ли sl_n. В статье Фейгина, Фурье и Литтельмана “PBW filtration and bases for irreducible modules in type A” описано построение некоторого однородного относительно этой градуировки базиса для произвольного неприводимого представления, причём элементы полученного базиса соответствуют целым точкам некоторого многогранника. Я сформулирую результаты этой статьи и, насколько позволит время, приведу доказательства основных из них. Предварительно я напомню все необходимые сведения из теории представлений полупростых алгебр Ли.

November 7, Wednesday, 18:30, room 306

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Alexei Piskunov

Newton-Okounkov polytopes of symplectic flag varieties

Abstract:
This is continuation of the previous talk, however, we remind all necessary definitions and results. We use chain of subgroups Sp(2n) > Sp(2n-2) > to define a valuation on the variety of isotropic flags. Using this valuation we interpret reduction multiplicities of representations of Sp(2n) in terms of lattice points inside a convex polytope.

October 31, Wednesday, 18:30, room 306

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Alexei Piskunov

Reduction of Sp(2n) representations or what is “Sp(2n-1)”?

Russian version: Редукция представлений Sp(2n) или что такое «Sp(2n-1)»? (см. ниже анонс на русском)

Abstract:
A famous construction of Gelfand-Zetlin bases in irreducible representations of a classical group G(n) uses reduction of representations to G(n-1) regarded as a subgroup of G(n). However, this construction fails for symplectic groups as there is no group Sp(2n-1). Or is there?
Russian version: Известно, что для классических серий групп Ли G(n) можно получать информацию об их неприводимых представлениях с помощью редукции до представления G(n-1) и разложения на неприводимые с помощью таблиц Гельфанда-Цетлина. В случае симплектической группы эта операция приводит к возникновению в цепочке Sp(2n) > Sp(2n-2) > … намёка на промежуточную группу «Sp(2n-1)». Я расскажу общую конструкцию редукции представлений для симплектической группы и то, как можно геометрически интерпретировать эту группу. Также, если останется время, то я сформулирую одно утверждение про функцию кратности вхождения данного неприводимого представления в разложение при редукции.

 

October 17, Wednesday, 18:30, room 306

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Dmitry Rybin

Representations of the symmetric group S_n

Russian version: Некоторые представления группы S_n (см. ниже анонс на русском)

Abstract:
We construct a series of irreducible complex representations of the symmetric group S_n. All necessary definitions will be given in the talk. If time permits we mention Schur-Weyl duality and a close relation between representations of S_n and enumeration of surfaces of genus g obtained by gluing n pairs of sides of a 2n-gon.
Russian version: Будет доказана неприводимость серии комплексных представлений S_n. Все определения прозвучат по мере необходимости. В зависимости от скорости движения, планируется упомянуть о двойственности Шура-Вейля и о тесной связи с задачей подсчёта кол-ва поверхностей рода g, получающихся при склеивании n пар сторон 2n-угольника.

October 10, Wednesday, 18:30, room 306

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Alex Savchik

T- and H- polar lines and other projective invariants
Russian version: T- и H-поляры и другие проективные инварианты (см. ниже анонс на русском)

Matching different views of the same object is a fundamental problem in computer vision. In geometric terms, we have to find a projective transformation of a real projective plane that yields a bijection between two given configurations. To solve this problem we use projective invariants. For instance, for a configuration “an oval and a point inside” the suitable invariants are T- and H- polars that generalize (in two different ways) polar lines from ellipses to arbitrary ovals. The talk will be devoted to such projective invariants and their properties.
All necessary definitions will be given in the talk.

Russian version: Проективное сопоставление контуров на вещественной плоскости – одна из актуальных задач в области компьютерного зрения. Её решение возможно с опорой на проективно инвариантные элементы. Например, для конфигурации “овал и внутренняя точка” такими являются T- и H-поляры, двумя способами обобщающие поляру эллипса на случай произвольного овала. Доклад посвящен анализу подобных геометрических проективных инвариантов. Будет доказано существование не менее, чем 3 точек пересечения T- и H-поляр, а также приведены некоторые инвариантно определенные точки для конфигураций “овал и внешняя прямая” и “овал и две точки на его контуре” на проективной плоскости.

October 3, Wednesday, 18:30, room 306

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Mikhail Troshkin

2-transitivity of Galois groups in Schubert calculus

Galois group of a problem in enumerative geometry indicates how solutions to the problem are permuted as conditions are varied continuously along loops. In recent years, new theoretical and experimental results concerning Galois groups in Schubert calculus were obtained. We shall review them and present a proof that Galois groups for a certain class of Schubert problems (including every problem on Gr(2, n)) are 2-transitive, following the paper by F. Sotille and J.White.

The proof will use only the basic notions of algebraic geometry; the definitions of Galois group (in this context) and Schubert problems will be recalled in the talk.

September 26, Wednesday, 18:30, room 306

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Alexander Esterov

Solvability of equations by radicals and classification of lattice polytopes

The celebrated Abel–Ruffini theorem states that it is impossible to express the solutions of the general polynomial equation of degree d>4 in terms of its coefficients, using the arithmetic operations and arithmetic roots.
We shall present the proof of this theorem based on the technique of topological Galois theory (different from the well known “Arnold’s proof”). We shall also generalize this theorem to systems of polynomial equations: we shall see that the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of volume at most 4, which is essentially finite (i.e. we can explicitly list the 34 “elementary” lattice polytopes of volume 4 and the corresponding 34 “elementary” solvable systems of equations, from which all the others can be constructed).
The aim of the talk is (1) to demonstrate one instance of the relation between algebraic and convex geometry based on the notion of Newton polytopes and (2) to provide the background for a subsequent talk about the Galois theory of Schubert calculus.
No previous knowledge of Galois theory is required.

 

September 24, Monday, 14:00, room 413 (extra session)

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Svetlana Makarova (MIT)

14:00 Understanding geometric invariant theory

In the first part of the talk, I will define a moduli problem and explain how quotients by reductive groups appear in the picture. I will introduce the notions of good and geometric quotients. In the end (maybe in the last five minutes), I will mention the relation of GIT quotients to stacks and sketch two instances of how convex polytopes appear in the theory — without going into too much detail.

15:30 GIT and derived categories

In the second part of the talk, I will present a new (complicated) proof of the fullness of Kapranov’s exceptional collection on Grassmannians which uses the theory developed by Sam and Halpern—Leistner in their recent paper.

September 19, Wednesday, 18:30, room 306 (note the change of time)

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Valentina Kiritchenko

Polytopes in representation theory

I will talk about classical and new polytopes that arise in representation theory. Lattice points in these polytopes count basis vectors in irreducible representations of linear groups such as GL_n(\C), SO_n(\C) and Sp_{2n}(\C). In particular, I will review construction of Gelfand-Zetlin bases and the corresponding polytopes. Theory of Newton-Okounkov convex bodies provides an alternative (and probably simpler) method for constructing the same polytopes. For example, I will outline a construction of Feigin-Fourier-Littelmann-Vinberg polytopes as Newton-Okounkov polytopes of flag varieties. I will also discuss interpretation of Schubert calculus in terms of polytopes.

No preliminary knowledge of representation theory is required, all necessary definitions will be given during the talk.

September 12, Wednesday, 15:30, room 306

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Evgeny Smirnov

Schubert varieties and Schubert polynomials

I am going to speak about Schubert varieties on a full flag variety. These varieties play an important role in enumerative geometry. The geometric problem of intersecting Schubert varieties (in general position) has its algebraic counterpart, dealing with Schubert polynomials. So a geometric question can be reduced to a purely algebraic/combinatorial one. I am planning to define Schubert polynomials combinatorially (in two different ways). Time permitting, I will speak about the results of Knutson and Miller (2005) who showed how Schubert polynomials appear in the context of Gröbner degeneration of Schubert varieties.

No preliminary knowledge of Schubert calculus is required, all necessary definitions will be given during the talk.