Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appear naturally in many independent fields of science and engineering. A substantial amount of work has been devoted to understanding all possible synchronization configurations on a given network. In this setting, a key problem is to determine the total number of such configurations. Through an algebraic formulation, for tree and cycle graphs, we provide an upper bound on this number using the birationally invariant intersection index of a system of rational functions on a toric variety.

A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart) h * -polynomial. This conjecture can be viewed as a strengthening of a previously disproved conjecture which stated that any Gorenstein lattice polytope has a unimodal h * -polynomial. The first counterexamples to unimodality for Gorenstein lattice polytopes were given in even dimensions greater than five by Mustaţǎ and Payne, and this was extended to all dimensions greater than five by Payne. While there exist numerous examples in support of the conjecture that IDP reflexives are h * -unimodal, its validity has not yet been considered for families of reflexive lattice simplices that closely generalize Payne's counterexamples. The main purpose of this work is to prove that the former conjecture does indeed hold for a natural generalization of Payne's examples. The second purpose of this work is to extend this investigation to a broader class of lattice simplices, for which we present new results and open problems.

Peak sets of a permutation record the indices of its peaks. These sets have been studied in a variety of contexts, including recent work by Billey, Burdzy, and Sagan, which enumerated permutations with prescribed peak sets. In this article, we look at a natural analogue of the peak set of a permutation, instead recording the values of the peaks. We define the "pinnacle set" of a permutation w to be the set {w(i) : i is a peak of w}. Although peak sets and pinnacle sets mark the same phenomenon, these objects differ in notable ways. In the work below, we characterize admissible pinnacle sets and study various enumerative questions related to these objects.

An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal h * -vector. Although various sufficient conditions have been found, necessary conditions remain a challenge. In this paper, we consider integrally closed reflexive simplices and discuss an operation that preserves reflexivity, integral closure, and unimodality of the h * -vector, providing one explanation for why unimodality occurs in this setting. We also discuss the failure of proving unimodality in this setting using weak Lefschetz elements.

No abstract