Congruences in modular, Jacobi, Siegel, and mock modular forms with applications

We study congruences in the coefficients of modular and other automorphic forms. Ramanujan famously found congruences for the partition function like p(5n+4) = 0 mod 5. For a wide class of modular forms, we classify the primes for which there can be analogous congruences in the coefficients of the Fourier expansion. We have several applications. We describe the Ramanujan congruences in the counting functions for overparitions, overpartition pairs, crank differences, and Andrews' two-coloured generalized Frobenius partitions. We also study Ramanujan congruences in the Fourier coefficients of certain ratios of Eisenstein series. We also determine the exact number of holomorphic modular forms with Ramanujan congruences when the weight is large enough. In a chapter based on joint work with Olav Richter, we study Ramanujan congruences in the coefficients of Jacobi forms and Siegel modular forms of degree two. Finally, the last chapter contains a completely unrelated result about harmonic weak Maass forms.

Minimalist Hardware Architectures for Agent Tracking and Guidance

The philosophy of minimalism in robotics promotes gaining an understanding of sensing and computational requirements for solving a task. This minimalist approach lies in contrast to the common practice of first taking an existing sensory motor system, and only afterwards determining how to apply the robotic system to the task. While it may seem convenient to simply apply existing hardware systems to the task at hand, this design philosophy often proves to be wasteful in terms of energy consumption and cost, along with unnecessary complexity and decreased reliability. While impressive in terms of their versatility, complex robots such as the PR2 (which cost hundreds of thousands of dollars) are impractical for many common applications. Instead, if a specific task is required, sensing and computational requirements can be determined specific to that task, and a clever hardware implementation can be built to accomplish the task. Since this minimalist hardware would be designed around accomplishing the specified task, significant reductions in hardware complexity can be obtained. This can lead to huge advantages in battery life, cost, and reliability. Even if cost is of no concern, battery life is often a limiting factor in many applications. Thus, a minimalist hardware system is critical in achieving the system requirements. In this thesis, we will discuss an implementation of a counting, tracking, and actuation system as it relates to ergodic bodies to illustrate a minimalist design methodology.

Carbon-carbon bond formation methods for the preparation of shape-persistent architectures

Carbon-rich, conjugated organic scaffolding is a popular basis for functional materials, especially for electronic and photonic applications. However, synthetic methods for generating these types of materials lack diversity and, in many cases, efficiency; the insistence of investigators focusing on the properties of the end product, rather than the process in which it was created, has led to the current state of the relatively homogeneous synthetic chemistry of functional organic materials. Because of this, there is plenty of room for improvement at the most basic level. Problems endemic to the preparation of carbon-rich scaffolding can, in many cases, be solved with modern advances in synthetic methodology. We seek to apply this synthesis-focused paradigm to solve problems in the preparation of carbon-rich scaffolds. Herein, the development and utilization of three methodologies: iridium-catalyzed arene C-H borylation; zinc- mediated alkynylations; and Lewis acid promoted Mo nitride-alkyne metathesis, are presented as improvements for the preparation of carbon-rich architectures. In addition, X-ray crystallographic analysis of two classes of compounds are presented. First, an analysis of carbazole-containing arylene ethynylene macrocycles showcases the significance of alkyl chain identity on solid-state morphology. Second, a class of rigid zwitterionic metal-organic compounds display an unusual propensity to crystallize in the absence of inversion symmetry. Hirshfeld surface analysis of these crystalline materials demonstrates that subtle intermolecular interactions are responsible for the overall packing motifs in this class of compounds.