Convex Polytopes: Extremal Constructions and f-Vector Shapes

These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of f-vectors, and extremal constructions. The first lecture treats 3-dimensional polytopes; it includes a complete proof of the Koebe--Andreev--Thurston theorem, using the variational principle by Bobenko & Springborn (2004). In Lecture 2 we look at f-vector shapes of very high-dimensional polytopes. The third lecture explains a surprisingly simple construction for 2-simple 2-simplicial 4-polytopes, which have symmetric f-vectors. Lecture 4 sketches the geometry of the cone of f-vectors for 4-polytopes, and thus identifies the existence/construction of 4-polytopes of high ``fatness'' as a key problem. In this direction, the last lecture presents a very recent construction of ``projected products of polygons,'' whose fatness reaches 9-\eps.Comment: 73 pages, large file. Lecture Notes for PCMI Summer Course, Park City, Utah, 2004; revised and slightly updated final version, December 200

Affine Buildings and Tropical Convexity

The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational perspective. Our results include a convex hull algorithm for the Bruhat--Tits building of SL$_d(K)$ and techniques for computing with apartments and membranes. While the original inspiration was the work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel and Tevelev in algebraic geometry, our tropical algorithms will also be applicable to problems in other fields of mathematics.Comment: 22 pages, 4 figure

Computational Geometric and Algebraic Topology

Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity. At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together, to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field

Discrete Geometry

[no abstract available

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Evaluating Feynman Integrals Using D-modules and Tropical Geometry

Feynman integrals play a central role in the modern scattering amplitudes research program. Advancing our methods for evaluating Feynman integrals will, therefore, strengthen our ability to compare theoretical predictions with data from particle accelerators such as the Large Hadron Collider. Motivated by this, the present manuscript purports to study mathematical concepts related to Feynman integrals. In particular, we present both numerical and analytical algorithms for the evaluation of Feynman integrals. The content is divided into three parts. Part I focuses on the method of DEQs for evaluating Feynman integrals. An otherwise daunting integral expression is thereby traded for the comparatively simpler task of solving a system of DEQs. We use this technique to evaluate a family of two-loop Feynman integrals of relevance for dark matter detection. Part II situates the study of DEQs for Feynman integrals within the framework of D-modules, a natural language for studying PDEs algebraically. Special emphasis is put on a particular D-module called the GKZ system, a set of higher-order PDEs that annihilate a generalized version of a Feynman integral. In the course of matching the generalized integral to a Feynman integral proper, we discover an algorithm for evaluating the latter in terms of logarithmic series. Part III develops a numerical integration algorithm. It combines Monte Carlo sampling with tropical geometry, a particular offspring of algebraic geometry that studies "piecewise-linear" polynomials. Feynman's i*epsilon-prescription is incorporated into the algorithm via contour deformation. We present an open-source program named Feyntrop that implements this algorithm, and use it to numerically evaluate Feynman integrals between 1-5 loops and 0-5 legs in physical regions of phase space.Comment: Ph.D. thesis. Defended on the 11th of December 202

Combinatorial aspects of total positivity

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 115-119).In this thesis I study combinatorial aspects of an emerging field known as total positivity. The classical theory of total positivity concerns matrices in which all minors are nonnegative. While this theory was pioneered by Gantmacher, Krein, and Schoenberg in the 1930s, the past decade has seen a flurry of research in this area initiated by Lusztig. Motivated by surprising positivity properties of his canonical bases for quantum groups, Lusztig extended the theory of total positivity to arbitrary reductive groups and real flag varieties. In the first part of my thesis I study the totally non-negative part of the Grassmannian and prove an enumeration theorem for a natural cell decomposition of it. This result leads to a new q-analog of the Eulerian numbers, which interpolates between the binomial coefficients, the Eulerian numbers, and the Narayana numbers. In the second part of my thesis I introduce the totally positive part of a tropical variety, and study this object in the case of the Grassmannian. I conjecture a tight relation between positive tropical varieties and the cluster algebras of Fomin and Zelevinsky, proving the conjecture in the case of the Grassmannian. The third and fourth parts of my thesis explore a notion of total positivity for oriented matroids. Namely, I introduce the positive Bergman complex of an oriented matroid, which is a matroidal analogue of a positive tropical variety. I prove that this object is homeomorphic to a ball, and relate it to the Las Vergnas face lattice of an oriented matroid. When the matroid is the matroid of a Coxeter arrangement, I relate the positive Bergman complex and the Bergman complex to the corresponding graph associahedron and the nested set complex.by Lauren Kiyomi Williams.Ph.D

Advances in Discrete Differential Geometry

Differential Geometr