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

Geometric, Algebraic, and Topological Combinatorics

The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics" was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) Karim Adiprasito presented his very recent proof of the $g$-conjecture for spheres (as a talk and as a "Q\&A" evening session) (2) Federico Ardila gave an overview on "The geometry of matroids", including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz

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

Topological analysis of discrete scalar data

This thesis presents a novel computational framework that allows for a robust extraction and quantification of the Morse-Smale complex of a scalar field given on a 2- or 3- dimensional manifold. The proposed framework is based on Forman\u27s discrete Morse theory, which guarantees the topological consistency of the computed complex. Using a graph theoretical formulation of this theory, we present an algorithmic library that computes the Morse-Smale complex combinatorially with an optimal complexity of $O(n^2)$ and efficiently creates a multi-level representation of it. We explore the discrete nature of this complex, and relate it to the smooth counterpart. It is often necessary to estimate the feature strength of the individual components of the Morse-Smale complex -- the critical points and separatrices. To do so, we propose a novel output-sensitive strategy to compute the persistence of the critical points. We also extend this wellfounded concept to separatrices by introducing a novel measure of feature strength called separatrix persistence. We evaluate the applicability of our methods in a wide variety of application areas ranging from computer graphics to planetary science to computer and electron tomography.In dieser Dissertation präsentieren wir ein neues System zur robusten Berechnung des Morse-Smale Komplexes auf 2- oder 3-dimensionalen Mannigfaltigkeiten. Das vorgestellte System basiert auf Forman’s diskreter Morsetheorie und garantiert damit die topologische Konsistenz des berechneten Komplexes. Basierend auf einer graphentheoretischer Formulierung präesentieren wir eine Bibliothek von Algorithmen, die es erlaubt, den Morse-Smale Komplex mit einer optimalen Kompliztät von $O(n^2)$ kombinatorisch zu berechnen und effizient eine mehrskalige Repräsentation davon erstellt. Wir untersuchen die diskrete Natur dieses Komplexes und vergleichen ihn zu seinem kontinuierlichen Gegenstück. Es ist häufig notwendig, die Merkmalsstärke einzelner Bestandteile des Komplexes -- der kritischen Punkte und Separatrizen -- abzuschätzen. Hierfür stellen wir eine neue outputsensitive Strategie vor, um die Persistenz von kritischen Punkten zu berechen. Wir erweitern dieses fundierte Konzept auf Separatrizen durch die Einführung des Wichtigkeitsmaßes Separatrixpersistenz. Wir evaluieren die Anwendbarkeit unserer Methoden anhand vielfältiger Anwendungen aus den Gebieten der Computergrafik, Planetologie, Computer- und Elektronentomographie