Moduli of Tropical Plane Curves

We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus $g$, our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with $g$ interior lattice points. It has dimension $2g+1$ unless $g \leq 3$ or $g = 7$. We compute these spaces explicitly for $g \leq 5$.Comment: 31 pages, 25 figure

Lower bounds on the dilation of plane spanners

(I) We exhibit a set of 23 points in the plane that has dilation at least $1.4308$, improving the previously best lower bound of $1.4161$ for the worst-case dilation of plane spanners. (II) For every integer $n\geq13$, there exists an $n$-element point set $S$ such that the degree 3 dilation of $S$ denoted by $\delta_0(S,3) \text{ equals } 1+\sqrt{3}=2.7321\ldots$ in the domain of plane geometric spanners. In the same domain, we show that for every integer $n\geq6$, there exists a an $n$-element point set $S$ such that the degree 4 dilation of $S$ denoted by $\delta_0(S,4) \text{ equals } 1 + \sqrt{(5-\sqrt{5})/2}=2.1755\ldots$ The previous best lower bound of $1.4161$ holds for any degree. (III) For every integer $n\geq6$, there exists an $n$-element point set $S$ such that the stretch factor of the greedy triangulation of $S$ is at least $2.0268$.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2 table

Algorithmic and Combinatorial Results on Fence Patrolling, Polygon Cutting and Geometric Spanners

The purpose of this dissertation is to study problems that lie at the intersection of geometry and computer science. We have studied and obtained several results from three different areas, namely–geometric spanners, polygon cutting, and fence patrolling. Specifically, we have designed and analyzed algorithms along with various combinatorial results in these three areas. For geometric spanners, we have obtained combinatorial results regarding lower bounds on worst case dilation of plane spanners. We also have studied low degree plane lattice spanners, both square and hexagonal, of low dilation. Next, for polygon cutting, we have designed and analyzed algorithms for cutting out polygon collections drawn on a piece of planar material using the three geometric models of saw, namely, line, ray and segment cuts. For fence patrolling, we have designed several strategies for robots patrolling both open and closed fences

Classification of empty 4-simplices and other lattice polytopes

RESUMEN: Un d-politopo es la envolvente convexa de un conjunto finito de puntos en R^d. En particular, si un d-politopo está generado por exactamente d + 1 puntos se dice que es un símplice o un d-símplice. Además, si tomamos los puntos con coordenadas enteras, se dice que el politopo es reticular. A lo largo de esta tesis doctoral se estudian los politopos reticulares y, más concretamente, se estudian dos tipos de estos que son los politopos reticulares vacíos (cuyos únicos puntos reticulares son los vértices) y los politopos reticulares huecos, politopos reticulares que no poseen puntos reticulares en su interior relativo, es decir, todos sus puntos reticulares se encuentran en la frontera. Los politopos huecos, también vacíos, aparecen como el ejemplo más sencillo de politopos reticulares al no tener puntos enteros en el interior de su envolvente convexa. El principal resultado de la tesis doctoral es la clasificación de símplices vacíos en dimensión 4. Mientras los casos en dimensión 1 y 2 son triviales y el caso de dimensión 3 estaba concluido desde 1964 con el trabajo de White [Whi64], con este trabajo se completa esta clasificación en dimensión 4. Artículos como el de Mori, Morrison y Morrison [MMM88] en 1988 consiguen describir algunas familias de 4-símplices vacíos de volumen primo en términos de quíntuplas. Otros trabajos como el de Haase y Ziegler [HZ00] en el 2000, obtienen resultados parciales de esta clasificación. En particular, en ese trabajo se conjeturó una lista completa de 4-símplices vacíos con anchura mayor que dos, la cual se prueba completa en esta tesis. Empleando técnicas de geometría convexa, geometría de números y resultados previos sobre la relación entre la anchura de un politopo y su volumen, somos capaces de establecer unas cotas superiores para los 4-símplices vacíos que deseamos clasificar. Con estas cotas para el volumen de los símplices y una gran cantidad de computación de estos politopos reticulares en dimensión 4 somos capaces de completar la clasificación, explicando el método general utilizado para describir las familias de símplices vacíos que aparecen en la clasificación.ABSTRACT: A d-polytope is the convex hull of a finite set of points in R^d. In particular, if a d-polytope is generated by exactly d + 1 points, it is said to be a simplex or a d-simplex. In addition, if we take the points with integer coordinates, the polytope is a lattice polytope. Throughout this thesis, lattice polytopes are studied and, more specifically, two types of these, which are empty lattice polytopes (whose only integer points are its vertices) and hollow polytopes, lattice polytopes that do not have integer points in their interior, that is, all their integer points are in their facets. Hollow polytopes, also empty, appear as the simplest example of lattice polytopes because they have no integer points inside their convex hull. The main result of the thesis is the classification of empty simplices in dimension 4. While cases in dimension 1 and 2 are trivial and the case of dimension 3 has been completed since 1964 with the work of White [Whi64], this work completes this classification in dimension 4. Papers such as Mori, Morrison and Morrison [MMM88] in 1988 manage to describe some families of empty 4-simplices of prime volume in terms of quintuples. Other works, such as Haase and Ziegler [HZ00] in 2000, obtain partial results tor this classification. In particular, this work conjecture a complete list of empty 4-simplices of width greater than two, which is verified in this thesis. With convex geometry tools, geometry of numbers and previous results that rely on the relationship between the width of a polytope and its volume, we are able to to set upper bounds for the volume of hollow 4-simpolices, that we want to classify. With these upper bounds for the volume of the simplices and a lot of computation of these lattice polytopes in dimension 4 we are able to complete the classification, explaining the general method used to describe the families of empty simplices that appear in the classification.This thesis has been developed under the following scholarships and project grants: MTM2014-54207-P, MTM2017-83750-P and BES-2015-073128 of the Spanish Ministry of Economy and Competitiveness

Higher secondary polytopes and regular plabic graphs

Given a configuration $A$ of $n$ points in $\mathbb{R}^{d-1}$, we introduce the higher secondary polytopes $\Sigma_{A,1},\dots, \Sigma_{A,n-d}$, which have the property that $\Sigma_{A,1}$ agrees with the secondary polytope of Gelfand--Kapranov--Zelevinsky, while the Minkowski sum of these polytopes agrees with Billera--Sturmfels' fiber zonotope associated with (a lift of) $A$. In a special case when $d=3$, we refer to our polytopes as higher associahedra. They turn out to be related to the theory of total positivity, specifically, to certain combinatorial objects called plabic graphs, introduced by the second author in his study of the totally positive Grassmannian. We define a subclass of regular plabic graphs and show that they correspond to the vertices of the higher associahedron $\Sigma_{A,k}$, while square moves connecting them correspond to the edges of $\Sigma_{A,k}$. Finally we connect our polytopes to soliton graphs, the contour plots of soliton solutions to the KP equation, which were recently studied by Kodama and the third author. In particular, we confirm their conjecture that when the higher times evolve, soliton graphs change according to the moves for plabic graphs