Thesis advisor: Ian Biringer / We consider the space $\rootedH2$ of all complete hyperbolic surfaces without boundary with a basepoint equipped with the pointed Gromov-Hausdorff topology. Continuous paths within $\rootedH2$ arising from certain deformations on a hyperbolic surface and concrete geometric constructions are studied. These include changing some Fenchel-Nielsen parameters of a subsurface, pinching a simple closed geodesic to a cusp, and inserting an infinite strip along a proper bi-infinite geodesic. We then use these paths to show that $\rootedH2$ is path-connected and that it is locally weakly connected at points whose underlying surfaces are either the hyperbolic plane or hyperbolic surfaces of the first kind. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Cette thèse est consacrée à l'étude de certains processus aléatoires à valeurs dans les arbres continus. Nous définissons d'abord un cadre conceptuel pour cette étude, en construisant une topologie polonaise sur l'espace des R-arbres localement compacts, complets et munis d'une mesure borélienne localement finie. Cette topologie, dite de Gromov-Hausdorff-Prokhorov, permet alors la définition de processus de Markov à valeurs arbre. Nous donnons ensuite une nouvelle construction du processus d'élagage d'Abraham-Delmas-Voisin, qui est un exemple de processus qui prend ses valeurs dans les arbres de Lévy. Notre construction, qui dévoile une nouvelle structure généalogique des arbres de Lévy, est trajectorielle, et permet d'identifier explicitement les transitions du processus d'élagage. Nous appliquons cette description à l'étude de certains temps d'arrêt, comme le premier temps auquel le processus franchit une hauteur donnée. Nous décrivons le processus à cet instant grâce à une nouvelle décomposition de type spinal. Enfin, nous nous intéressons à la fragmentation d'Aldous-Pitman de l'arbre brownien d'Aldous. En particulier, nous étudions, à la suite d'Abraham et Delmas, l'effet de cette fragmentation sur les sous-arbres discrets de l'arbre brownien. Le nombre de coupures nécessaires avant d'isoler la racine, convenablement renormalisé, converge vers une variable aléatoire de Rayleigh ; nous donnons un théorème central limite qui précise les fluctuations autour de cette limite / In this thesis, we study continuum tree-valued processes. First, we define an abstract framework for these processes, by constructing a metric on the space of locally compact, complete R-trees, endowed with a locally finite Borel measure. This topology, called Gromov-Hausdorff-Prokhorov topology, allows for the definition of tree-valued Markov processes. We then give a new construction of the pruning process of Abraham-Delmas-Voisin, which is an example of a Lévy tree-valued process. Our construction reveals a new genealogical structure of Lévy trees. Furthermore, it is a path wise construction, which describes the transitions of the process explicitly. We apply this description to the study of certain stopping times, such as the first moment the process crosses a given height. We describe the process at that time through a new spinal decomposition. Finally, we focus on the Aldous-Pitman fragmentation of Aldous's Brownian tree. Following Abraham and Delmas, we study the effect of the fragmentation on discrete subtrees of the Brownian tree. The number of cuts needed to isolate the root, suitably renormalized, converges towards a Rayleigh-distributed random variable; we prove a Central Limit Theorem describing the fluctuations around this limit

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Neste trabalho apresento a teoria de Gromov-Witten, cohomologia quÃntica e mapas estÃveis e uso estas ferramentas para obter alguns resultados enumerativos. Em particular, provo a fÃrmula de Kontsevich para curvas racionais projetivas planas de grau d. FaÃo um estudo introdutÃrio dos espaÃos de Mumford-Knudsen e construo os espaÃos de Kontsevich a fim de definir os invariantes de Gromov-Witten. Estes sÃo usados para definir o anel de cohomologia quÃntica. Em seguida, aplico a teoria geral para o caso do plano projetivo e, usando a associatividade do produto quÃntico, obtenho a fÃrmula de Kontsevich. TambÃm estudo a fronteira do espaÃo modulli de mapas estÃveis e descrevo o grupo de Picard destes. Com isso, seguindo as ideias de Pandharipand, especialmente o algoritmo por este desenvolvido, calculo alguns nÃmeros caracterÃsticos de curvas no espaÃo projetivo. / In this work, I present the Gromov-Witten theory, quantum cohomology and stable maps and use these tools to obtain some enumerative results. In particular, I proof the Kontsevich formula to projective rational plane curves of degree d. I do an introductory study of Mumford-Knudsen spaces and construct the Kontsevich spaces in order to define gromov-witten invariants. These are used to define the quantum cohomology ring. Next, I apply the general theory to the case of the projective plane and, using the the associativity of the quantum product, I obtain the Kontsevich formula. I also study the boundary of the modulli space of stable maps and describe its Picard group. Following the ideas of Pandharipand, especially the algorithm he developed, I calculate some characteristic numbers of curves in the projective space.

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It is conjectured that the SYZ map equals to the inverse mirror map. In dimension two this conjecture is proved, and in dimension three supporting evidences of the equality are studied in various examples. Since the SYZ map is expressed in terms of open Gromov-Witten invariants, this conjectural equality established an enumerative meaning of the inverse mirror map. / Moreover a computational method of open Gromov-Witten invariants for toric Calabi-Yau manifolds is invented. As an application, the Landau-Ginzburg mirrors of compact semi-Fano toric surfaces are computed explicitly. / This thesis gives a procedure to carry out SYZ construction of mirrors with quantum corrections by Fourier transform of open Gromov-Witten invariants. Applying to toric Calabi-Yau manifolds, one obtains the Hori-Iqbel-Vafa mirror together with a map from the Kahler moduli to the complex moduli of the mirror, called the SYZ map. / Lau, Siu Cheong. / Adviser: N.C. Leung. / Source: Dissertation Abstracts International, Volume: 73-06, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 143-148). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Kuranishi atlases were introduced by McDuff and Wehrheim as a means to build a virtual fundamental cycle on moduli spaces of J-holomorphic curves and resolve some of the challenges in this field. This thesis considers genus zero Gromov-Witten invariants on a general closed symplectic manifold. We complete the construction of these invariants using Kuranishi atlases. To do so, we show that Gromov-Witten moduli spaces admit a smooth enough Kuranishi atlas to define a virtual fundamental class in any virtual dimension. In the process, we prove a stronger gluing theorem. Once we have defined genus zero Gromov-Witten invariants, we show that they satisfy the Gromov-Witten axioms of Kontsevich and Manin, a series of main properties that these invariants are expected to satisfy. A key component of this is the introduction of the notion of a transverse subatlas, a useful tool for working with Kuranishi atlases.

Building on the work of Conrad Plaut and Valera Berestovskii regarding uniform spaces and the covering spectrum of Christina Sormani and Guofang Wei developed for geodesic spaces, the author defines and develops discrete homotopy theory for metric spaces, which can be thought of as a discrete analog of classical path-homotopy and covering space theory. Given a metric space, X, this leads to the construction of a collection of covering spaces of X - and corresponding covering groups - parameterized by the positive real numbers, which we call the [epsilon]-covers and the [epsilon]-groups. These covers and groups evolve dynamically as the parameter decreases, changing topological type at specific parameter values which depend on the topology and local geometry of X. This leads to the definition of a critical spectrum for metric spaces, which is the set of all values at which the topological type of the covers change. Several results are proved regarding the critical spectrum and its connections to topology and local geometry, particularly in the context of geodesic spaces, refinable spaces, and Gromov-Hausdorff limits of compact metric spaces. We investigate the relationship between the critical spectrum and covering spectrum in the case when X is geodesic, connections between the geometry of the [epsilon]-groups and the metric and topological structure of the [epsilon]-covers, as well as the behavior of the [epsilon]-covers and critical values under Gromov-Hausdorff convergence.

We will compute relative Gromov--Witten invariants of maximal contact order by applying the virtual localization formula to the moduli space of relative stable maps. In particular, we will enumerate genus 0 stable maps to the Hirzebruch surface 𝔽₁ = ℙ(𝒪_ℙ¹ ⊕ 𝒪_ℙ¹ (1)) relative to the divisor 𝐷 = 𝐵 + 𝐹, where 𝐵 is the base and 𝐹 the fiber of the projective bundle. We will provide an explicit description of the connected components of the fixed locus of the moduli space 𝑀̅₀,𝑛 (𝔽₁ ; 𝐷|𝛽 ; 𝜇) using decorated colored graphs and further determine the weight decomposition of their virtual normal bundles. This thesis contains explicit computations for 𝜇 = (3) and 𝛽 = 3𝐹 + 𝐵), and additionally 𝜇 = (4) and 𝛽 ∈ {4𝐹 + 𝐵, 4𝐹 + 2𝐵}. The same methodology however can be applied to any other ramification pattern 𝜇 and curve class 𝛽.

The need to measure similarity between two objects is everywhere. It is not always clear what it means for two objects to be similar. The definition changes depending on the area of application. However, similarity between two objects is generally defined as an inverse function to the distance between them. Also it is not always easy to apply distance functions on objects directly. Sometimes, we have to transform them or embed them in another space first before we can calculate distance and subsequently similarity. We introduce three similarity algorithms/measures to quantify similarity between objects in different applications. First, we propose the first non brute force algorithm to calculate the Gromov hyperbolicity constant. We present several approximate and exact algorithms to solve this problem. For example, we provide an exact algorithm to compute the hyperbolicity constant in time O (n3:686) for a discrete metric space. We also show that hyperbolicity at a fixed base-point cannot be computed in O(n2:05) time, unless there exists a faster algorithm for (max,min) matrix multiplication than currently known. Then, we present a new system to find proteins similar in functionality. We employ text mining techniques to map text similarity to similarity in functionality. We use manually curated data from Swiss-Prot to train and build our system. The result is a search engine that given a query protein, reports the top similar proteins in functionality with 99% accuracy. The system is tested extensively using GO annotations. We used this system, that predicts similarity in function, to enhance protein annotations. In particular, we were able to predict that some GO annotations should be added to some proteins. After careful literature reviews we were able to con rm many of those predictions, for example, in one case, we have 96% prediction accuracy. We also present a new algorithm for measuring the similarity between GPS traces. Our algorithm is robust against subsampling and supersampling. We perform experiments to compare this new similarity measure with the two main approaches that have been used so far: Dynamic Time Warping (DTW) and the Euclidean distance and our algorithm outperforms both of them in most of the cases.

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