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# Mathematics - Theses, Dissertations, and other Required Graduate Degree Essays

Receive updates for this collection## Die Freude an der Gestalt: Methods, Figures, and Practices in Early Nineteenth Century Geometry

As recounted by later historians, modern geometry began with Jean Victor Poncelet, whose contributions then spread to Germany alongside an opposition between geometric methods that came to be exemplified by the antagonism of Julius Plücker, an analytic geometer, and Jakob Steiner, a synthetic geometer. To determine the participants, arguments, and qualities of this perceived divide, we drew upon historical accounts from the late nineteenth and early twentieth centuries. Several themes emerged from the historical perspective, which we investigated within the original sources. Our questions centred on how geometers distinguished methods, when opposition arose, in what ways geometry disseminated from Poncelet to Plücker and Steiner, and whether this geometry was "modern" as claimed. Our search for methodological debates led to Poncelet's proposal that within pure geometry the figure was never lost from view, while it could be obscured by the calculations of algebra. We examined his argument through a case study that revealed visual attention within constructive problem solving, regardless of method. Further, geometers manipulated and represented figures through textual descriptions and coordinate equations. In these same texts, Poncelet and Joseph-Diez Gergonne instigated a debate on the principle of duality. Rather than dismiss their priority dispute as external to mathematics, we consider the texts involved as a medium for communicating geometry in which Poncelet and Gergonne developed strategies for introducing new geometry to a conservative audience. This conservative audience did not include Plücker and Steiner, who adapted new vocabulary, techniques and objects. Through comparing their common research, we found they differentiated methods based on personal considerations. Plücker practiced a "pure analytic geometry" that avoided calculation. Steiner admired "synthetic geometry'' because of its organic unity. These qualities contradicted descriptions of analytic geometry as computational or synthetic geometry as ad-hoc. Finally, we turned to claims for novelty in the context of contemporary French books on geometry. Most of these books point to a pedagogical orientation, where the methodological divide was grounded in student prerequisites and "modern'' implied the use of algebra in geometry. By contrast, research publications exhibited evolving forms of geometry that evaded dichotomous categorization.

## Symmetric Differential Forms on the Barth Sextic Surface

This thesis concerns the existence of regular symmetric differential 2-forms on the Barth sextic surface, here denoted by X. This surface has 65 nodes, the maximum possible for a sextic hypersurface in P^3. This project is motivated by a recent work of Bogmolov and De Oliveira where it is shown that a hypersurface in P^3 with many nodes compared to its degree contains only finitely many genus zero and one curves. We find that there are symmetric differential 2-forms on X that are regular everywhere outside the nodes. We also find that none of these extend to a regular form on the minimal resolution of X. Using these forms we can prove that any genus 0 curve on X must pass through at least one node, and we determine the curves passing through just a select set of nodes.

## Solving variational problems and partial differential equations mapping between manifolds via the closest point method

Many applications from fields such as mathematical physics, image processing, computer vision and medical imaging require computation of maps between manifolds. A numerical framework is introduced for solving variational problems and partial differential equations that map from a source manifold M to a target manifold N . The numerics rely on the closest point representations of M and N. Using the closest point representation produces simple algorithms for handling complex surface geometries, since standard Cartesian numerical methods can be used. The framework is illustrated with harmonic maps and a straightforward algorithm is given for this case. Harmonic maps are important in applications such as texture mapping, brain image regularization and colour image denoising. Moreover, the harmonic mapping energy is part of numerous energy functionals. The algorithm is justified theoretically and shown to be first order accurate. It is implemented in two applications: removing noise from texture maps and colour image enhancement.

## On Laplace-Borel resummation of Dyson-Schwinger equations

On Laplace-Borel Resummation of Dyson-Schwinger Equations Abstract: In this work we conduct a complex analytic study of Dyson-Schwinger equations, the quantum equations of motion. Focusing on a particular family of those functional equations we consider the class of formal solutions G(x,L), series expansions in the coupling constant x and the energy scale L whose formal and real analytic aspects have already been studied. Taking the point of view of complex analysis we are able to shed some new light on the structure of these solutions and provide useful tools to consider asymptotic questions. This thesis is built around two functions. The anomalous dimension $\gamma_1$ which is closely tied to the energy scaling properties of quantum field theory and the Green function G(x,L), the actual solution to the Dyson-Schwinger equation. We study their dual aspects as formal power series and analytic functions in the variable x and L. Our tool of choice is the Laplace-Borel resummation method which proves suitable to take care of the divergent series occurring naturally in quantum field theory. Our main results consist in: i) conducting a Laplace-Borel analysis of the anomalous dimension $\gamma_1$; ii) constructing a Laplace-Borel solution to our Dyson-Schwinger equation by using the renormalization group equation.

## Simple eigenvalues of graphs and digraphs

The spectra of graphs and their relation to graph properties have been well-studied. For digraphs, in contrast, there are relatively few results. The adjacency matrix of a digraph is usually difficult to work with; it is not always diagonalizable and the interlacing theorem does not hold (in general) for adjacency matrices of digraphs. All acyclic digraphs have the same spectrum as the empty graph. This motivates the need to work with a different matrix which captures the adjacency of the digraph. To this end, we introduce the Hermitian adjacency matrix. Another way to extract more information out of the spectrum is by restricting to specific classes of digraphs. In this thesis, we look at vertex-transitive digraphs with simple eigenvalues. Intuitively, the property of having many simple eigenvalues tends to coincide with having few automorphisms. For example, the only vertex-transitive graph with all eigenvalues simple is K_2. In the case of graphs, we restrict to the cubic vertex-transitive case, where we find combinatorial properties of graphs with multiple simple eigenvalues. We also explore the eigenvectors of vertex-transitive digraphs with all eigenvalues distinct.

## Variants of the Consecutive Ones Property: Algorithms, Computational Complexity and Applications to Genomics

Genome mapping problems in bioinformatics can be modelled as problems of finding sequences of vertices in hypergraphs, subject to consecutivity constraints. These problems are related to the \emph{consecutive ones property}, a well-studied structural property on binary matrices. Many variants of this property have been introduced to include subtleties in the model, such as upper bounds on the number of times a vertex may appear in a sequence, the distance of the input from having the property, and confidence values for the consecutivity constraints. Most problems involving these variants are intractable, and efficient solutions call for restrictions on the structure of the input, exponential time algorithms, or approximations. The following document discusses these problems, from both a theoretical perspective, and from the genomics point of view.We encounter two main classes of problems, divided into models which account for repeated elements in genomes, and those which do not. Orthogonally, we divide the problems into decision and optimization questions. For models with repeats, we discuss when the given input can be used to reconstruct the genome map of interest, and if we can discard a minimal set of encoded consecutivity information from the model to obtain an input which can be used to reconstruct this genome map. We also discuss the problem of ambiguity introduced by repeats, and introduce the concept of \emph{repeat spanning intervals} in order to address them. We show that the problem of optimizing over the set of repeat spanning intervals is NP-hard in general, and give an algorithm when the intervals are small. In models without repeated elements, we discuss the problem of optimization byfinding a solution that minimizes the distortion in the consecutivity information, by generalizing the concepts of bandwidth and minimum linear arrangement to hypergraphs. We design approximation algorithms for two versions of the latter problem, with an approximation ratio of $O\left(\sqrt{\log n}\log\log n\right)$.Finally, we provide details of implementations of some of the methods developed for genome mapping and scaffolding on ancestral genomes. We include results on real data for the genome of the Black Death agent, and for ancestral \textit{Anopheles} mosquitoes.

## Using Viral Dynamics to connect clinical markers of disease progression to sequence evolution during HIV-1 infection

HIV-1 remains a global health challenge, with over 35 million people infected. The high rates of turnover and evolutionary adaptability exhibited by HIV-1 pose a particular challenge to HIV-1 vaccine development. We developed a dynamic model of HIV-1 infection that uses equilibration, adaptation, and inheritance to model the initial infection and successive generations of viral lineages. The model allows viruses to generate new lineages in proportion to their viral load. These lineages compete for immune cells to infect. We use this model to demonstrate how viruses with a sufficiently high mutation rate could overcome the immune system, even when most changes are expected to be detrimental to viral fitness. We have calibrated our model to match averages of CD4+ T cells/mm^3 and HIV-1 RNA/ml derived from 91 HIV-infected individuals studied longitudinally during various disease stages. We also explore the use of phylogenies to validate the underlying composition of viral load.

## On excluded minors and biased graph representations of frame matroids

A biased graph is a graph in which every cycle has been given a bias, either balanced or unbalanced. Biased graphs provide representations for an important class of matroids, the frame matroids. As with graphs, we may take minors of biased graphs and of matroids, and a family of biased graphs or matroids is minor-closed if it contains every minor of every member of the family. For any such class, we may ask for the set of those objects that are minimal with respect to minors subject to not belonging to the class - i.e., we may ask for the set of excluded minors for the class. A frame matroid need not be uniquely represented by a biased graph. This creates complications for the study of excluded minors. Hence this thesis has two main intertwining lines of investigation: (1) excluded minors for classes of frame matroids, and (2) biased graph representations of frame matroids. Trying to determine the biased graphs representing a given frame matroid leads to the necessity of determining the biased graphs representing a given graphic matroid. We do this in Chapter 3. Determining all possible biased graph representations of non-graphic frame matroids is more difficult. In Chapter 5 we determine all biased graphs representa- tions of frame matroids having a biased graph representation of a certain form, subject to an additional connectivity condition. Perhaps the canonical examples of biased graphs are group-labelled graphs. Not all biased graphs are group-labellable. In Chapter 2 we give two characterisations of those biased graphs that are group labellable, one topological in nature and the other in terms of the existence of a sequence of closed walks in the graph. In contrast to graphs, which are well-quasi-ordered by the minor relation, this characterisation enables us to construct infinite antichains of biased graphs, even with each member on a fixed number of vertices. These constructions are then used to exhibit infinite antichains of frame matroids, each of whose members are of a fixed rank. In Chapter 4, we begin an investigation of excluded minors for the class of frame ma- troids by seeking to determine those excluded minors that are not 3-connected. We come close, determining a set E of 18 particular excluded minors and drastically narrowing the search for any remaining such excluded minors.

## Generalized thrackles and graph embeddings

A thrackle on a surface X is a graph of size e and order n drawn on X such that every two distinct edges of G meet exactly once either at their common endpoint, or at a proper crossing. An unsolved conjecture of Conway (1969) asserts that e≤n for every thrackle on a sphere. Until now, the best known bound is e≤1.428n. By using discharging rules we show that e≤1.4n. Furthermore we show that the following are equivalent: G has a drawing on X where every two edges meet an odd number of times (a generalized thrackle); G has a drawing on X where every two edges meet exactly once (a one-thrackle); G has a special embedding on a surface whose genus differs from the genus of X by at most one.

## Barycentric Rational Interpolation and Spectral Methods

For the numerical solution of differential equations spectral methods typically give excellent accuracy with relatively few points (small N), but certain numerical issues arise with larger N. This thesis focuses on spectral collocation methods, also known as pseudo-spectral methods, that rely on interpolation at collocation points. A relatively new class of interpolants will be considered, namely the Floater-Hormann family of rational interpolants. These interpolants and their properties will be studied, including their use in differentiation by means of differentiation matrices based on rational interpolants in the barycentric form. Then, consideration will be given to the solution of singularly perturbed boundary value problems through the use of boundary layer resolving coordinate transformations. Finally, coupled systems of singularly perturbed boundary value problems will be investigated, though only with the standard Chebyshev collocation method.