Wednesday, May 29, 17:00 and 18:30, room 306

17:00 Yuri Rud’ko: The number of lattice polytopes of a given volume
We will study the following question: what is the number of different lattice convex polytopes of a given volume A in R^d. This question is rather well-studied. There are a few results of Arnold; Konyagin and Sevostyanov; Andrews and so on. I am going to tell about the best possible estimate (by Vershik and Barany) for the order of this number. It is rather close to the previously discussed Arnold’s result for the number of plane convex lattice polytopes but uses new methods to go through in higher dimentions.

18:30 Alexander Iuran: Covering minima and lattice-point-free convex bodies This is the first in a series of talks devoted to the study of convex bodies and polyhedra without interior lattice points. The topic originates from Minkowski’s “gometry of numbers’ and relates many fields in mathematics and its applicationss from number theory to optimization.
We will prove several inequalities about “covering minima” of a convex body, quantities which describe the ability of its integer translations to cover all the space. This allows to give bounds of O(dim^2) on the width of a convex body without lattice points. Even though the volume of such a body is unbounded, we will prove that it has a projection whose volume is not much greater than the determinant of the lattice.

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.