This thesis introduces two new algorithms to find hypergeometric solutions of second order regular singular differential operators with rational function or polynomial coefficients. Algorithm 3.2.1 searches for solutions of type: exp(∫ r dx) ⋅ ₂F₁ (a₁,a₂;b₁;f) and Algorithm 5.2.1 searches for solutions of type exp(∫ r dx) (r₀ ⋅ ₂F₁(a₁,a₂;b₁;f) + r₁ ⋅ ₂F´₁ (a₁,a₂;b₁;f)) where f, r, r₀, r₁ ∈ ℚ̅(̅x̅)̅ and a₁,a₂,b₁ ∈ ℚ and denotes the Gauss hypergeometric function. The algorithms use modular reduction, Hensel lifting, rational function reconstruction, and rational number reconstruction to do so. Numerous examples from different branches of science (mostly from combinatorics and physics) showed that the algorithms presented in this thesis are very effective. Presently, Algorithm 5.2.1 is the most general algorithm in the literature to find hypergeometric solutions of such operators. This thesis also introduces a fast algorithm (Algorithm 4.2.3) to find integral bases for arbitrary order regular singular differential operators with rational function or polynomial coefficients. A normalized (Algorithm 4.3.1) integral basis for a differential operator provides us transformations that convert the differential operator to its standard forms (Algorithm 5.1.1) which are easier to solve. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / May 25, 2017. / Differential Operators, Hypergeometric Solutions, Integral Bases / Includes bibliographical references. / Mark van Hoeij, Professor Directing Dissertation; Robert A. van Engelen, University Representative; Amod S. Agashe, Committee Member; Ettore Aldrovandi, Committee Member; Paolo B. Aluffi, Committee Member.

In this manuscript, we propose space-time spectral element methods to solve problems arising from fluid dynamics and materials science. Many engineering applications require one to solve complex problems, such as flows containing multi-scale structure in either space or time or both. It is straightforward that high-order methods are always more accurate and efficient than low-order ones for solving smooth problems. For example, spectral element methods can achieve a given level of accuracy with significantly fewer degrees of freedom compared to methods with algebraic convergence rates, e.g., finite difference methods. However, when it comes to complex problems, a high order method should be augmented with, e.g., a level set method or an artificial viscosity method, in order to address the issues caused by either sharp interfaces or shocks in the solution. Complex problems considered in this work are problems with solutions exhibiting multiple scales, i.e., the Stefan problem, nonlinear hyperbolic problems, and problems with smooth solutions but forces exhibiting disparate temporal scales, such as advection, diffusion and reaction processes. Correspondingly, two families of space-time spectral element methods are introduced in order to achieve spectral accuracy in both space and time. The first category of space-time methods are the fully implicit space-time discontinuous Galerkin spectral element methods. In the fully implicit space-time methods, time is treated as an additional dimension, and the model equation is rewritten into a space-time formulation. The other category of space-time methods are specialized for problems exhibiting multiple time scales: multi-implicit space-time spectral element methods are developed. The method of lines approach is employed in the multi-implicit space-time methods. The model is first discretized by a discontinuous spectral element method in space, and the resulting ordinary differential equations are then solved by a new multi-implicit spectral deferred correction method. A novel fully implicit space-time discontinuous Galerkin (DG) spectral element method is presented to solve the Stefan problem in an Eulerian coordinate system. This method employs a level set procedure to describe the time-evolving interface. To deal with the prior unknown interface, a backward transformation and a forward transformation are introduced in the space-time mesh. By combining an Eulerian description with a Lagrangian description, the issue of dealing with the implicitly defined arbitrary shaped space-time elements is avoided. The backward transformation maps the unknown time-varying interface in the fixed frame of reference to a known stationary interface in the moving frame of reference. In the moving frame of reference, the transformed governing equations, written in the space-time framework, are discretized by a DG spectral element method in each space-time slab. The forward transformation is used to update the level set function and then to project the solution in each phase onto the new corresponding time-dependent domain. Two options for calculating the interface velocity are presented, and both options exhibit spectral accuracy. Benchmark tests in one spatial dimension indicate that the method converges with spectral accuracy in both space and time for the temperature distribution and the interface velocity. The interrelation between the interface position and the temperature makes the Stefan problem a nonlinear problem; a Picard iteration algorithm is introduced in order to solve the nonlinear algebraic system of equations and it is found that just a few iterations lead to convergence. We also apply the fully implicit space-time DG spectral element method to solve nonlinear hyperbolic problems. The space-time method is combined with two different approaches for treating problems with discontinuous solutions: (i) space-time dependent artificial viscosity is introduced in order to capture discontinuities/shocks, and (ii) the sharp discontinuity is tracked with space-time spectral accuracy, as it moves through the grid. To capture the discontinuity whose location is initially unknown, an artificial viscosity term is strategically introduced, and the amount of artificial viscosity varies in time within a given space-time slab. It is found that spectral accuracy is recovered everywhere except in the "troublesome element(s)'' where the unresolved steep/sharp gradient exists. When the location of a discontinuity is initially known, a space-time spectrally accurate tracking method has been developed so that the spectral accuracy of the position of the discontinuity and the solution on either side of the discontinuity is preserved. A Picard iteration method is employed to handle nonlinear terms. Within each Picard iteration, a linear system of equations is solved, which is derived from the space-time DG spectral element discretization. Spectral accuracy in both space and time is first demonstrated for the Burgers' equation with a smooth solution. For tests with discontinuities, the present space-time method enables better accuracy at capturing the shock strength in the element containing shock when higher order polynomials in both space and time are used. Moreover, the spectral accuracy of the shock speed and location is demonstrated for the solution of the inviscid Burgers' equation obtained by the shock tracking method, and the sensitivity of the number of Picard iterations to the temporal order is discussed. The dynamics of many physical and biological systems involve two or more processes with a wide difference of characteristic time scales, e.g., problems with advection, diffusion and reaction processes. The computational cost of solving a coupled nonlinear system of equations is expensive for a fully implicit (i.e., "monolithic") space-time method. Thus, we develop another type of a space-time spectral element method, which is referred to as the multi-implicit space-time spectral element method. Rather than coupling space and time together, the method of lines is used to separate the discretization of space and time. The model is first discretized by a discontinuous spectral element method in space and the resulting ordinary differential equations are then solved by a new multi-implicit spectral deferred correction method. The present multi-implicit spectral deferred correction method treats processes with disparate temporal scales independently, but couples them iteratively by a series of deferred correction steps. Compared to lower order operator splitting methods, the splitting error in the multi-implicit spectral deferred correction method is eliminated by exploiting an iterative coupling strategy in the deferred correction procedure. For the spectral element discretization in space, two advective flux reconstructions are proposed: extended element-wise flux reconstruction and non-extended element-wise flux reconstruction. A low-order I-stable building block time integration scheme is introduced as an explicit treatment for the hyperbolic terms in order to obtain a stable and efficient building block for the spectrally accurate space-time scheme along with these two advective flux reconstructions. In other words, we compare the extended element-wise reconstruction with I-stable building block scheme with the non-extended element-wise reconstruction with I-stable building block scheme. Both options exhibit spectral accuracy in space and time. However, the solutions obtained by extended element-wise flux reconstruction are more accurate than those yielded by non-extended element-wise flux reconstruction with the same number of degrees of freedom. The spectral convergence in both space and time is demonstrated for advection-diffusion-reaction problems. Two different coupling strategies in the multi-implicit spectral deferred correction method are also investigated and both options exhibit spectral accuracy in space and time. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / July 5, 2017. / Includes bibliographical references. / Mark Sussman, Professor Co-Directing Dissertation; M. Yousuff Hussaini, Professor Co-Directing Dissertation; William Dewar, University Representative; Nick Cogan, Committee Member; Xiaoming Wang, Committee Member.

In this dissertation, we study three important cases in noncommutative geometry. We first observe the standard noncommutative object, noncommutative torus, in noncommutative motives. We work with the category of holomorphic bundles on a noncommutative torus, which is known to be equivalent to the heart of a nonstandard t-structure on coherent sheaves of an elliptic curve. We then introduce a notion of (weak) t-structure in dg categories. By lifting the nonstandard t-structure to the t-structure that we defined, we find a way of seeing a noncommutative torus in noncommutative motives. By applying the t-structure to a noncommutative torus and describing the cyclic homology of the category of holomorphic bundle on the noncommutative torus, we finally show that the periodic cyclic homology functor induces a decomposition of the motivic Galois group of the Tannakian category generated by the associated auxiliary elliptic curve. In the second case, we generalize the results of Laca, Larsen, and Neshveyev on the GL2-Connes-Marcolli system to the GLn-Connes-Marcolli systems. We introduce and define the GLn-Connes-Marcolli systems and discuss the existence and uniqueness questions of the KMS equilibrium states. Using the ergodicity argument and Hecke pair calculation, we classify the KMS states at different inverse temperatures β. Specifically, we show that in the range of n − 1 < β ≤ n, there exists only one KMS state. We prove that there are no KMS states when β < n − 1 and β ̸= 0, 1, . . . , n − 1,, while we actually construct KMS states for integer values of β in 1 ≤ β ≤ n − 1. For β > n, we characterize the extremal KMS states. In the third case, we push the previous results to more abstract settings. We mainly study the connected Shimura dynamical systems. We give the definition of the essential and superficial KMS states. We further develop a set of arithmetic tools to generalize the results in the previous case. We then prove the uniqueness of the essential KMS states and show the existence of the essential KMS stats for high inverse temperatures. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / July 3, 2017. / CM Systems, Connected Shimura Varieties, Motives, Noncommutative Goemetry, Noncommutative tori / Includes bibliographical references. / Matilde Marcolli, Professor Co-Directing Dissertation; Paolo Aluffi, Professor Co-Directing Dissertation; Eric Chicken, University Representative; Philip Bowers, Committee Member; Kathleen Petersen, Committee Member.

: Let M be a hyperbolic manifold. The SL2(C) character variety of M is essentially the set of all representations ρ : π1(M) → SL2(C) up to trace equivalence. This algebraic set is connected to many geometric properties of the manifold M. We examine the effect of symmetries of M on its character variety. We compute the SL2(C) and PSL2(C) character varieties for an infinite family of two-bridge hyperbolic knots with symmetry. We explore the effect the symmetry has on the character variety and exploit this symmetry to factor the character variety. We then find the geometric genus of both components of the character variety. We compute the SL2(C) character variety for the Borromean ring complement in S^3. Further, we explore how the symmetries effect this character variety. Finally, we prove some general results about the structure of character varieties of links with symmetries. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / July 6, 2017. / Includes bibliographical references. / Kathleen Petersen, Professor Directing Dissertation; Kristine Harper, University Representative; Sam Ballas, Committee Member; Philip Bowers, Committee Member; Eriko Hironaka, Committee Member.

This dissertation proposes a Riemannian approach for computing geodesics for closed curves in elastic shape space. The application of two Riemannian unconstrained optimization algorithms, Riemannian Steepest Descent (RSD) algorithm and Limited-memory Riemannian Broyden-Fletcher-Goldfarb-Shanno (LRBFGS) algorithm are discussed in this dissertation. The application relies on the definition and computation for basic differential geometric components, namely tangent spaces and tangent vectors, Riemannian metrics, Riemannian gradient, as well as retraction and vector transport. The difference between this Riemannian approach to compute closed curve geodesics as well as accurate geodesic distance, the existing Path-Straightening algorithm and the existing Riemannian approach to approximate distances between closed shapes, are also discussed in this dissertation. This dissertation summarizes the implementation details and techniques for both Riemannian algorithms to achieve the most efficiency. This dissertation also contains basic experiments and applications that illustrate the value of the proposed algorithms, along with comparison tests to the existing alternative approaches. It has been demonstrated by various tests that this proposed approach is superior in terms of time and performance compared to a state-of-the-art distance computation algorithm, and has better performance in applications of shape distance when compared to the distance approximation algorithm. This dissertation applies the Riemannian geodesic computation algorithm to calculate Karcher mean of shapes. Algorithms that generate less accurate distances and geodesics are also implemented to compute shape mean. Test results demonstrate the fact that the proposed algorithm has better performance with sacrifice in time. A hybrid algorithm is then proposed, to start with the fast, less accurate algorithm and switch to the proposed accurate algorithm to get the gradient for Karcher mean problem. This dissertation also applies Karcher mean computation to unsupervised learning of shapes. Several clustering algorithms are tested with the distance computation algorithm and Karcher mean algorithm. Different versions of Karcher mean algorithm used are compared with tests. The performance of clustering algorithms are evaluated by various performance metrics. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2018. / June 29, 2018. / Includes bibliographical references. / Kyle A. Gallivan, Professor Co-Directing Dissertation; Pierre-Antoine Absil, Professor Co-Directing Dissertation; Gordon Erlebacher, University Representative; Giray Okten, Committee Member; Mark Sussman, Committee Member.

Determinantal varieties parametrize spaces of matrices of given ranks. The main results of this dissertation are computations of intersection-theoretic invariants of determinantal varieties. We focus on the Chern-Mather and Chern-Schwartz-MacPherson classes, on the characteristic cycles, and on topologically motivated invariants such as the local Euler obstruction. We obtain explicit formulas in both the ordinary and the torus-equivariant setting, and formulate a conjecture concerning the effectiveness of the Chern-Schwartz-MacPherson classes of determinantal varieties. We also prove a vanishing property for the Chern-Schwartz-MacPherson classes of general group orbits. As applications we obtain formulas for the sectional Euler characteristic of determinantal varieties and the microlocal indices of their intersection cohomology sheaf complexes. Moreover, for a close embedding we define the equivariant version of the Segre class and prove an equivariant formula for the Chern-Schwartz-MacPherson classes of hypersurfaces of projective varieties. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2018. / April 11, 2018. / Chern classes, determinantal variety, equivariant Chern classes, local Euler obstruction / Includes bibliographical references. / Paolo Aluffi, Professor Directing Dissertation; Jorge Piekarewicz, University Representative; Ettore Aldrovandi, Committee Member; Kate Petersen, Committee Member; Mark van Hoeij, Committee Member.

We extend Culler and Shalen's work on constructing essential surfaces in 3-manifolds to orbifolds. A consequence of this work is that every valuation on the canonical component that detects an essential surface, detects an essential surface that is preserved by every orientation preserving symmetry on the manifold. This Theorem applies to orientable hyperbolic manifolds, with orientation preserving symmetry group, whose quotient by this group is an orbifold with a flexible cusp, which is the case for most hyperbolic 3-manifolds. We then look at a family of two bridge knots where our theorem shows it is impossible for every essential surface to be detected on the canonical component. We then prove that all surfaces that are preserved by the orientation preserving symmetries of these knots are detected by ideal points on the canonical component of the character variety by calculating the canonical component of the A-polynomial for the family of knots. We then prove that every essential surface in these knot that is not detected on the canonical component of the character variety is detected on another component. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2018. / June 28, 2018. / Character Varieties, Essential Surfaces / Includes bibliographical references. / Kathleen Petersen, Professor Directing Dissertation; Dennis Duke, University Representative; Wolfgang Heil, Committee Member; Samuel Ballas, Committee Member.

It is known that the category of Waldhausen categories is a closed symmetric multicategory and algebraic K-theory is a multifunctor from the category of Waldhuasen categories to the category of spectra. By assigning to any Waldhausen category the fundamental groupoid of the 1-type of its K-theory spectrum, we get a functor from the category of Waldhausen categories to the category of Picard groupoids, since stable 1-types are classified by Picard groupoids. We prove that this functor is a multifunctor to a corresponding multicategory of Picard groupoids. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2018. / April 10, 2018. / closed structure, K-theory, multifunctor, Picard groupoid, Stable quadratic modules, Waldhausen / Includes bibliographical references. / Ettore Aldrovandi, Professor Directing Dissertation; John Rawling, University Representative; Amod Agashe, Committee Member; Paolo Aluffi, Committee Member; Kate Petersen, Committee Member; Mark van Hoeij, Committee Member.

Our aim is to study the affine dimension of some smooth manifolds. In Chapter 1, we review the notions of Minkowski and Hausdorff dimension, and compare them with the lesser studied affine dimension. In Chapter 2, we focus on understanding the affine dimension of curves. In Section 2.1, we review the existing results for the affine dimension of a strictly convex curve in the plane, and in Section 2.2, we classify the smooth curves in ℝn based on affine dimension. In Chapter 3, we classify the smooth hypersurfaces in ℝ3 with non-negative Gaussian curvature based on affine dimension, and in Chapter 4 we provide a lower and upper bound for the affine dimension of smooth, convex hypersurfaces in ℝn. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2018. / April 10, 2018. / Includes bibliographical references. / Richard Oberlin, Professor Directing Dissertation; Mike Ormsbee, University Representative; Alexander Reznikov, Committee Member; Martin Bauer, Committee Member.

Quantifying shape variation within a group of individuals, identifying morphological contrasts between populations and categorizing these groups according to morphological similarities and dissimilarities are central problems in developmental evolutionary biology and genetics. In this dissertation, we present an approach to optimal shape categorization through the use of a new family of metrics for shapes represented by a finite collection of landmarks. We develop a technique to identify metrics that optimally differentiate and categorize shapes using Monte Carlo based optimization methods. We discuss the theory and the practice of the method and apply it to the categorization of 62 mice offsprings based on the shape of their skull. We also create a taxonomic classification tree for multiple species of fruit flies given the shape of their wings. The results of these experiments validate our method. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2018. / January 16, 2018. / Global Optimization, Metric Learning, Monte Carlo Optimization, Quasi Monte Carlo, Statistical Shape Analysis, Stochastic Optimization / Includes bibliographical references. / Washington Mio, Professor Co-Directing Dissertation; Giray Okten, Professor Co-Directing Dissertation; Sudhir Aggarwal, University Representative; Nick Cogan, Committee Member; Harsh Jain, Committee Member.