Monday, May 15, 2017
Freie Universität Berlin
Takustr. 9
14195 Berlin
room 005
Lecture - 14:15
Abstract:
The lattice simplices in real n-dimensional euclidean space formed via the convex hull of the standard basis vectors together with a weakly decreasing vector of negative integers have an important place in algebraic geometry since they contain simplices that play a central role in problems from enumerative geometry and mirror symmetry. The importance of the lattice combinatorics of these simplices in regards to such problems motivated the examination of their Ehrhart h*-polynomials from the algebro-geometric standpoint. Simultaneously, it is worthwhile to ask if such simplices capture ubiquitous examples of h*-polynomials, thereby motivating their study from a purely combinatorial standpoint. In this talk, we will discuss an arithmetic formula for the h*-polynomials of such simplices, and we will use this formula to recover some classic examples of h*-polynomials with desirable combinatorial properties. Our desired combinatorial examples will arise via the representations of integers in various numeral systems. We will see that n-simplices within this family associate via their normalized volumes to the n^th place value of positional numeral systems, and that the h*-polynomials of these simplices admit a combinatorial formula via descent-like statistics on the numeral strings encoding the nonnegative integers within these systems. With these methods, we will recover such ubiquitous h*-polynomials as the Eulerian polynomials and the binomial coefficients arising from the factoradic and binary numeral systems, respectively. We will also see that the binary case generalizes to any base-r numeral system such that the corresponding h*-polynomials are all real-rooted and unimodal.
Colloquium - 16:00
Abstract:
The h^*-polynomial are meaningful invariants in the study of lattice
polytopes, and characterizing them is an incredibly hard challenge.
Luckily in the last decades some relations among their coefficients have
been proved, and all of them have some dependence either on the degree
or the dimension of the lattice polytopes. In a recent work with A.
Higashitani we prove that this is not always the case, by proving a
relation which is "universal". In particular we prove that Scott's
inequality is true in each dimension and degree, if interpreted correctly.