We generalize Forman’s discrete Morse theory, on one end by developing a discrete analogue of Morse-Bott theory for CW complexes, motivated by Morse-Bott theory in the smooth setting. On the other, motivated by J-N. Corvellec’s Morse theory for continuous functionals, we generalize Forman’s discrete Morse-floer theory by considering a vector field more general than the one extracted from a discrete Morse function, and defining a boundary operator from which the Betti numbers of the CW complex are obtained. We also do some Conley theory analysis.

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

In 1970, Raoul Bott published The Periodicity Theorem for the Classical Groups and Some of Its Applications, in which he uses this famous result as a guideline to present some important areas and tools of Algebraic Topology. This dissertation aims to use the path Bott presented in his article as a guideline to address certain topics on Algebraic Topology. We start this incursion developing important tools used in Homotopy Theory such as spectral sequences and Eilenberg-MacLane spaces, exploring how they can be combined to aid in computation of homotopy groups. We then study important results of Morse Theory, a tool which was in the centre of Botts proof of the Periodicity Theorem. We also develop two extensions: Morse-Bott Theory, and the applications of such results to the loopspace of a manifold. We end by giving an introduction to generalised cohomology theories and K-Theory. / Em 1970, Raoul Bott publicou o artigo The Periodicity Theorem for the Classical Groups and Some of Its Applications no qual usava esse famoso resultado como um guia para apresentar importantes áreas e ferramentas da Topologia Algébrica. O presente trabalho usa o mesmo caminho traçado por Bott em seu artigo como roteiro para explorar tópicos importantes da Topologia Algébrica. Começamos esta incursão desenvolvendo ferramentas importantes da Teoria de Homotopia como sequências espectrais e espaços de Eilenberg-MacLane, explorando como estes podem ser combinados para auxiliar em cálculos de grupos de homotopia. Passamos então a estudar resultados importantes de Teoria de Morse, uma ferramenta que estava no centro da demonstração de Bott do Teorema da Periodicidade. Desenvolvemos ainda, duas extensões: Teoria de Morse-Bott e aplicações destes resultados ao espaço de laços de uma variedade. Terminamos com uma introdução a teorias de cohomologia generalizadas e K-Teoria.

Let G be a finite graph with vertex set V and edge set E. A graph complex on G is an abstract simplicial complex consisting of subsets of E. In particular, we may interpret such a complex as a family of subgraphs of G. The subject of this thesis is the topology of graph complexes, the emphasis being placed on homology, homotopy type, connectivity degree, Cohen-Macaulayness, and Euler characteristic. We are particularly interested in the case that G is the complete graph on V. Monotone graph properties are complexes on such a graph satisfying the additional condition that they are invariant under permutations of V. Some well-studied monotone graph properties that we discuss in this thesis are complexes of matchings, forests, bipartite graphs, disconnected graphs, and not 2-connected graphs. We present new results about several other monotone graph properties, including complexes of not 3-connected graphs and graphs not coverable by p vertices. Imagining the vertices as the corners of a regular polygon, we obtain another important class consisting of those graph complexes that are invariant under the natural action of the dihedral group on this polygon. The most famous example is the associahedron, whose faces are graphs without crossings inside the polygon. Restricting to matchings, forests, or bipartite graphs, we obtain other interesting complexes of noncrossing graphs. We also examine a certain "dihedral" variant of connectivity. The third class to be examined is the class of digraph complexes. Some well-studied examples are complexes of acyclic digraphs and not strongly connected digraphs. We present new results about a few other digraph complexes, including complexes of graded digraphs and non-spanning digraphs. Many of our proofs are based on Robin Forman's discrete version of Morse theory. As a byproduct, this thesis provides a loosely defined toolbox for attacking problems in topological combinatorics via discrete Morse theory. In terms of simplicity and power, arguably the most efficient tool is Forman's divide and conquer approach via decision trees, which we successfully apply to a large number of graph and digraph complexes. / QC 20100622

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

[pt] A geometria diferencial descreve de maneira intuitiva os objetos suaves no espaço. Porém, com a evolução da modelagem geométrica por computador, essa ferramenta se tornou ao mesmo tempo necessária e difícil de se descrever no mundo discreto. A teoria de Morse ficou importante pela ligação que ela cria entre a topologia e a geometria diferenciais. Partindo de um ponto de vista mais combinatório, a teoria de Morse discreta de Forman liga de forma rigorosa os objetos discretos à topologia deles, abrindo essa teoria para estruturas discretas. Este trabalho propõe uma definição construtiva de funções de Morse geométricas no mundo discreto e do complexo de Morse-Smale correspondente, onde a geometria é definida como a amostragem de uma função suave nos vértices da estrutura discreta. Essa construção precisa de cálculos de homologia que se tornaram por si só uma melhoria significativa dos métodos existentes. A decomposição de Morse- Smale resultante pode ser eficientemente computada e usada para aplicações de cálculo da persistência, geração de grafos de Reeb, remoção de ruído e mais. . . / [en] Differential geometry provides an intuitive way of understanding smooth objects in the space. However, with the evolution of geometric modeling by computer, this tool became both necessary and difficult to transpose to the discrete setting. The power of Morse theory relies on the link it created between differential topology and geometry. Starting from a combinatorial point of view, Forman´s discrete Morse theory relates rigorously discrete objects to their topology, opening Morse theory to discrete structures. This work proposes a constructive definition of geometric discrete Morse functions and their corresponding discrete Morse-Smale complexes, where the geometry is defined as a smooth function sampled on the vertices of the discrete structure. This construction required some homology computations that turned out to be a significant improvement over existing methods by itself. The resulting Morse-Smale decomposition can then be efficiently computed, and used for applications to persistence computation, Reeb graph generation, noise removal. . .

In 1970, Raoul Bott published The Periodicity Theorem for the Classical Groups and Some of Its Applications, in which he uses this famous result as a guideline to present some important areas and tools of Algebraic Topology. This dissertation aims to use the path Bott presented in his article as a guideline to address certain topics on Algebraic Topology. We start this incursion developing important tools used in Homotopy Theory such as spectral sequences and Eilenberg-MacLane spaces, exploring how they can be combined to aid in computation of homotopy groups. We then study important results of Morse Theory, a tool which was in the centre of Botts proof of the Periodicity Theorem. We also develop two extensions: Morse-Bott Theory, and the applications of such results to the loopspace of a manifold. We end by giving an introduction to generalised cohomology theories and K-Theory. / Em 1970, Raoul Bott publicou o artigo The Periodicity Theorem for the Classical Groups and Some of Its Applications no qual usava esse famoso resultado como um guia para apresentar importantes áreas e ferramentas da Topologia Algébrica. O presente trabalho usa o mesmo caminho traçado por Bott em seu artigo como roteiro para explorar tópicos importantes da Topologia Algébrica. Começamos esta incursão desenvolvendo ferramentas importantes da Teoria de Homotopia como sequências espectrais e espaços de Eilenberg-MacLane, explorando como estes podem ser combinados para auxiliar em cálculos de grupos de homotopia. Passamos então a estudar resultados importantes de Teoria de Morse, uma ferramenta que estava no centro da demonstração de Bott do Teorema da Periodicidade. Desenvolvemos ainda, duas extensões: Teoria de Morse-Bott e aplicações destes resultados ao espaço de laços de uma variedade. Terminamos com uma introdução a teorias de cohomologia generalizadas e K-Teoria.

Orientador: Djairo Guedes de Figueiredo / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T23:11:17Z (GMT). No. of bitstreams: 1 Monteiro_Evandro_D.pdf: 681089 bytes, checksum: 5ec4729a2d7b386329193adf424f6b42 (MD5) Previous issue date: 2011 / Resumo: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: The complete abstract is available with the full electronic digital thesis or dissertations / Doutorado / Matematica / Doutor em Matemática

In recent decades, scientific data has become available in increasing sizes and precision. Therefore techniques to analyze and summarize the ever increasing datasets are of vital importance. A common form of scientific data, resulting from simulations as well as observational sciences, is in the form of scalar-valued function on domains of interest. The Morse-Smale complex is a topological data-structure used to analyze and summarize the gradient behavior of such scalar functions. This thesis deals with efficient parallel algorithms to compute the Morse-Smale complex as well as its application to datasets arising from cosmological sciences as well as structural biology. The first part of the thesis discusses the contributions towards efficient computation of the Morse-Smale complex of scalar functions de ned on two and three dimensional datasets. In two dimensions, parallel computation is made possible via a paralleizable discrete gradient computation algorithm. This algorithm is extended to work e ciently in three dimensions also. We also describe e cient algorithms that synergistically leverage modern GPUs and multi-core CPUs to traverse the gradient field needed for determining the structure and geometry of the Morse-Smale complex. We conclude this part with theoretical contributions pertaining to Morse-Smale complex simplification. The second part of the thesis explores two applications of the Morse-Smale complex. The first is an application of the 3-dimensional hierarchical Morse-Smale complex to interactively explore the filamentary structure of the cosmic web. The second is an application of the Morse-Smale complex for analysis of shapes of molecular surfaces. Here, we employ the Morse-Smale complex to determine alignments between the surfaces of molecules having similar surface architecture.

L'obtention de ce mémoire a été rendue possible par le soutien financier du FRQNT et du CRSNG. / Ce mémoire est une étude détaillée de certains aspects de la théorie de Morse et des complexes de chaînes qui en découlent : le complexe de Morse, le complexe de Milnor et le complexe de Barraud-Cornea. À l’aide de différentes techniques de la topologie différentielle et de la théorie de Morse, dont les bases forment les premiers chapitres de ce texte, nous ferons la construction détaillée de ces trois complexes avant de démontrer leurs équivalences deux à deux. Ce mémoire synthétise et met en parallèle trois branches de la théorie de Morse en ne supposant que des connaissances du niveau d’un étudiant de début maîtrise. / In this thesis, we study aspects of Morse theory and the chain complexes that derive from it : the Morse complex, the Milnor complex and the Barraud-Cornea complex. Using different techniques from differential topology and Morse theory, which will be presented in the first chapters, we carefully build these complexes before proving their equivalence. This thesis synthesises and compares three points of view in Morse theory in a document accessible to beginning graduate students.