Wednesday, May 22, 17:00 and 18:30, room 306

17:00 Ksenia Gadakhabadze: Ehrhart polynomials of irrational polytopes

It is classically known that, stretching a lattice polytope k times, the number of lattice points in it depends on k polynomially (the Ehrhart polynomial). Amazingly, if some vertices of a polytope are irrational, the aforementioned number of lattice points still may happen to depend polynomially on k. Polytopes of this kind are called pseudo-lattice. We shall present examples of such polytopes and the complete classification of pseudo-lattice coordinate triangles, recently obtained by Richard Stanley and coauthors.

18:30 Roman Krutovskiy, Grigory Yurgin: Сlassification of Gorenstein polytopes

The goal of our two joint talks is to obtain the classification of Gorenstein polytopes of degree two, which is equivalent to classifying all Gorenstein toric Del Pezzo varieties.
On the first half (May 15, 18:30) we introduce the notion of a Gorenstein polytope and start our classification using methods of commutative algebra. On the second half (May 22, 18:30) we finish the classification using combinatorial approach and the results of the first half.
The talks are independent form the preceding mini-talk (and do not require the knowledge of toric varieties). All necessary defenitions and facts from commutative algebra will be given during the talks.

Wednesday, May 15, 17:00 and 18:30, room 306

17:00 Boris Nazarov: Batyrev’s correspondence between lattice simplices and finite torus subgroups

This is an addendum to the preceding talk on the classification of small lattice polytopes.

17:30 Roman Krutovskiy: Gorenstein polytopes and toric Del Pezzo varieties

The material of this mini-talk is not necessary for understanding of the subsequent main talks. Its aim is to provide an algebraically geometric motivation for classification of all Gorenstein polytopes of degree two. I will briefly remind some general properties of toric varieties. Especially, I will give a description of the canonical class in terms of the underlying fan. Finally, I will formulate a theorem which establishes conditions on the underlying polytope of a projective toric variety X which guarantee that X is a Gorenstein Del Pezzo variety. Basic knowledge of toric varieties is expected.

18:30 Roman Krutovskiy, Grigory Yurgin: Сlassification of Gorenstein polytopes

The goal of our two joint talks is to obtain the classification of Gorenstein polytopes of degree two, which is equivalent to classifying all Gorenstein toric Del Pezzo varieties. On the first half (May 15, 18:30) we introduce the notion of a Gorenstein polytope and start our classification using methods of commutative algebra. On the second half (May 22, 18:30) we finish the classification using combinatorial approach and the results of the first half. The talks are independent form the preceding mini-talk (and do not require the knowledge of toric varieties). All necessary defenitions and facts from commutative algebra will be given during the talks.   

Wednesday, April 24, 17:00 and 18:30, room 306

17:00  Arina Voorhaar: Thom polynomials and the method of restriction equations

The global behavior of singularities is governed by their so-called Thom polynomials. Once the Thom polynomial of a singularity \mu is known, one can compute the cohomology class represented by the \mu-points of a map. One of the methods to compute the Thom polynomial of a singularity is due to R.Rimanyi and consists in solving a certain system of linear equations. We will discuss this method and will compute some Thom polynomials using it.  Basic familiarity with singular homology is strongly recommended.  I will introduce all the other necessary notions during the talk and give some examples.

18:30 Boris Nazarov: About lattice polytopes with a given Ehrhart polynomial

In the first part, I will prove that the number of unimodular equivalence classes of lattice polytopes with fixed normalized volume and degree is finite as well as the number of lattice polytopes with fixed h-polynomial.  Later, we will classify all lattice polytopes with normalized volume less or equal than 3. 

Wednesday, April 17, 18:30, room 306

Alexander Sharipov

Introduction to Ehrhart theory

  I will give definitions and examples of Ehrhart and h- polynomials, talk on their geometric properties, discuss connection with combinatorics and, finally, prove some fundamental theorems (like Stanley’s Nonnegativity and Ehrhart-Macdonald Reciprocity). NB: Notions and facts from Ehrhart theory will be extensively used in many subsequent talks

Wednesday, April 10, 17:00 and 18:30, room 306

17:00  Igor Makhlin (Skoltech)

Gelfand-Tsetlin degenerations

PBW degenerations of flag varieties were introduced by Feigin et al. in 2010. Weighted versions of this construction were shown to provide other flat degenerations of the flag variety, including the toric variety associated with the FFLV polytope. I will recall the key definitions and results and then I will explain how another particularly well known toric degeneration can be obtained in a similar representation theoretic context. The toric degeneration in question is the one associated with the Gelfand-Tsetlin polytope.

18:30 Ilias Suvanov

Statistics and classification of small polytopes

Since there are finitely many lattice polytopes of a given volume and dimension, the next natural questions are (1) to estimate asymptotically the number of such polytopes for large volume and (2) to completely classify such polytopes for small volume.
The question (1) was answered by Arnold in two dimensions and later addressed by Vershik and many others in general. We shall discuss Arnold’s estimate.
The question (2) will be discussed in a slightly more general setting of mixed volumes. Namely, we shall recall the notion of the mixed volume of a tuple of polytopes and classify triples of lattice polytopes (P1,P2,P3) in dimension 3 with a mixed volume up to 4. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which, we will prove is finite for every fixed value of the mixed volume. Using the algorithm by Averkov, Borger and Sorpunov, we will enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This in particular produces a classification of generic trivariate systems of polynomial equations solvable by radicals.

Wednesday, April 3, 18:30, room 306

Dmitry Leonkin

Finitely many lattice polytopes of a given volume

In our quest for classification of small lattice polytopes it’s absolutely crucial to prove finiteness of equivalency classes of polytopes with fixed volume (or with fixed positive number of interior points). We will prove that every lattice polytope in R^n of volume <= V is integrally equivalent to a lattice polytope, contained in lattice cube of side length at most n*n!*V. To deduce the result about polytopes with fixed number of interior points we will investigate boundaries for volume of polytope with given amount of interior lattice points. The talk is based on papers by Lagarias, Ziegler and Hensley.

No Convex and Algebraic Geometry seminar on March 27

There will be no seminar on Wednesday, March 27, due to the spring examination period.

Wednesday, March 20, 18:30, room 306

Evgeny Goncharov

Symplectic resolutions of certain hypersurfaces in A^5

The hypersurfaces we are going to consider are obtained by the general procedure of constructing Coulomb branches (an object of interest to physicists that was rigorously defined mathematically in a series of papers by A. Braverman, M. Finkelberg and H. Nakajima). The general procedure is quite involved but for the case we will consider (quiver with one vertex and l loops with dim V = 2, dim W = 1) there is a theorem that describes the generators (and gives a way to find relations between them) of the coordinate ring of the Coulomb branch (it is an algebraic variety). We will apply the theorem to find the Coulomb branch which turns out to be a hypersurface in A^5.
Then I will explain how to (naturally) define a symplectic form on the smooth locus of this hypersurface. For l=1 the Coulomb branch turns out to be Sym^2 A^2 and its symplectic resolution (resolution of singularities such that the pullback of the symplectic form on the smooth locus extends to define a symplectic form on the resolution) has been known for a long time (it is given by the Hilbert scheme Hilb^2 A^2). I will show that a naive attempt to construct a symplectic resolution for a general l does not work and discuss the general properties of our hypersurface (compute the Hilbert polynomial, etc).
We will then note that a certain construction in a Lie algebra sp_{2l} (Slodowy slice to the subsubregular orbit of the nilpotent of type (2l-2, 1, 1)) has precisely the same properties and compute the Slodowy slice explicitly to see that the resulting variety is the same. The symplectic resolution of the Slodowy slice, however, is easy to describe. This argument gives a new family of symplectic hypersurfaces (apart from the known Kleinian singularities in A^3). Time permitting we will discuss what happens if one varies the dimensions of the vector spaces V and W in the quiver we started with.

Wednesday, March 13, 17:30, room 110

This week, instead of a regular talk, we invite you to the Mathematical physics seminar (note the change of time and place):

17:30, room 110

Xin Fang (University of Cologne)

Toric degenerations of flag varieties and their applications

Toric degenerations translate geometric properties of varieties to combinatorial ones of polytopes or polyhedral cones. In this talk I will introduce a general framework to construct toric degenerations of flag varieties via birational sequences and Newton-Okounkov bodies. Most of the known toric degenerations turn out to be concrete examples of this construction. As an application, I will explain how to get the exact value of the Gromov width of a coadjoint orbit.

March 6, Wednesday, 18:30, room 306

Fedor Selyanin

LATTICE VERTEX POLYTOPES WITH INTERIOR LATTICE POINTS

Consider a convex polytope with lattice vertices and at least one interior lattice point. We will prove that the number of boundary lattice points is bounded from above by a function of the dimension and the number of interior lattice points. We will find the boundaries for zero symmetric convex polytopes, then for arbitrary lattice polytopes and compare them.
This is the starting point to many results on classification of small lattice polytopes that will be presented in subsequent talks.