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This thesis consists of an introduction and five research papers on topics related to conformal mapping, the Loewner equation and its applications, and Bergman-type spaces of holomorphic functions. The first two papers are devoted to the study of integral means of derivatives of conformal mappings. In Paper I, we present improved upper estimates of the universal means spectrum of conformal mappingsof the unit disk. These estimates rely on inequalities  obtained by Hedenmalm and Shimorin using Bergman space techniques, and on computer calculations. Paper II is a survey of recent results on the universal means spectrum, with particular emphasis on Bergman spacetechniques.Paper III concerns Bergman-type spaces of holomorphic functions in subsets of $\textbf{C}^d$ and their reproducing kernel functions. By expanding the norm of a function in a Bergman space along the zero variety of a polynomial, we obtain a series expansion of reproducing kernel functions in terms of kernels associated with lower-dimensionalspaces of holomorphic functions. We show how this general approach can be used to explicitly compute kernel functions for certain weighted Bergman and Bargmann-Fock spaces defined in domains in $\textbf{C}^2$.The last two papers contribute to the theory of Loewner chains and theirapplications in the analysis of planar random growth model defined in terms of compositions of conformal maps.In Paper IV, we study Loewner chains generated by unimodular L\'evy processes.We first establish the existence of a capacity scaling limit for the associated growing hulls in terms of whole-plane Loewner chains driven by a time-reversed process. We then analyze the properties of Loewner chains associated with a class of two-parameter compound Poisson processes, and we describe the dependence of the geometric properties of the hulls on the parameters of the driving process. In Paper V, we consider a variation of the Hastings-Levitov growth model, with anisotropic growth. We again establish results concerning scaling limits, when the number of compositions increases and the basic conformal mappings tends to the identity. We show that the resulting limit sets can be associated with solutions to the Loewner equation.We also prove that, in the limit, the evolution of harmonic measure on the boundary is deterministic and is determined by the flow associated with an ordinary differential equation, and we give a description of the fluctuations around this deterministic limit flow. / <p>QC 20100414</p>

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 5 June 2017. / The holographic computation of extremal correlators is often frustrated by divergences. The interpretation of these divergences is incomplete. The primary goal of this study is to develop a better understanding of these divergences. Towards this end, working within the AdS/CFT correspondence we review the computation of correlators. In the field theory we review well known matrix model techniques useful to study the planar limit, as well as methods exploiting group representation theory that are useful for the computation of correlators in large N but non-planar limits. On the gravity side of the correspondence, we describe in detail the computation of two- and three point functions of a scalar field coupled to gravity on the Euclidian AdSd+1 space, three-point functions of two giant gravitons and one pointlike graviton as well as correlators of Kaluza-Klein gravitons. A key observation of this study is that extremal correlators are mapped to scattering amplitudes of particles with parallel momenta. These are naturally accompanied by involve collinear divergences. Therefore, we suggest that the divergences in the computation of extremal correlators are linked to collinear divergences. A lot more work is needed to establish this connection. / LG2017

Modern microscopy and automation technologies enable experiments which can produce millions of images each day. The valuable information is often sparse, and requires clever methods to find useful data. In this thesis a general image classification tool for fluorescence microscopy images was developed usingfeatures extracted from a general Convolutional Neural Network (CNN) trained on natural images. The user selects interesting regions in a microscopy image and then, through an iterative process, using active learning, continually builds a training data set to train a classifier that finds similar regions in other images. The classifier uses conformal prediction to find samples that, if labeled, would most improve the learned model as well as specifying the frequency of errors the classifier commits. The result show that with the appropriate choice of significance one can reach a high confidence in true positive. The active learning approach increased the precision with a downside of finding fewer examples.

Orientador: José Geraldo Pereira / Banc: Roldão da Rocha Junior / Banca:Marcos Vinícius Borges Teixeira Lima / Banca: Bruto Max Pimentel Escobar / Banca: Yuri N. Obukhov / Resumo: Na presença de uma constante cosmológica, interpretada como uma entidade puramente geométrica, a ausência de matéria é representada pelo espaço de de Sitter. Como consequência, a relatividade especial de Poincaré não é mais válida e deve ser substituída por uma relatividade especial baseada no grupo de de Sitter, o que produz modificações em todas as áreas da física. Nesta tese, vamos estudar os fundamentos da relatividade especial de de Sitter, bem como algumas implicações para o problema da energia escura, para a propagação de raios gama de altíssimas energias e para o espalhamento Compton. Como um subproduto desses estudos, desenvolvemos um novo método de se obter equações de campo com invariância conforme, o qual faz uso dos invariantes de Casimir do grupo de de Sitter. Usando esse novo método, fazemos um estudo crítico das equações que descrevem um campo fundamental de spin-2 / Abstract: The presence of a cosmological constant, interpreted as a purely geometric entity, absence of matter ir represented by a de Sitter spacetime. As a consequence, ordinary Poincaré specil relativity is no longer valid and must be replaced by a special relativity based on the de Sitter group, which produces concomitant changes in all areas of physics. In this thesis, we are going to explore the implications for the dark energy problem, for the propagation of ultra high-energy gamma rays, and for the Compton scattering formula. As a byoproduct of these studies, we developed a new method for obtaining conformal invariant fiels equations, which makes use of the Casimir operators of the de Sitter group. Using this new method, we make a critical review of the fiels equations describing a fundamental spin-2 field / Doutor

Neste trabalho são apresentados alguns conceitos básicos da Teoria de Liouville e N=1 Super Liouville, enfatizando o cálculo das funções de três pontos dessas teorias.Uma introdução a Teoria de Campos Conformes (CFT) e a Supersimetria também sao incluídas, as quais constituem ferramentas básicas da presente pesquisa. / In this dissertation we present some basic features about Liouville and N=1 Super Liouville Theory, and focus in the computation of their three point functions. Additionally, we include an introduction to Conformal Field Theories (CFT) and Supersymmetry, which are the basic tools of the present research.

O principal objetivo deste trabalho é estudar a quantização covariante da corda bosônica e da supercorda RNS, explorando as simetrias envolvidas, ou seja, as simetrias BRST e conforme no caso da corda bosônica e as generalizações correspondentes para a corda fermiônica. Em particular, discutimos alguns aspectos perturbativos da teoria bosônica e a construção de operadores de vértice da corda fermiônica. / The main goal of this work is to study the covariant quantization of the bosonic and RNS string theories by exploiting the involved symmetries, namely, the BRST and conformal invariance for the bosonic string and the corresponding supersymmetric generalizations for the fermionic case. In particular, we discuss some perturbative aspects of the bosonic theory and the construction of vertex operators for the fermionic string.

Thesis advisor: Krzysztof Kempa / The resonant modes of a sequence of periodically spaced microstrip resonators is studied. The system is analyzed as transmission line with periodic capacitive gaps, as a waveguide with apertures via normal mode expansion, and through a derivation of the static fields in the gap between two microstrip resonators via conformal mapping. FDTD simulations are also performed to numerically calculate the resonant modes of the sequence and also its absorption spectrum when it contains a lossy dielectric. It is found, as expected, that when the gap size is large, the microstrip resonators are uncoupled and there resonant modes are unperturbed. As the gap size narrows, the resonators become strongly coupled, and changing boundary conditions perturb the resonant modes upwards in frequency. Moreover, an additional resonant mode is observed that does not correspond to any uncoupled mode. / Thesis (BS) — Boston College, 2011. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: College Honors Program. / Discipline: Physics Honors Program. / Discipline: Physics.

Neste trabalho são apresentados alguns conceitos básicos da Teoria de Liouville e N=1 Super Liouville, enfatizando o cálculo das funções de três pontos dessas teorias.Uma introdução a Teoria de Campos Conformes (CFT) e a Supersimetria também sao incluídas, as quais constituem ferramentas básicas da presente pesquisa. / In this dissertation we present some basic features about Liouville and N=1 Super Liouville Theory, and focus in the computation of their three point functions. Additionally, we include an introduction to Conformal Field Theories (CFT) and Supersymmetry, which are the basic tools of the present research.

We consider a series of problems regarding quantum Hall edges, focusing on both dynamics and the mathematical structure of edge states. We begin in Chapter 3 with a limiting case of the Laughlin state placed in a very steep confining potential, but which is weak compared to the interactions. We find that the eigenstates have a Jack polynomial structure and an energy spectrum which is extremely different from the well-known Luttinger liquid edge. In Chapter 5 we analyse the inner products of edge state wavefunctions, using an effective description given by a large-N expansion ansatz proposed by J. Dubail, N. Read and E. Rezayi, PRB 86, 245310 (2012). As noted by these authors, the terms in this ansatz can be constrained using symmetry, a procedure we perform to high orders. We then check the conjecture by calculating overlaps exactly for small system sizes and comparing the numerics with our high-order expansion to find excellent agreement. Finally, Chapter 6 considers the behaviour of quantum Hall edges close to the Luttinger liquid fixed point that occurs in the low energy, large system limit. We construct effective Hamiltonians using a local field theory description and then consider the effect of bulk symmetries on this edge. The symmetry analysis produces remarkable simplifications which allow for very accurate descriptions of the low-energy edge physics even relatively far away from the Luttinger liquid fixed point.