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.