IN SEARCH OF THE DIVINE: SPIRITUALITY AND PRAYER IN DIVERSE VIOLIN REPERTOIRE FROM BIBER TO IVES

Composers from all eras and of all ethnicities explore spirituality and prayer by using one or a combination of the following ideas: having a spiritual concept in mind when composing certain pieces, quoting hymns, being influenced by their own personal beliefs, or portraying spiritual figures and ideas in their works. Some musical works are inspired by spirituality; others, as in the case of Bloch's Nigun, even serve as prayers themselves. These recitals gave me the opportunity to approach a wide variety of musical styles while discovering my own mode for expression. The unaccompanied violin works throughout this project trace a distinct lineage from the baroque to the twentieth century. Biber's appendix to the Rosary Sonatas, the Passacaglia for solo violin, is a crucial predecessor to Bach's monumental Chaconne. Eugene Ysaye was inspired to write the Six Sonatas, Op. 27 after he attended a performance of Bach's Sonatas and Partitas given by Josef Szigeti. Ysaye's second solo sonata blatantly quotes Bach's Partita No.3 in E major throughout the first movement. Every movement also contains quotations from and variations on the plainchant Dies Irae. Although each of the solo violin works presented in this project may be viewed as virtuosic concert pieces, each piece allows the performer to transcend the technical hurdles-and perhaps even utilize them-to serve a higher, artistic and spiritual purpose while alone on the concert stage. Each of the sonata works in this project requires a close, equal collaboration between violinist and pianist, rather than displaying the violinist as soloist.

OPERA IN AMERICA: MUSIC OF, BY, AND FOR THE PEOPLE­: A PERFORMANCE DISSERTATION

Opera in America: Music of, by, and for the people is a study of the relationship between American popular culture and opera in the United States. Four performance projects demonstrate the on-going exchange between the operatic community-including its composer, singers, and patrons-and the country's popular entertainment industry with its broad audience base. Numerous examples of artistic cross pollination between lowbrow and highbrow music will illustrate the artistic and social consequences created by this artistic amalgamation. Program #1, By George! By Ira! By Gershwin!, is a retrospective of Gershwin's vocal music representing a blending of popular and serious music in both style and form. The concert includes selections from Porgy and Bess, a work considered by many musicologists as the first American opera. Program #2, Shadowboxer, is a premiere performance of an opera by Frank Proto and John Chenault. For this newly commissioned work, I serve as Assistant Director to Leon Major. Shadowboxer provides a clear example of opera utilizing popular culture both musically and dramatically to tell the true story of American hero and legendary boxer, Joe Louis. Program #3, Just a Song at Twilight, is an original theatrical music piece featuring music, letters, diaries, and journals of the Gilded Age, an era when opera was synonymous with popular entertainment. Special attention is focused on tum-of­ the-century singers who performed in both opera and vaudeville. Program #4 is a presentation of Dominick Argento's Miss Manners on Music and illustrates the strong relationship that can exist between opera and American popular entertainment. Originally conceived as a song cycle, I have staged the work as a one-act opera sung and acted by soprano Carmen Balthrop. This piece is based on the writings of pop icon and newspaper columnist Judith Martin, otherwise known as Miss Manners. All four performances are recorded in audio and video formats.

Sparse Signal Representation in Digital and Biological Systems

Theories of sparse signal representation, wherein a signal is decomposed as the sum of a small number of constituent elements, play increasing roles in both mathematical signal processing and neuroscience. This happens despite the differences between signal models in the two domains. After reviewing preliminary material on sparse signal models, I use work on compressed sensing for the electron tomography of biological structures as a target for exploring the efficacy of sparse signal reconstruction in a challenging application domain. My research in this area addresses a topic of keen interest to the biological microscopy community, and has resulted in the development of tomographic reconstruction software which is competitive with the state of the art in its field. Moving from the linear signal domain into the nonlinear dynamics of neural encoding, I explain the sparse coding hypothesis in neuroscience and its relationship with olfaction in locusts. I implement a numerical ODE model of the activity of neural populations responsible for sparse odor coding in locusts as part of a project involving offset spiking in the Kenyon cells. I also explain the validation procedures we have devised to help assess the model's similarity to the biology. The thesis concludes with the development of a new, simplified model of locust olfactory network activity, which seeks with some success to explain statistical properties of the sparse coding processes carried out in the network.

Existence and weak-strong uniqueness for the Navier-Stokes-Smoluchowski system over moving domains

This dissertation concerns the well-posedness of the Navier-Stokes-Smoluchowski system. The system models a mixture of fluid and particles in the so-called bubbling regime. The compressible Navier-Stokes equations governing the evolution of the fluid are coupled to the Smoluchowski equation for the particle density at a continuum level. First, working on fixed domains, the existence of weak solutions is established using a three-level approximation scheme and based largely on the Lions-Feireisl theory of compressible fluids. The system is then posed over a moving domain. By utilizing a Brinkman-type penalization as well as penalization of the viscosity, the existence of weak solutions of the Navier-Stokes-Smoluchowski system is proved over moving domains. As a corollary the convergence of the Brinkman penalization is proved. Finally, a suitable relative entropy is defined. This relative entropy is used to establish a weak-strong uniqueness result for the Navier-Stokes-Smoluchowski system over moving domains, ensuring that strong solutions are unique in the class of weak solutions.

Spectral Frame Analysis and Learning through Graph Structure

This dissertation investigates the connection between spectral analysis and frame theory. When considering the spectral properties of a frame, we present a few novel results relating to the spectral decomposition. We first show that scalable frames have the property that the inner product of the scaling coefficients and the eigenvectors must equal the inverse eigenvalues. From this, we prove a similar result when an approximate scaling is obtained. We then focus on the optimization problems inherent to the scalable frames by first showing that there is an equivalence between scaling a frame and optimization problems with a non-restrictive objective function. Various objective functions are considered, and an analysis of the solution type is presented. For linear objectives, we can encourage sparse scalings, and with barrier objective functions, we force dense solutions. We further consider frames in high dimensions, and derive various solution techniques. From here, we restrict ourselves to various frame classes, to add more specificity to the results. Using frames generated from distributions allows for the placement of probabilistic bounds on scalability. For discrete distributions (Bernoulli and Rademacher), we bound the probability of encountering an ONB, and for continuous symmetric distributions (Uniform and Gaussian), we show that symmetry is retained in the transformed domain. We also prove several hyperplane-separation results. With the theory developed, we discuss graph applications of the scalability framework. We make a connection with graph conditioning, and show the in-feasibility of the problem in the general case. After a modification, we show that any complete graph can be conditioned. We then present a modification of standard PCA (robust PCA) developed by Cand\`es, and give some background into Electron Energy-Loss Spectroscopy (EELS). We design a novel scheme for the processing of EELS through robust PCA and least-squares regression, and test this scheme on biological samples. Finally, we take the idea of robust PCA and apply the technique of kernel PCA to perform robust manifold learning. We derive the problem and present an algorithm for its solution. There is also discussion of the differences with RPCA that make theoretical guarantees difficult.

Multiscale Modeling and Simulation of Stepped Crystal Surfaces

A primary goal of this dissertation is to understand the links between mathematical models that describe crystal surfaces at three fundamental length scales: The scale of individual atoms, the scale of collections of atoms forming crystal defects, and macroscopic scale. Characterizing connections between different classes of models is a critical task for gaining insight into the physics they describe, a long-standing objective in applied analysis, and also highly relevant in engineering applications. The key concept I use in each problem addressed in this thesis is coarse graining, which is a strategy for connecting fine representations or models with coarser representations. Often this idea is invoked to reduce a large discrete system to an appropriate continuum description, e.g. individual particles are represented by a continuous density. While there is no general theory of coarse graining, one closely related mathematical approach is asymptotic analysis, i.e. the description of limiting behavior as some parameter becomes very large or very small. In the case of crystalline solids, it is natural to consider cases where the number of particles is large or where the lattice spacing is small. Limits such as these often make explicit the nature of links between models capturing different scales, and, once established, provide a means of improving our understanding, or the models themselves. Finding appropriate variables whose limits illustrate the important connections between models is no easy task, however. This is one area where computer simulation is extremely helpful, as it allows us to see the results of complex dynamics and gather clues regarding the roles of different physical quantities. On the other hand, connections between models enable the development of novel multiscale computational schemes, so understanding can assist computation and vice versa. Some of these ideas are demonstrated in this thesis. The important outcomes of this thesis include: (1) a systematic derivation of the step-flow model of Burton, Cabrera, and Frank, with corrections, from an atomistic solid-on-solid-type models in 1+1 dimensions; (2) the inclusion of an atomistically motivated transport mechanism in an island dynamics model allowing for a more detailed account of mound evolution; and (3) the development of a hybrid discrete-continuum scheme for simulating the relaxation of a faceted crystal mound. Central to all of these modeling and simulation efforts is the presence of steps composed of individual layers of atoms on vicinal crystal surfaces. Consequently, a recurring theme in this research is the observation that mesoscale defects play a crucial role in crystal morphological evolution.