Computational and Theoretical Issues of Multiparameter Persistent Homology for Data Analysis

The basic goal of topological data analysis is to apply topology-based descriptors to understand and describe the shape of data. In this context, homology is one of the most relevant topological descriptors, well-appreciated for its discrete nature, computability and dimension independence. A further development is provided by persistent homology, which allows to track homological features along a oneparameter increasing sequence of spaces. Multiparameter persistent homology, also called multipersistent homology, is an extension of the theory of persistent homology motivated by the need of analyzing data naturally described by several parameters, such as vector-valued functions. Multipersistent homology presents several issues in terms of feasibility of computations over real-sized data and theoretical challenges in the evaluation of possible descriptors. The focus of this thesis is in the interplay between persistent homology theory and discrete Morse Theory. Discrete Morse theory provides methods for reducing the computational cost of homology and persistent homology by considering the discrete Morse complex generated by the discrete Morse gradient in place of the original complex. The work of this thesis addresses the problem of computing multipersistent homology, to make such tool usable in real application domains. This requires both computational optimizations towards the applications to real-world data, and theoretical insights for finding and interpreting suitable descriptors. Our computational contribution consists in proposing a new Morse-inspired and fully discrete preprocessing algorithm. We show the feasibility of our preprocessing over real datasets, and evaluate the impact of the proposed algorithm as a preprocessing for computing multipersistent homology. A theoretical contribution of this thesis consists in proposing a new notion of optimality for such a preprocessing in the multiparameter context. We show that the proposed notion generalizes an already known optimality notion from the one-parameter case. Under this definition, we show that the algorithm we propose as a preprocessing is optimal in low dimensional domains. In the last part of the thesis, we consider preliminary applications of the proposed algorithm in the context of topology-based multivariate visualization by tracking critical features generated by a discrete gradient field compatible with the multiple scalar fields under study. We discuss (dis)similarities of such critical features with the state-of-the-art techniques in topology-based multivariate data visualization

Exploring 3D Shapes through Real Functions

This thesis lays in the context of research on representation, modelling and coding knowledge related to digital shapes, where by shape it is meant any individual object having a visual appareance which exists in some two-, three- or higher dimensional space. Digital shapes are digital representations of either physically existing or virtual objects that can be processed by computer applications. While the technological advances in terms of hardware and software have made available plenty of tools for using and interacting with the geometry of shapes, to manipulate and retrieve huge amount of data it is necessary to define methods able to effectively code them. In this thesis a conceptual model is proposed which represents a given 3D object through the coding of its salient features and defines an abstraction of the object, discarding irrelevant details. The approach is based on the shape descriptors defined with respect to real functions, which provide a very useful shape abstraction method for the analysis and structuring of the information contained in the discrete shape model. A distinctive feature of these shape descriptors is their capability of combining topological and geometrical information properties of the shape, giving an abstraction of the main shape features. To fully develop this conceptual model, both theoretical and computational aspects have been considered, related to the definition and the extension of the different shape descriptors to the computational domain. Main emphasis is devoted to the application of these shape descriptors in computational settings; to this aim we display a number of application domains that span from shape retrieval, to shape classification and to best view selection.Questa tesi si colloca nell\u27ambito di ricerca riguardante la rappresentazione, la modellazione e la codifica della conoscenza connessa a forme digitali, dove per forma si intende l\u27aspetto visuale di ogni oggetto che esiste in due, tre o pi? dimensioni. Le forme digitali sono rappresentazioni di oggetti sia reali che virtuali, che possono essere manipolate da un calcolatore. Lo sviluppo tecnologico degli ultimi anni in materia di hardware e software ha messo a disposizione una grande quantit? di strumenti per acquisire, rappresentare e processare la geometria degli oggetti; tuttavia per gestire questa grande mole di dati ? necessario sviluppare metodi in grado di fornirne una codifica efficiente. In questa tesi si propone un modello concettuale che descrive un oggetto 3D attraverso la codifica delle caratteristiche salienti e ne definisce una bozza ad alto livello, tralasciando dettagli irrilevanti. Alla base di questo approccio ? l\u27utilizzo di descrittori basati su funzioni reali in quanto forniscono un\u27astrazione della forma molto utile per analizzare e strutturare l\u27informazione contenuta nel modello discreto della forma. Una peculiarit? di tali descrittori di forma ? la capacit? di combinare propriet? topologiche e geometriche consentendo di astrarne le principali caratteristiche. Per sviluppare questo modello concettuale, ? stato necessario considerare gli aspetti sia teorici che computazionali relativi alla definizione e all\u27estensione in ambito discreto di vari descrittori di forma. Particolare attenzione ? stata rivolta all\u27applicazione dei descrittori studiati in ambito computazionale; a questo scopo sono stati considerati numerosi contesti applicativi, che variano dal riconoscimento alla classificazione di forme, all\u27individuazione della posizione pi? significativa di un oggetto

Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications

We study a class of Legendrian surfaces in contact five-folds by encoding their wavefronts via planar combinatorial structures. We refer to these surfaces as Legendrian weaves, and to the combinatorial objects as N-graphs. First, we develop a diagrammatic calculus which encodes contact geometric operations on Legendrian surfaces as multi-colored planar combinatorics. Second, we present an algebraic-geometric characterization for the moduli space of microlocal constructible sheaves associated to these Legendrian surfaces. Then we use these N-graphs and the flag moduli description of these Legendrian invariants for several new applications to contact and symplectic topology. Applications include showing that any finite group can be realized as a subfactor of a 3-dimensional Lagrangian concordance monoid for a Legendrian surface in the 1-jet space of the two-sphere, a new construction of infinitely many exact Lagrangian fillings for Legendrian links in the standard contact three-sphere, and performing rational point counts over finite fields that distinguish Legendrian surfaces in the standard five-dimensional Darboux chart. In addition, the manuscript develops the notion of Legendrian mutation, studying microlocal monodromies and their transformations. The appendix illustrates the connection between our N-graph calculus for Lagrangian cobordisms and Elias-Khovanov-Williamson's Soergel Calculus.Comment: 114 Pages, 105 Figure

Topological Data Analysis of High-dimensional Correlation Structures with Applications in Epigenetics

This thesis comprises a comprehensive study of the correlation of highdimensional datasets from a topological perspective. Derived from a lack of efficient algorithms of big data analysis and motivated by the importance of finding a structure of correlations in genomics, we have developed two analytical tools inspired by the topological data analysis approach that describe and predict the behavior of the correlated design. Those models allowed us to study epigenetic interactions from a local and global perspective, taking into account the different levels of complexity. We applied graph-theoretic and algebraic topology principles to quantify structural patterns on local correlation networks and, based on them, we proposed a network model that was able to predict the locally high correlations of DNA methylation data. This model provided with an efficient tool to measure the evolution of the correlation with the aging process. Furthermore, we developed a powerful computational algorithm to analyze the correlation structure globally that was able to detect differentiated methylation patterns over sample groups. This methodology aimed to serve as a diagnostic tool, as it provides with selected epigenetic biomarkers associated with a specific phenotype of interest. Overall, this work establishes a novel perspective of analysis and modulation of hidden correlation structures, specifically those of great dimension and complexity, contributing to the understanding of the epigenetic processes, and that is designed to be useful for non-biological fields too