This thesis presents a general background on discrete Morse theory, as developed by Robin Forman, as well as an introduction to computability and computational complexity. Since general point-set data equipped with a smooth structure can admit a triangulation, discrete Morse theory finds numerous applications in data analysis which can range from traffic control to geographical interpretation. Currently, there are various methods which convert point-set data to simplicial complexes or piecewise-smooth manifolds; however, this is not the focus of the thesis. Instead, this thesis will show that the Morse homology of such data is computable in the classical sense of Turing decidability, bound the complexity of finding the Morse homology of a given simplicial complex, and provide a measure for when this is more efficient than simplicial homology. / Master of Science / With the growing prevalence of data in the technological world, there is an emerging need to identify geometric properties (such as holes and boundaries) to data sets. However, it is often fruitless to employ an algorithm if it is known to be too computationally expensive (or even worse, not computable in the traditional sense). However, discrete Morse theory was originally formulated to provide a simplified manner of calculating these geometric properties on discrete sets. Therefore, this thesis outlines the general background of Discrete Morse theory and formulates the computational cost of computing specific geometric algorithms from the Discrete Morse perspective.

In this thesis we construct for a given smooth, generic Hamiltonian H on the 2n dimensional torus a chain-isomorphism between the Morse complex of the Hamiltonian action on the free loop space of the torus and the Floer-complex. Though both complexes are generated by the critical points of the Hamiltonian action, their boundary operators differ. Therefore the construction of the isomorphism is based on counting the moduli spaces of hybrid-type solutions which involves stating a new non-Lagrangian boundary value problem for Cauchy-Riemann type operators not yet studied in Floer theory. It is crucial for the statement that the torus is compact, possesses trivial tangent bundle and an additive structure. We finally want to note that the problem is completely symmetric.

No contexto de Renormalização Wilsoniana, os fluxos do grupo de renormalização (RG flows) são um conjunto de equações diferenciais que define como as constantes de acoplamento de uma teoria dependem de uma escala de energia. o conteúdo destes é semelhante a como sistemas termodinâmicos estão relacionados com a temperatura. Neste sentindo, é natural olhar para estruturas nos fluxos que demonstram um comportamento termodinâmico. A teoria matemática para estudar estas equações é chamada de sistemas dinâmicos e aplicações desta têm sido usadas no estudo de RG flows. Como exemplo o teorema-C de Zamolodchikov e os equivalentes teoremas em dimensões maiores mostram que existe uma função monotonicamente decrescente ao longo do fluxo e é uma propriedade que se assemelha à segunda lei da termodinâmica, estão relacionadas com a função de Lyapunov no contexto de sistemas dinâmicos e podem ser usadas para excluir a possibilidade de comportamentos assintóticos exóticos, como fluxos periódicos ou ciclos limites. Estudamos a teoria de bifurcação e a teoria de índice, que foram propostas como sendo úteis no estudo de RG flows: a primeira pode ser usada para explicar constantes cruzando pela marginalidade e a segunda para extrair informação global do espaço em que os fluxos vivem. Nesta dissertação, também olhamos para aplicações em RG flows holográficos e tentamos buscar relações entre as estruturas em teorias holográficas e as suas duais teorias de campos. / In the context of Wilsonian Renormalization, renormalization group (RG) flows are a set of differential equations that defines how the coupling constants of a theory depend on an energy scale. These equations closely resemble thermodynamical equations and how thermodynamical systems are related to temperature. In this sense, it is natural to look for structures in the flows that show a thermodynamics-like behaviour. The mathematical theory to study these equations is called Dynamical Systems, and applications of that have been used to study RG flows. For example, the classical Zamolodchikov\'s C-Theorem and its higher-dimensional counterparts, that show that there is a monotonically decreasing function along the flow and it is a property that resembles the second-law of thermodynamics, is related to the Lyapunov function in the context of Dynamical Systems. It can be used to rule out exotic asymptotic behaviours like periodic flows (also known as limit cycles). We also study bifurcation theory and index theories, which have been proposed to be useful in the study of RG flows, the former can be used to explain couplings crossing through marginality and the latter to extract global information about the space the flows lives in. In this dissertation, we also look for applications in holographic RG flows and we try to see if the structural behaviours in holographic theories are the same as the ones in the dual field theory side.

No contexto de Renormalização Wilsoniana, os fluxos do grupo de renormalização (RG flows) são um conjunto de equações diferenciais que define como as constantes de acoplamento de uma teoria dependem de uma escala de energia. o conteúdo destes é semelhante a como sistemas termodinâmicos estão relacionados com a temperatura. Neste sentindo, é natural olhar para estruturas nos fluxos que demonstram um comportamento termodinâmico. A teoria matemática para estudar estas equações é chamada de sistemas dinâmicos e aplicações desta têm sido usadas no estudo de RG flows. Como exemplo o teorema-C de Zamolodchikov e os equivalentes teoremas em dimensões maiores mostram que existe uma função monotonicamente decrescente ao longo do fluxo e é uma propriedade que se assemelha à segunda lei da termodinâmica, estão relacionadas com a função de Lyapunov no contexto de sistemas dinâmicos e podem ser usadas para excluir a possibilidade de comportamentos assintóticos exóticos, como fluxos periódicos ou ciclos limites. Estudamos a teoria de bifurcação e a teoria de índice, que foram propostas como sendo úteis no estudo de RG flows: a primeira pode ser usada para explicar constantes cruzando pela marginalidade e a segunda para extrair informação global do espaço em que os fluxos vivem. Nesta dissertação, também olhamos para aplicações em RG flows holográficos e tentamos buscar relações entre as estruturas em teorias holográficas e as suas duais teorias de campos. / In the context of Wilsonian Renormalization, renormalization group (RG) flows are a set of differential equations that defines how the coupling constants of a theory depend on an energy scale. These equations closely resemble thermodynamical equations and how thermodynamical systems are related to temperature. In this sense, it is natural to look for structures in the flows that show a thermodynamics-like behaviour. The mathematical theory to study these equations is called Dynamical Systems, and applications of that have been used to study RG flows. For example, the classical Zamolodchikov\'s C-Theorem and its higher-dimensional counterparts, that show that there is a monotonically decreasing function along the flow and it is a property that resembles the second-law of thermodynamics, is related to the Lyapunov function in the context of Dynamical Systems. It can be used to rule out exotic asymptotic behaviours like periodic flows (also known as limit cycles). We also study bifurcation theory and index theories, which have been proposed to be useful in the study of RG flows, the former can be used to explain couplings crossing through marginality and the latter to extract global information about the space the flows lives in. In this dissertation, we also look for applications in holographic RG flows and we try to see if the structural behaviours in holographic theories are the same as the ones in the dual field theory side.

The Leray-Serre spectral sequence is a fundamental tool for studying singular homology of a fibration E->B with typical fiber F. It expresses H (E) in terms of H (B) and H (F). One of the classic examples of a fibration is given by the free loop space fibration, where the typical fiber is given by the based loop space . The first part of this thesis constructs the Leray-Serre spectral sequence in Morse homology on Hilbert manifolds under certain natural conditions, valid for instance for the free loop space fibration if the base is a closed manifold. We extend the approach of Hutchings which is restricted to closed manifolds. The spectral sequence might provide answers to questions involving closed geodesics, in particular to spectral invariants for the geodesic energy functional. Furthermore we discuss another example, the free loop space of a compact G-principal bundle, where G is a connected compact Lie group. Here we encounter an additional difficulty, namely the base manifold of the fiber bundle is infinite-dimensional. Furthermore, as H ( P) = HF (T P) and H ( Q) =HF (T Q), where HF denotes Floer homology for periodic orbits, the spectral sequence for P -> Q might provide a stepping stone towards a similar spectral sequence defined in purely Floer-theoretic terms, possibly even for more general symplectic quotients. Hutchings’ approach to the Leray-Serre spectral sequence in Morse homology couples a fiberwise negative gradient flow with a lifted negative gradient flow on the base. We study the Morse homology of a vector field that is not of gradient type. The central issue in the Hilbert manifold setting to be resolved is compactness of the involved moduli spaces. We overcome this difficulty by utilizing the special structure of the vector field. Compactness up to breaking of the corresponding moduli spaces is proved with the help of Gronwall-type estimates. Furthermore we point out and close gaps in the standard literature, see Section 1.4 for an overview. In the second part of this thesis we introduce a Lagrangian Floer homology on cotangent bundles with varying Lagrangian boundary condition. The corresponding complex allows us to obtain the Leray-Serre spectral sequence in Floer homology on the cotangent bundle of a closed manifold Q for Hamiltonians quadratic in the fiber directions. This corresponds to the free loop space fibration of a closed manifold of the first part. We expect applications to spectral invariants for the Hamiltonian action functional. The main idea is to study pairs of Morse trajectories on Q and Floer strips on T Q which are non-trivially coupled by moving Lagrangian boundary conditions. Again, compactness of the moduli spaces involved forms the central issue. A modification of the compactness proof of Abbondandolo-Schwarz along the lines of the Morse theory argument from the first part of the thesis can be utilized.

Orientadora: Profa. Dra. Mariana Rodrigues da Silveira / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2014. / Neste trabalho apresentamos um estudo das Teorias de Morse e Morse-Bott no contexto de sistemas dinâmicos. Consideramos uma variedade Riemanniana suave e fechada M de dimensão finita. Dada f : M ! R uma função de Morse-Smale, associamos a f o complexo de cadeia de Morse-Smale-Witten, que recupera a homologia da variedade M (Teorema de Homologia de Morse). Mais geralmente, qualquer função de Morse- Bott-Smale f :M !R pode ser associada ao complexo de cadeia de Morse-Bott-Smale, que é um multicomplexo que se reduz ao complexo de cadeia de Morse-Smale-Witten quando f é uma função de Morse. O Teorema de Homologia de Morse-Bott mostra que a homologia deste multicomplexo também coincide com a homologia de M sua prova tem como caso particular uma prova para o Teorema da Homologia de Morse. / In this work we present a study of Morse and Morse-Bott theories in the context of dynamical systems. We consider a Riemannian smooth, closed n-dimensional manifold M. Given a Morse-Smale function f :M !R, we associate f to the Morse-Smale-Witten chain complex, which recovers the homology of the manifold M (Morse Homology Theorem). More generally, any Morse-Bott-Smale function f :M !R can be associated to the Morse-Bott-Smale chain complex, which is a multicomplex that coincides with the Morse-Smale-Witten complex when f is a Morse function. The Morse-Bott Homology Theorem shows that the homology of thismulticomplex also coincides with the homology of M and its proof has as a particular case a proof for the Morse Homology Theorem.

In this thesis we construct for a given smooth, generic Hamiltonian H on the 2n dimensional torus a chain-isomorphism between the Morse complex of the Hamiltonian action on the free loop space of the torus and the Floer-complex. Though both complexes are generated by the critical points of the Hamiltonian action, their boundary operators differ. Therefore the construction of the isomorphism is based on counting the moduli spaces of hybrid-type solutions which involves stating a new non-Lagrangian boundary value problem for Cauchy-Riemann type operators not yet studied in Floer theory. It is crucial for the statement that the torus is compact, possesses trivial tangent bundle and an additive structure. We finally want to note that the problem is completely symmetric.

The Leray-Serre spectral sequence is a fundamental tool for studying singular homology of a fibration E->B with typical fiber F. It expresses H (E) in terms of H (B) and H (F). One of the classic examples of a fibration is given by the free loop space fibration, where the typical fiber is given by the based loop space . The first part of this thesis constructs the Leray-Serre spectral sequence in Morse homology on Hilbert manifolds under certain natural conditions, valid for instance for the free loop space fibration if the base is a closed manifold. We extend the approach of Hutchings which is restricted to closed manifolds. The spectral sequence might provide answers to questions involving closed geodesics, in particular to spectral invariants for the geodesic energy functional. Furthermore we discuss another example, the free loop space of a compact G-principal bundle, where G is a connected compact Lie group. Here we encounter an additional difficulty, namely the base manifold of the fiber bundle is infinite-dimensional. Furthermore, as H ( P) = HF (T P) and H ( Q) =HF (T Q), where HF denotes Floer homology for periodic orbits, the spectral sequence for P -> Q might provide a stepping stone towards a similar spectral sequence defined in purely Floer-theoretic terms, possibly even for more general symplectic quotients. Hutchings’ approach to the Leray-Serre spectral sequence in Morse homology couples a fiberwise negative gradient flow with a lifted negative gradient flow on the base. We study the Morse homology of a vector field that is not of gradient type. The central issue in the Hilbert manifold setting to be resolved is compactness of the involved moduli spaces. We overcome this difficulty by utilizing the special structure of the vector field. Compactness up to breaking of the corresponding moduli spaces is proved with the help of Gronwall-type estimates. Furthermore we point out and close gaps in the standard literature, see Section 1.4 for an overview. In the second part of this thesis we introduce a Lagrangian Floer homology on cotangent bundles with varying Lagrangian boundary condition. The corresponding complex allows us to obtain the Leray-Serre spectral sequence in Floer homology on the cotangent bundle of a closed manifold Q for Hamiltonians quadratic in the fiber directions. This corresponds to the free loop space fibration of a closed manifold of the first part. We expect applications to spectral invariants for the Hamiltonian action functional. The main idea is to study pairs of Morse trajectories on Q and Floer strips on T Q which are non-trivially coupled by moving Lagrangian boundary conditions. Again, compactness of the moduli spaces involved forms the central issue. A modification of the compactness proof of Abbondandolo-Schwarz along the lines of the Morse theory argument from the first part of the thesis can be utilized.

Ce mémoire présente une nouvelle méthode d’étudier des fonctions de Morse sur une variété compacte. Plus précisément, les croisements entre les lignes de flot de pseudo-gradients associés à des fonctions de Morse permettent de définir géométriquement des morphismes entre les complexes de Morse, morphismes qui ne peuvent généralement pas être obtenus par une homotopie. Cette nouvelle classe de morphismes mène à la définition d’une catégorie triangulée. La question centrale est de savoir si tout objet de cette catégorie est décomposable en cône itéré de fonctions de Morse parfaites. En effet, une telle décomposition simplifierait l’étude de la dynamique d’une fonction de Morse en l’interprétant plutôt comme plusieurs fonctions parfaites. Une seconde question d’importance porte sur une condition de généricité globale à laquelle est soumise cette catégorie triangulée. Nous étudions la possibilité de s’en soustraire en proposant une méthode de déformations des fonctions de Morse. / This master’s thesis introduces a new way to sudy Morse functions on a compact manifold. More specifically, crossings between flows of pseudo-gradients associated to Morse functions allow one to define geometric realisations of morphisms between the Morse complexes. This new class of morphisms leads to the definition of a triangulated category. The main question is to determine if every object of this category admits an iterated cone decomposition. Such a decomposition would greatly simplify the study of the dynamic of a Morse function by interpreting it as many perfect Morse functions. A second topic concerns the global genericity condition to which this category is subject. We study a way, through deformation of Morse functions, to avoid such a constraint.

Dans ce travail, nous définissons des objets composés de disques complexes marqués reliés entre eux par des segments de droite munis d’une longueur. Nous construisons deux séries d’espaces de module de ces objets appelés clus- ters, une qui sera dite non symétrique, la version ⊗, et l’autre qui est dite symétrique, la version •. Cette construction permet des choix de perturba- tions pour deux versions correspondantes des trajectoires de Floer introduites par Cornea et Lalonde ([CL]). Ces choix devraient fournir une nouvelle option pour la description géométrique des structures A∞ et L∞ obstruées étudiées par Fukaya, Oh, Ohta et Ono ([FOOO2],[FOOO]) et Cho ([Cho]). Dans le cas où L ⊂ (M, ω) est une sous-variété lagrangienne Pin± mono- tone avec nombre de Maslov ≥ 2, nous définissons une structure d’algèbre A∞ sur les points critiques d’une fonction de Morse générique sur L. Cette struc- ture est présentée comme une extension du complexe des perles de Oh ([Oh]) muni de son produit quantique, plus récemment étudié par Biran et Cornea ([BC]). Plus généralement, nous décrivons une version géométrique d’une catégorie de Fukaya avec seul objet L qui se veut alternative à la description (relative) hamiltonienne de Seidel ([Sei]). Nous vérifions la fonctorialité de notre construction en définissant des espaces de module de clusters occultés qui servent d’espaces sources pour des morphismes de comparaison. / We define objects made of marked complex disks connected by metric line seg- ments and construct two sequences of moduli spaces of these objects, referred as the ⊗ version (nonsymmetric) and the • version (symmetric). This allows choices of coherent perturbations over the corresponding versions of the Floer trajectories proposed by Cornea and Lalonde ([CL]). These perturbations are intended to lead to an alternative geometric description of the (obstructed) A∞ and L∞ structures studied by Fukaya, Oh, Ohta and Ono ([FOOO2],[FOOO]) and Cho ([Cho]). Given a Pin± monotone lagrangian submanifold L ⊂ (M, ω) with mini- mal Maslov number ≥ 2, we define an A∞ -algebra structure from the critical points of a generic Morse function on L. We express this structure as a cochain complex extending the pearl complex introduced by Oh ([Oh]) and further ex- plicited by Biran and Cornea ([BC]), equipped with its quantum product. This could also be seen as an alternative geometric description of a Fukaya cate- gory of (M, ω) with L as its only object, a hamiltonian relative version appear- ing in [Sei]. Using spaces of quilted clusters, we verify, using more general quilted cluster spaces, that this defines a functor from a homotopy category of Pin± monotone lagrangian submanifolds hL mono,± (M, ω) to the homotopy category of cochain complexes hK(Λ-mod) where Λ is an appropriate Novikov ring.