FG Diskrete Mathematik / Geometrie

16 Items

Recent Submissions
Miquel dynamics, Clifford lattices and the Dimer model

Affolter, Niklas C. (2021-05-02)

Miquel dynamics was introduced by Ramassamy as a discrete time evolution of square grid circle patterns on the torus. In each time step every second circle in the pattern is replaced with a new one by employing Miquel’s six circle theorem. Inspired by this dynamics we consider the local Miquel move, which changes the combinatorics and geometry of a circle pattern. We prove that the circle cente...

Tropical bisectors and diameters of simplicial complexes

Criado Gallart, Francisco (2021)

Tropical geometry is a discrete analogue of algebraic geometry where addition is replaced by taking the minimum (or maximum) and multiplication is replaced by addition. It is an active research eld where tropical analogues of geometric concepts, like linear spaces, varieties, polytopes or volume, are in different stages of development. The interest of these analogues is twofold: their combinat...

Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

Petrera, Matteo ; Vermeeren, Mats (2020-11-09)

We investigate the relation between pluri-Lagrangian hierarchies of 2-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings in Petrera and Suris (Nonlinear Math. Phys. 24(suppl. 1):121–145, 2017) for ordinary differential equations. We consider hierarchies of 2-dimensional Lagrangia...

Triangulations, discriminants, and Teichmüller theory

Löwe, Robert (2021)

A famous construction of Gel’fand, Kapranov and Zelevinsky associates to each finite point configuration A its secondary fan. This polyhedral fan stratifies the space of height functions by means of the induced regular subdivisions of A. A key result says that the secondary fan arises as the normal fan of a convex polytope, the secondary polytope of A. We apply the theory of regular triangulati...

Randomized construction of complexes with large diameter

Criado, Francisco ; Newman, Andrew (2020-09-23)

We consider the question of the largest possible combinatorial diameter among pure dimensional and strongly connected (d-1)-dimensional simplicial complexes on n vertices, denoted H_s(n, d). Using a probabilistic construction we give a new lower bound on H_s(n, d) that is within an O(d^2) factor of the upper bound. This improves on the previously best known lower bound which was within a factor...

Packings, sausages and catastrophes

Henk, Martin ; Wills, Jörg M. (2020-05-30)

In this survey we give an overview about some of the main results on parametric densities, a concept which unifies the theory of finite (free) packings and the classical theory of infinite packings.

Forbidden patterns in tropical plane curves

Joswig, Michael ; Tewari, Ayush Kumar (2020-08-19)

Tropical curves in R^2 correspond to metric planar graphs but not all planar graphs arise in this way. We describe several new classes of graphs which cannot occur. For instance, this yields a full combinatorial characterization of the tropically planar graphs of genus at most five.

Realizability of tropical plane curves and tropical incidence geometry

Tewari, Ayush Kumar (2021)

Tropical plane curves are one of the building blocks in the study of tropical algebraic geometry. A lot of work has been done to understand and establish connections between tropical and classical algebraic geometry. The first step in this direction is to consider the case of smooth tropical curves. This in turn comes with a nice connection to lattice polytopes and their unimodular triangulatio...

Convex geometry of numbers: covering, successive minima and Banach-Mazur distance

Xue, Fei (2019)

This thesis addresses several classical problems in convex geometry of numbers, including the lattice point covering problem, successive-minima-type inequalities and the Banach-Mazur distance of convex bodies. In the first chapter we will introduce basic concepts, definitions and results which provide the background for the problems in this thesis. Other concepts which are more specific or lim...

Subspace concentration of geometric measures

Pollehn, Hannes (2019)

In this work we study geometric measures in two different extensions of the Brunn-Minkowski theory. The first part of this thesis is concerned with problems in Lp Brunn-Minkowski theory, that is based on the concept of p-addition of convex bodies, which was first introduced by Firey for p = 1 and later considered for all real p by Lutwak et al. The interplay of the volume and other functionals ...

Tropicalization of del Pezzo surfaces

Ren, Qingchun ; Shaw, Kristin ; Sturmfels, Bernd (2016-03-30)

We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (-1)-curves. On these tropical surfaces, the boundary divisors are represented by trees at infinity. These trees are glued together according to the Petersen, Clebsch and Schläfli graphs, respectively. There are 27 trees on each tropical ...

Matroidal subdivisions, Dressians and tropical Grassmannians

Schröter, Benjamin (2018)

In dieser Arbeit untersuchen wir verschiedene Aspekte von tropischen linearen Räumen und deren Modulräumen, den tropischen Grassmannschen und Dressschen. Tropische lineare Räume sind dual zu Matroidunterteilungen. Motiviert durch das Konzept der Splits, dem einfachsten Fall einer polytopalen Unterteilung, wird eine neue Klasse von Matroiden eingeführt, die mit Techniken der polyedrischen Geomet...

Algebraic statistics of Gaussian mixtures

Améndola Cerón, Carlos Enrique (2017)

In this work we study the statistical models known as Gaussian mixtures from an algebraic point of view. First, we illustrate how algebraic techniques can be useful to address funda- mental questions on the shape of Gaussian mixture densities, namely the problem of determining the maximum number of modes a mixture of Gaussians can have, depending on the number of components and the dimensi...

Computer vision and computer algebra

Wagner, André (2017)

In Multiview-Geometrie, einem Teilgebiet der Computer-Vision, werden Bilder mit Kameras von 3D-Objekten aus verschiedenen Perspektiven aufgenommen. Diese Arbeit studiert Multiview-Geometrie mit Methoden aus der Computer-Algebra. Hauptprobleme des Maschinellen Sehens sind die Objektrekonstruktion und die Bestimmung der Kameraparameter. In diesen beiden fundamentalen Fragestellungen erzielen wir ...

Combinatorics of tropical linear programming

Loho, Georg (2017)

The main topic of this work is the study of systems of inequalities where only the operations 'min' and '+' are used and which we call tropical linear inequality systems. An important algorithmic problem is to determine a point which fulfills all the inequalities. This problem, referred as the tropical feasibility problem, occurs in connection with scheduling as well as mean payoff games. It is...

DGD gallery: storage, sharing, and publication of digital research data

Joswig, Michael ; Mehner, Milan ; Sechelmann, Stefan ; Techter, Jan ; Bobenko, Alexander I. (2016)

We describe a project, called the DGD Gallery, whose goal is to store geometric data and to make it publicly available. The DGD Gallery offers an online web service for the storage, sharing, and publication of digital research data.