The amount of data that our digital society collects is unprecedented. This represents a valuable opportunity to improve our quality of life by gaining insights about complex problems related to neuroscience, medicine and biology among others. Topological methods, in combination with classical statistical ones, have proven to be a precious resource in understanding and visualizing data. Multidimensional persistence is a method in topological data analysis which allows a multi-parameter analysis of a dataset through an algebraic object called multidimensional persistence module. Multidimensional persistence modules are complicated and contain a lot of information about the input data. This thesis deals with the problem of algorithmically describing multidimensional persistence modules and extracting information that can be used in applications. The information we extract, through invariants, should not only be efficiently computable and informative but also robust to noise. In Paper A we describe in an explicit and algorithmic way multidimensional persistence modules. This is achieved by studying the multifiltration of simplicial complexes defining multidimensional persistence modules. In particular we identify the special structure underlying the modules of n-chains of such multifiltration and exploit it to write multidimensional persistence modules as the homology of a chain complex of free modules. Both the free modules and the homogeneous matrices in such chain complex can be directly read off the multifiltration of simplicial complexes. Paper B deals with identifying stable invariants for multidimensional persistence. We introduce an algebraic notion of noise and use it to compare multidimensional persistence modules. Such definition allows not only to specify the properties of a dataset we want to study but also what should be neglected. By disregarding noise the, so called, persistent features are identified. We also propose a stable discrete invariant which collects properties of persistent features in a multidimensional persistence module. / <p>QC 20150525</p>

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<p>This dissertation places intersection homology and local homology within the framework of persistence, which was originally developed for ordinary homology by Edelsbrunner, Letscher, and Zomorodian. The eventual goal, begun but not completed here, is to provide analytical tools for the study of embedded stratified spaces, as well as for high-dimensional and possibly noisy datasets for which the number of degrees of freedom may vary across the parameter space. Specifically, we create a theory of persistent intersection homology for a filtered stratified space and prove several structural theorems about the pair groups asso- ciated to such a filtration. We prove the correctness of a cubic algorithm which computes these pair groups in a simplicial setting. We also define a series of intersec- tion homology elevation functions for an embedded stratified space and characterize their local maxima in dimension one. In addition, we develop a theory of persistence for a multi-scale analogue of the local homology groups of a stratified space at a point. This takes the form of a series of local homology vineyards which allow one to assess the homological structure within a one-parameter family of neighborhoods of the point. Under the assumption of dense sampling, we prove the correctness of this assessment at a variety of radius scales.</p> / Dissertation

We formulate a novel approach to infer conditional independence models or Markov structure of a multivariate distribution. Specifically, our objective is to place informative prior distributions over graphs (decomposable and unrestricted) and sample efficiently from the induced posterior distribution. We also explore the idea of factorizing according to complete sets of a graph; which implies working with a hypergraph that cannot be retrieved from the graph alone. The key idea we develop in this paper is a parametrization of hypergraphs using the geometry of points in $R^m$. This induces informative priors on graphs from specified priors on finite sets of points. Constructing hypergraphs from finite point sets has been well studied in the fields of computational topology and random geometric graphs. We develop the framework underlying this idea and illustrate its efficacy using simulations. / Dissertation

<p>The work presented in this dissertation includes the study of cohomology and cohomological operations within the framework of Persistence. Although Persistence was originally defined for homology, recent research has developed persistent approaches to other algebraic topology invariants. The work in this document extends the field of persistence to include cohomology classes, cohomology operations and characteristic classes. </p><p>By starting with presenting a combinatorial formula to compute the Stiefel-Whitney homology class, we set up the groundwork for Persistent Characteristic Classes. To discuss persistence for the more general cohomology classes, we construct an algorithm that allows us to find the Poincar'{e} Dual to a homology class. Then, we develop two algorithms that compute persistent cohomology, the general case and one for a specific cohomology class. We follow this with defining and composing an algorithm for extended persistent cohomology. </p><p>In addition, we construct an algorithm for determining when a cohomology class is decomposible and compose it in the context of persistence. Lastly, we provide a proof for a concise formula for the first Steenrod Square of a given cohomology class and then develop an algorithm to determine when a cohomology class is a Steenrod Square of a lower dimensional cohomology class.</p> / Dissertation

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Computing reliable numerical approximations of invariant sets for nonlinear systems is the core problem for computer assisted study of dynamical systems. In the case of conservative systems the problem is complicated by the fact that there is no phase space dissipation to drive orbits onto attractors. In this dissertation we discuss several contributions to the field of computer assisted study of invariant dynamics in conservative systems. / text

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

This thesis presents an algorithm for producing 4–manifold triangulations with boundary an arbitrary orientable, closed, triangulated 3–manifold. The research is an extension of Costantino and Thurston’s work on determining upper bounds on the number of 4–dimensional simplices necessary to construct such a triangulation. Our first step in this bordism construction is the geometric partitioning of an initial 3–manifold M using smooth singularity theory. This partition provides handle attachment sites on the 4–manifold Mx[0,1] and the ensuing handle attachments eliminate one of the boundary components of Mx[0,1], yielding a 4-manifold with boundary exactly M. We first present the construction in the smooth case before extending the smooth singularity theory to triangulated 3–manifolds. / Graduate