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

Computing the Volume of a Union of Balls: a Certified Algorithm

Balls and spheres are amongst the simplest 3D modeling primitives, and computing the volume of a union of balls is an elementary problem. Although a number of strategies addressing this problem have been investigated in several communities, we are not aware of any robust algorithm, and present the first such algorithm. Our calculation relies on the decomposition of the volume of the union into convex regions, namely the restrictions of the balls to their regions in the power diagram. Theoretically, we establish a formula for the volume of a restriction, based on Gauss' divergence theorem. The proof being constructive, we develop the associated algorithm. On the implementation side, we carefully analyse the predicates and constructions involved in the volume calculation, and present a certified implementation relying on interval arithmetic. The result is certified in the sense that the exact volume belongs to the interval computed using the interval arithmetic. Experimental results are presented on hand-crafted models presenting various difficulties, as well as on the 58,898 models found in the 2009-07-10 release of the Protein Data Bank

Homological mirror symmetry for a Calabi-Yau hypersurface in projective space

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 365-369).This thesis is concerned with Kontsevich's Homological Mirror Symmetry conjecture. In Chapter 1, which is based on [1], we consider the n-dimensional pair of pants, which is defined to be the complement of n + 2 generic hyperplanes in CPn. The pair of pants is conjectured to be mirror to the Landau-Ginzburg model (Cn+2 , W), where W = z1...zn+2 We construct an immersed Lagrangian sphere in the pair of pants, and show that its endomorphism A.. algebra in the Fukaya category is quasi-isomorphic to the endomorphism dg algebra of the structure sheaf of the origin in the mirror,.giving some evidence for the Homological Mirror Symmetry conjecture in this case. In Chapter 2, which is based on [2], we build on these results to prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d =/> 3.by Nicholas Sheridan.Ph.D

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Dynamical systems applied to consciousness and brain rhythms in a neural network

This thesis applies the great advances of modern dynamical systems theory (DST) to consciousness. Consciousness, or subjective experience, is faced here in two different ways: from the global dynamics of the human brain and from the integrated information theory (IIT), one of the currently most prestigious theories on consciousness. Before that, a study of a numerical simulation of a network of individual neurons justifies the use of the Lotka-Volterra model for neurons assemblies in both applications. All these proposals are developed following this scheme: • First, summarizing the structure, methods and goal of the thesis. • Second, introducing a general background in neuroscience and the global dynamics of the human brain to better understand those applications. • Third, conducting a study of a numerically simulated network of neurons. This network, which displays brain rhythms, can be employed, among other objectives, to justify the use of the Lotka-Volterra model for applications. • Fourth, summarizing concepts from the mathematical DST such as the global attractor and its informational structure, in addition to its particularization to a Lotka-Volterra system. • Fifth, introducing the new mathematical concepts of model transform and instantaneous parameters that allow the application of simple mathematical models such as Lotka-Volterra to complex empirical systems as the human brain. • Sixth, using the model transform, and specifically the Lotka-Volterra transform, to calculate global attractors and informational structures in global dynamics of the human brain. • Seventh, knowing the probably most prestigious theory on consciousness, the IIT developed by G. Tononi. • Eighth, using informational structures to develop a continuous version of IIT. And ninth, establishing some final conclusions and commenting on new open questions from this work. These nine points of this scheme correspond to the nine chapters of this thesis