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.