A recent construction of Hacking relates the classification of stable vector bundles on a surface of general type with geometric genus 0 and the boundary of the moduli space of deformations of the surface. The goal of this thesis is to analyze this relation for Godeaux surfaces. To do this, first, we give a description of some boundary components of the moduli space of Godeaux surfaces. Second, we explicitly construct certain exceptional vector bundles of rank 2 on Godeaux surfaces, stable with respect to the canonical class. Finally, we examine the relation between such boundary components and exceptional vector bundles of rank two on Godeaux surfaces.

In 2002, Giroux showed that every contact structure had a corresponding open book decomposition. This was the converse to a previous construction of Thurston and Winkelnkemper, and made open books a vital tool in the study of contact three-manifolds. We extend these results to contact orbifolds, i.e. spaces that are locally diffeomorphic to the quotient of a contact manifold and a compatible finite group action. This involves adapting some of the main concepts and constructions of three dimensional contact geometry to the orbifold setting.

In this thesis, we examine the relationship between Weyl group multiple Dirichlet series over the rational function field and their p-parts, which we define using the Chinta–Gunnells method [10]. We show that these series may be written as the finite sum of their p-parts (after a certain variable change), with “multiplicities” that are character sums. Because the p-parts and global series are closely related, this result follows from a series of local results concerning the p-parts. In particular, we give an explicit recurrence relation on the coefficients of the p-parts, which allows us to extend the results of Chinta, Friedberg, and Gunnells [9]. Additionally, we show that the p-parts of Chinta and Gunnells [10] agree with those constructed using the crystal graph technique of Brubaker, Bump, and Friedberg [4,5] (in the cases when both constructions apply).

In this thesis we study the central limit theorem (CLT) for nonuniformly hyperbolic dynamical systems. We examine cases in which polynomial decay of correlations leads to a CLT with a non-standard scaling factor. We also formulate an explicit expression for the the diffusion constant σ in situations where a return time function on the system is a certain class of supermartingale. We then demonstrate applications by exhibiting the CLT for the return time function in four classes of dynamical billiards, including one previously unproven case, the skewed stadium, as well as for the linked twist map. Finally, we introduce a new class of billiards which we conjecture are ergodic, and we provide numerical evidence to support that claim.

In this thesis, we give a topological interpretation of knot contact homology, by considering intersections of a particular class of chains of open strings with the knot itself. In doing so, we provide evidence toward a differential graded algebra structure on the algebra generated by chains of open strings.

This thesis investigates two possible versions of a "spectral curve" construction for compact constant mean curvature (CMC) surfaces of genus g > 1 in [special characters omitted]. The first version uses the holonomy spectral curve which was originally formulated for tori in [special characters omitted]. In order to make sense of the definition of this curve for a higher genus surface M, we must assume that the holonomy is abelian, and in this case it is shown that M must be a branched immersion factoring holomorphically through a CMC torus which can be located naturally in the Jacobian of M. The second version uses a curve defined as a double cover of M branched at the zeroes of the Hopf differential Q which coincides with that used originally by Hitchin to analyze the moduli space of stable bundles over M. We propose a method of defining a CMC immersion of this curve which has abelian holonomy and therefore, by the earlier result, factors through a naturally defined CMC torus. Along with the non-abelian holonomy of a certain meromorphic connection around the zeroes of Q, this data might provide effective moduli for M.

Fix an integer d>0. In 2008, Chantal David and Tom Weston showed that, on average, an elliptic curve over Q picks up a nontrivial p-torsion point defined over a finite extension K of the p-adics of degree at most d for only finitely many primes p. This dissertation is an extension of that work, investigating the frequency with which a principally polarized abelian surface A over Q with real multiplication by Q adjoin a squared-root of 5 has a nontrivial p-torsion point defined over K. Averaging by height, the main result shows that A picks up a nontrivial p-torsion point over K for only finitely many p. The proof of our main theorem primarily rests on three lemmas. The first lemma uses the reduction-exact sequence of an abelian surface defined over an unramified extension K of Q p to give a mod p2 condition for detecting when A has a nontrivial p-torsion point defined over K. The second lemma employs crystalline Dieudonné theory to count the number of isomorphism classes of lifts of abelian surfaces over Fp to Z/p2 that satisfy the condition from our first lemma. Finally, the third lemma addresses the issue of the assumption in the first lemma that K is an unramified extension of Qp. Specifically, it shows that if A has a nontrivial p-torsion point over a ramified extension K of Qp and p - 1 > d then this p-torsion point is actually defined over the maximal unramified subextension of K. We then combine these algebraic results to reduce the main analytic calculation to a series of straightforward estimates.

<p> A theorem by Kurtz on convergence of Markov jump processes is presented as it relates to the use of the chemical master equation. Necessary mathematical background in the theory of stochastic processes is developed, as well as requirements of the mathematical model necessitated by results in the physical sciences. Applicability and usefulness of the master equation for this type of combinatorial model in chemistry is discussed, as well as analytical connections and modern applications in multiple research fields.</p>

<p> In previous papers about searches on star graphs several patterns have been made apparent; the speed up only occurs when graphs are ''tuned'' so that their time step operators have degenerate eigenvalues, and only certain initial states are effective. More than that, the searches are never faster than order square root of N time. In this thesis the problem is defined rigorously, the causes for all of these patterns are identified, sufficient and necessary conditions for quadratic-speed searches for any connected subgraph are demonstrated, the tolerance of these conditions is investigated, and it is shown that (unfortunately) we can do no better than order square root of N time. Along the way, a useful formalism is established that may be useful in future work involving highly symmetric graphs.</p><p> The tools and techniques so derived are then used to demonstrate that tree graphs can be used for the computation of Boolean functions. The philosophy of Farhi's work on the continuous-time NAND tree is applied to a discrete-time walk with any (AND, OR, NAND, or NOR) gate at each vertex. Tentative results show that the vast majority of possible Boolean functions on <i>N</i> bits can be calculated in order square root of N time.</p>

<p>Network analysis provides tools for addressing fundamental applications in graphs such as webpage ranking, protein-function prediction, and product categorization and recommendation. As real-world networks grow to have millions of nodes and billions of edges, the scalability of network analysis algorithms becomes increasingly important. Whereas many standard graph algorithms rely on matrix-vector operations that require exploring the entire graph, this thesis is concerned with graph algorithms that are local (that explore only the graph region near the nodes of interest) as well as the localized behavior of global algorithms. We prove that two well-studied matrix functions for graph analysis, PageRank and the matrix exponential, stay localized on networks that have a skewed degree sequence related to the power-law degree distribution common to many real-world networks. Our results give the first theoretical explanation of a localization phenomenon that has long been observed in real-world networks. We prove our novel method for the matrix exponential converges in sublinear work on graphs with the specified degree sequence, and we adapt our method to produce the first deterministic algorithm for computing the related heat kernel diffusion in constant-time. Finally, we generalize this framework to compute any graph diffusion in constant time. </p>