Discrete Geometry

The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions

Tube algebras, excitations statistics and compactification in gauge models of topological phases

We consider lattice Hamiltonian realizations of ($d$+1)-dimensional Dijkgraaf-Witten theory. In (2+1)d, it is well-known that the Hamiltonian yields point-like excitations classified by irreducible representations of the twisted quantum double. This can be confirmed using a tube algebra approach. In this paper, we propose a generalization of this strategy that is valid in any dimensions. We then apply the tube algebra approach to derive the algebraic structure of loop-like excitations in (3+1)d, namely the twisted quantum triple. The irreducible representations of the twisted quantum triple algebra correspond to the simple loop-like excitations of the model. Similarly to its (2+1)d counterpart, the twisted quantum triple comes equipped with a compatible comultiplication map and an $R$-matrix that encode the fusion and the braiding statistics of the loop-like excitations, respectively. Moreover, we explain using the language of loop-groupoids how a model defined on a manifold that is $n$-times compactified can be expressed in terms of another model in $n$-lower dimensions. This can in turn be used to recast higher-dimensional tube algebras in terms of lower dimensional analogues.Comment: 71 page

Nonperturbative Quantum Gravity

Asymptotic safety describes a scenario in which general relativity can be quantized as a conventional field theory, despite being nonrenormalizable when expanding it around a fixed background geometry. It is formulated in the framework of the Wilsonian renormalization group and relies crucially on the existence of an ultraviolet fixed point, for which evidence has been found using renormalization group equations in the continuum. "Causal Dynamical Triangulations" (CDT) is a concrete research program to obtain a nonperturbative quantum field theory of gravity via a lattice regularization, and represented as a sum over spacetime histories. In the Wilsonian spirit one can use this formulation to try to locate fixed points of the lattice theory and thereby provide independent, nonperturbative evidence for the existence of a UV fixed point. We describe the formalism of CDT, its phase diagram, possible fixed points and the "quantum geometries" which emerge in the different phases. We also argue that the formalism may be able to describe a more general class of Ho\v{r}ava-Lifshitz gravitational models.Comment: Review, 146 pages, many figure

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Dilation, Transport, Visibility and Fault-Tolerant Algorithms

Connecting some points in the plane by a road network is equivalent to constructing a finite planar graph G whose vertex set contains a predefined set of vertices (i. e., the possible destinations in the road network). The dilation between two vertices p and q of graph G is defined as the Euclidean length of a shortest path in G from p to q, divided by the Euclidean distance from p to q. That is, given a point set P, the goal is to place some additional crossing vertices C such that there exists a planar graph G = (P ∪ C, E) whose dilation is small. Here, the dilation of G is defined as the maximum dilation between two vertices in G. We show that, except for some special point sets P, there is a lower bound Δ(P) > 1, depending on P, on the dilation of any finite graph containing P in its vertex set. The transportation problem is the problem of finding a transportation plan that minimizes the total transport cost. We are given a set of suppliers, and each supplier produces a fixed amount of some commodity, say, bread. Furthermore, there is a set of customers, and each customer has some demand of bread, such that the total demand equals the amount of bread the suppliers produce. The task is to assign each unit of bread produced to some customer, such that the total transportation cost becomes a minimum. A first idea is to assign each unit of bread to the client to which the transport cost of this unit is minimal. Clearly, this gives rise to a transportation plan which minimizes the total transportation cost. However, it is likely that not every customer will obtain the required amount of bread. Therefore, we need to use a different algorithm for distributing the supplier's bread. We show that if the bread produced by the suppliers is given by a continuous probability density function and the set of customers is discrete, then every optimal transport plan can be characterized by a unique additively weighted Voronoi diagram for the customers. When managing the construction process of a building by a digital model of the building, it is necessary to compute essential parts between walls of the building. Given two walls A and B, the essential part between A and B is the set of line segments s where one endpoint belongs to A, the other endpoint belongs to B, and s does not intersect A or B. We give an algorithm that computes, in linear time, the essential parts between A and B. Our algorithm is based on computing the visibility polygon of A and of B, and two shortest paths connecting points of A with points of B. We conclude the thesis by giving fault-tolerant algorithms for some fundamental geometric problems. We assume that a basic primitive operation used by an algorithm fails with some small probability p. Depending on the results of the primitive operations, it is possible that the algorithm will not work correctly. For example, one faulty comparison when executing a sorting algorithm can result in some numbers being placed far away from their true positions. An algorithm is called tolerant, if with high probability a good answer is given, if the error probability p is small. We provide tolerant algorithms that find the maximum of n numbers, search for a key in a sorted sequence of n keys, sort a set of n numbers, and solve Linear Programming in R2

A computational geometric approach for an ensemble-based topological entropy calculation in two and three dimensions

From the stirring of dye in viscous fluids to the availability of essential nutrients spreading over the surface of a pond, nature is rife with examples of mixing in two-dimensional fluids. The long-time exponential growth rate of a thin filament of dye stretched by the fluid is a well-known proxy for the quality of mixing in two dimensions. This growth rate in turn gives a lower bound on the flow's topological entropy, a measure quantifying the complexity of chaotic dynamics. In the real-world study of mixing, topological entropy may be hard to compute; the velocity field may not be known or may be expensive to recover or approximate, thus limiting our knowledge of the governing system and underlying mechanics driving the mixing. Central to this study are two questions: \emph{How can stretching rates in two-dimensional planar flows best be computed using only trajectory data?}, and \emph{Can a method for computing stretching rates in higher dimensions from only trajectory data be developed?}. In this spirit, we introduce the Ensemble-based Topological Entropy Calculation (E-tec), a method to derive a lower-bound on topological entropy that requires only finite number of system trajectories, like those obtained from ocean drifters, and no detailed knowledge of the velocity field. E-tec is demonstrated to be computationally more efficient than other competing methods in two dimensions that accommodate trajectory data. This is accomplished by considering the evolution of a ``rubber band" wrapped around the data points and evolving with their trajectories. E-tec records the growth of this band as the collective motion of trajectories strike, deform, and stretch it. This exponential growth rate acts as a lower bound on the topological entropy. In this manuscript, I demonstrate convergence of E-tec's approximation with respect to both the number of trajectories (ensemble size) and the duration of trajectories in time. Driving the efficiency of E-tec in two dimensions is the use of computational geometry tools. Not only this, by computing stretching rates in this new computational geometry framework, I extend E-tec to three dimensions using two methods. First, I consider a two-dimensional rubber sheet stretched around a collection of points in a three-dimensional flow. Similar to the band-stretching component of two-dimensional E-tec, a three-dimensional triangulation is used to record the growth of the sheet as it is stretched and deformed by points evolving in time. Second, I calculate the growth rates of one-dimensional rubber strings as they are stretched by the edges of this dynamic, moving triangulation

Homotopy Theory

Algebraic topology in general and homotopy theory in particular is in an exciting period of growth and transformation, driven in part by strong interactions with algebraic geometry, mathematical physics, and representation theory, but also driven by new approaches to our classical problems. This workshop was a forum to present and discuss the latest result and ideas in homotopy theory and the connections to other branches of mathematics. Central themes of the workshop were derived algebraic geometry, homotopical invariants for ring spectra such as topological Hochschild homology, interactions with modular representation theory, group actions on spaces and the closely-related study of the classifying spaces of groups

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop