Gamma-polynomials of flag homology spheres

Chapter 1 contains the main definitions used in this thesis. It also includes some basic theory relating to these fundamental concepts, along with examples. Chapter 1 includes an original result, Theorem 1.5.4, answering a question of Postnikov-Reiner-Williams, which characterises the normal fans of nestohedra. Chapter 2 contains the content of the paper [2], of which Theorem 2.0.6 is the main result. As mentioned, [2] shows that the Nevo and Petersen conjecture holds for simplicial complexes in sd(Σd−1). . Chapter 3 includes the content of the paper [1], where we show that the Nevo and Petersen conjecture holds for the dual simplicial complexes to nestohedra in Theorem 3.0.4. Chapter 4 contains the content of the paper [3] in which we prove Conjecture 0.0.4 in Theorem 4.1.2 by showing that tree shifts lower the γ-polynomial of graph-associahedra. Chapter 4 also includes Theorem 4.2.1, which shows that flossing moves also lower the γ-polynomial of graph-associahedra. In Chapter 5 we include smaller results that have been made. This chapter includes a result proving Gal’s conjecture for edge subdivisions of the order complexes of Gorenstein* complexes, and shows that this result can be attributed to the work of Athanasiadis in [4]. Chapter viii INTRODUCTION 5 also includes some work we have done towards answering Question 14.3 of [26] for interval building sets

gamma-vectors of edge subdivisions of the boundary of the cross polytope

For any flag simplicial complex $\Theta$ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex $\Gamma(\Theta)$ (dependent on the sequence of subdivisions) whose $f$-vector is the $\gamma$-vector of $\Theta$. This proves that the $\gamma$-vector of any such simplicial complex satisfies the Frankl-F\"{u}redi-Kalai inequalities, partially solving a conjecture by Nevo and Petersen \cite{np}. We show that when $\Theta$ is the dual simplicial complex to a nestohedron, and the sequence of subdivisions corresponds to a flag ordering as defined in \cite{ai}, that $\Gamma(\Theta)$ is equal to the flag simplical complex defined there.Comment: 18 pages, 1 figur

Examples and counterexamples in Ehrhart theory

This article provides a comprehensive exposition about inequalities that the coefficients of Ehrhart polynomials and $h^*$-polynomials satisfy under various assumptions. We pay particular attention to the properties of Ehrhart positivity as well as unimodality, log-concavity and real-rootedness for $h^*$-polynomials. We survey inequalities that arise when the polytope has different normality properties. We include statements previously unknown in the Ehrhart theory setting, as well as some original contributions in this topic. We address numerous variations of the conjecture asserting that IDP polytopes have a unimodal $h^*$-polynomial, and construct concrete examples that show that these variations of the conjecture are false. Explicit emphasis is put on polytopes arising within algebraic combinatorics. Furthermore, we describe and construct polytopes having pathological properties on their Ehrhart coefficients and roots, and we indicate for the first time a connection between the notions of Ehrhart positivity and $h^*$-real-rootedness. We investigate the log-concavity of the sequence of evaluations of an Ehrhart polynomial at the non-negative integers. We conjecture that IDP polytopes have a log-concave Ehrhart series. Many additional problems and challenges are proposed.Comment: Comments welcome

Enumerative Combinatorics

Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds

Generalized parking function polytopes

A classical parking function of length $n$ is a list of positive integers $(a_1, a_2, \ldots, a_n)$ whose nondecreasing rearrangement $b_1 \leq b_2 \leq \cdots \leq b_n$ satisfies $b_i \leq i$. The convex hull of all parking functions of length $n$ is an $n$-dimensional polytope in $\mathbb{R}^n$, which we refer to as the classical parking function polytope. Its geometric properties have been explored in (Amanbayeva and Wang 2022) in response to a question posed in (Stanley 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of $\mathbf{x}$-parking functions for $\mathbf{x}=(a,b,\dots,b)$, which we refer to as $\mathbf{x}$-parking function polytopes. We explore connections between these $\mathbf{x}$-parking function polytopes, the Pitman-Stanley polytope, and the partial permutahedra of (Heuer and Striker 2022). In particular, we establish a closed-form expression for the volume of $\mathbf{x}$-parking function polytopes. This allows us to answer a conjecture of (Behrend et al. 2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.Comment: 29 pages, 3 figures, comments welcome

A friendly introduction to Fourier analysis on polytopes

This book is an introduction to the nascent field of Fourier analysis on polytopes, and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the field. We assume a familiarity with Linear Algebra, and some Calculus. Of the many applications, we have chosen to focus on: (a) formulations for the Fourier transform of a polytope, (b) Minkowski and Siegel's theorems in the geometry of numbers, (c) tilings and multi-tilings of Euclidean space by translations of a polytope, (d) Computing discrete volumes of polytopes, which are combinatorial approximations to the continuous volume, (e) Optimizing sphere packings and their densities, and (f) use iterations of the divergence theorem to give new formulations for the Fourier transform of a polytope, with an application. Throughout, we give many examples and exercises, so that this book is also appropriate for a course, or for self-study.Comment: 204 pages, 46 figure

Hepp's bound for Feynman graphs and matroids

We study a rational matroid invariant, obtained as the tropicalization of the Feynman period integral. It equals the volume of the polar of the matroid polytope and we give efficient formulas for its computation. This invariant is proven to respect all known identities of Feynman integrals for graphs. We observe a strong correlation between the tropical and transcendental integrals, which yields a method to approximate unknown Feynman periods.Comment: 26 figures, comments very welcom

Aspects of Scattering Amplitudes and Moduli Space Localization

We propose that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with $n$ punctures, $\mathcal{M}_{0,n}$, compute tree-level scattering amplitudes in quantum field theories with a finite spectrum of particles. The relevant cohomology groups are twisted by representations of the fundamental group $\pi_1(\mathcal{M}_{0,n})$ that describes how punctures braid around each other on the Riemann surface. Such a structure can be used to link the space of Riemann surfaces with the space of kinematic invariants. Intersection numbers of said cohomology classes—whose representatives we call twisted forms—can be shown to fully localize on the boundaries of $\mathcal{M}_{0,n}$, which are in a one-to-one correspondence with trivalent trees that have an interpretation as Feynman diagrams. In this work we develop systematic approaches towards accessing such boundary information. We prove that when twisted forms are logarithmic, their intersection numbers have a simple expansion in terms of trivalent Feynman diagrams weighted by residues, allowing only for massless propagators on the internal and external lines. It is also known that in the massless limit intersection numbers have a different localization formula on the support of so-called scattering equations. Nevertheless, for physical applications one also needs to study non-logarithmic forms as they are responsible for propagation of massive states. We utilize the natural fibre bundle structure of $\mathcal{M}_{0,n}$—which allows for a direct access to the boundaries—to introduce recursion relations for intersection numbers that "integrate out" puncture-by-puncture. The resulting recursion involves only linear algebra of certain matrices describing braiding properties of $\mathcal{M}_{0,n}$ and evaluating one-dimensional residues, thus paving a way for explicit analytic computations of scattering amplitudes. Together with a reformulation of the tree-level S-matrix of string theory in terms of twisted forms, the results of this work complete a unified geometric framework for studying scattering amplitudes from genus-zero Riemann surfaces. We show that a web of dualities between different homology and cohomology groups allows for deriving a host of identities among various types of amplitudes computed from the moduli space, which in this setup become a consequence of linear algebra. Throughout this work we emphasize that algebraic computations can be supplemented—or indeed replaced—by combinatorial, geometric, and topological ones