Advances in Discrete Differential Geometry

Differential Geometr

Multi-Black-Hole Configurations as Models for Inhomogeneous Cosmologies

On the largest scales, the Universe is homogeneous and isotropic, whereas on smaller scales, various structures immediately begin to emerge. The transition from an inhomogeneous spacetime to the homogeneous and isotropic Friedmann universe is not sufficiently understood yet. Modern cosmology rests on the hypothesis that the LambdaCDM-model applies and, indeed, this model is very successful. On the other hand, as the precision of observations steadily increases, it is more than likely that inhomogeneities will no longer be negligible in the future. For this reason, the study of inhomogeneous cosmological models is reasonable. In this thesis, we consider the question which Friedmann universe is the best fit to a particular given inhomogeneous spacetime, which is known as the fitting problem. We consider models in which matter is replaced by a discrete configuration of black holes, that is, we concentrate on vacuum solutions to Einstein's equations. Since the full system of the field equations is too complicated to find an exact time-dependent solution for the whole spacetime, we restrict ourselves to approximative models as well as solutions to the initial value problem. In the former case, we reconsider Swiss-cheese and Lindquist-Wheeler models. In both models, the spacetime around a mass is described by the Schwarzschild metric. In the latter case, we determine the spatial metric of a space-like hypersurface. We limit our attention to time-symmetric initial data characterised by the vanishing of the extrinsic curvature. In this case, we are able to find a solution for an arbitrary number of black holes using the conformal method. Clearly, it is not reasonable to assume that every configuration of black holes leads to a spacetime which may be approximated well by a Friedmann solution. Such an approximation should be possible if the masses are distributed somehow uniformly. The aim of this thesis is to clarify this statement and to provide criteria which allow quantitative statements about the degree of uniformity. We determine the parameters of the fitted dust universe, in particular the scale factor. Our considerations are supported by several example configurations. In particular, we provide a new method based on Lie sphere geometry to construct various configurations with a high degree of uniformity in a surprisingly simple fashion. Moreover, we provide a generalisation to an approximative inhomogeneous model given by Lindquist and Wheeler. In this case, it is possible to determine the parameters of the fitted Friedmann universe even if we do not know the exact solution. Under certain conditions, this model becomes similar to a Swiss-cheese model, allowing us to formulate first expectations on the time evolution, which is otherwise mostly disregarded within the framework of this thesis

Advances in Discrete Differential Geometry

Differential Geometr

Some problems in combinatorial topology of flag complexes

In this work we study simplicial complexes associated to graphs and their homotopical and combinatorial properties. The main focus is on the family of flag complexes, which can be viewed as independence complexes and clique complexes of graphs. In the first part we study independence complexes of graphs using two cofibre sequences corresponding to vertex and edge removals. We give applications to the connectivity of independence complexes of chordal graphs and to extremal problems in topology and we answer open questions about the homotopy types of those spaces for particular families of graphs. We also study the independence complex as a space of configurations of particles in the so-called hard-core models on various lattices. We define, and investigate from an algorithmic perspective, a special family of combinatorially defined homology classes in independence complexes. This enables us to give algorithms as well as NP-hardness results for topological properties of some spaces. As a corollary we prove hardness of computing homology of simplicial complexes in general. We also view flag complexes as clique complexes of graphs. That leads to the study of various properties of Vietoris-Rips complexes of graphs. The last result is inspired by a problem in face enumeration. Using methods of extremal graph theory we classify flag triangulations of 3-manifolds with many edges. As a corollary we complete the classification of face vectors of flag simplicial homology 3-spheres

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum