Algebraic and geometric properties of lattice polytopes.
Prof. Dr. C. Haase (FU), Akiyoshi Tasuchiya (Japan)
Freie Universität Berlin. Research Alumni Programm.
(1) The reflexive dimension of a lattice polytope.
The reflexive polytope is one of the keywords belonging to the current trends in research of lattice polytopes. Haase and Melinkov showed that every lattice polytope is a face of some reflexive polytope. This lead us to consider which lattice polytopes are facets of reflexive polytopes. In particular, I conjecture that every 0/1-polytope is a facet of some reflexive polytope. Recently, we were able to prove this conjecture for the following classes of 0/1-polytopes:
(i) 0/1-polytopes up to dimension 4;
(ii) order polytopes and hypersimplices;
(iii) the stable set polytope of perfect graphs.
One of the goals of the Research Alumni Program is to generalize these results. Concretely, I will consider this problem for 0/1-polytopes of dimension 5 and for compressed polytopes. The class of compressed polytopes contains (ii) and (iii).
(2) The unimodality of the h*-vector of an alcoved polytope.
One of the most popular open problems is Stanley's unimodality conjecture. I will consider this conjecture for alcoved polytopes. Alcoved polytopes have good triangulations and this property should be useful to attack Stanley's conjecture. One of the goals of the Research Alumni Program is to solve this conjecture for order polytopes and hypersimplices. Note that these polytopes are alcoved polytopes.
These two problems are important in the study of lattice polytopes. In particular, (1) is a geometric problem and (2) is an algebraic problem.