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.