August 1
(Monday) ～ August 5 (Friday), 2016
ABSTRACT
August, 1 (Mon) | August, 2 (Tue) | August, 3 (Wed) | August, 4 (Thu) | August, 5 (Fri) |
13:15--14:15 | Akiyoshi Tsuchiya (Osaka University)
"Gorenstein Fano polytopes arising from two poset polytopes" |
It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. In this talk, we discuss whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In particular, we consider order polytopes and chain polytopes. We introduce several classes of normal Gorenstein Fano polytopes arising from order polytopes and chain polytopes, and we investigate combinatorial properties, especially, the Ehrhart polynomials and the volume of these polytopes. This talk is joint work with Takayuki Hibi. | |
14:30--15:30 | Robert Davis (Michigan State University)
"Pattern-Avoiding Polytopes" |
The permutohedron and the Birkhoff polytope are two well-studied polytopes related to many areas of mathematics. This talk will discuss generalizations of these polytopes by considering subpolytopes whose vertices correspond to certain avoidance classes of permutations. We explore their combinatorial structure as well as their Ehrhart polynomials and Ehrhart series. In two specific cases we will identify when the polytopes have symmetric and unimodal h^{*}-vectors, whose proofs show an elegant interplay among pattern avoidance, poset topology, discrete geometry, and physics. | |
15:45--16:45 | Liam Solus (MIT and IST Austria)
"On h^{*}-unimodality and the integer decomposition property for reflexive simplices" |
A wide open question is whether or not all reflexive polytopes exhibiting the integer decomposition property (IDP) have unimodal Ehrhart h*-polynomials. To broaden the family of examples available at the intersection of the relevant properties, we identify families of reflexive simplices exhibiting both IDP and h*-unimodality. Our approach relies on Conrads' encoding of a reflexive simplex in terms of an arithmetic sequence satisfying some divisibility conditions. In this talk, we will identify an arithmetic condition on these sequences that is necessary for a reflexive simplex to be IDP. We will see that satisfying this necessary condition is enough to ensure h^{*}-unimodality for many families of reflexive simplices, including some which are not IDP. This talk is based on on-going work with Ben Braun and Rob Davis. |