In the early 70's Parshin introduced his notion of the multidimensional residues of meromorphic top-forms on algebraic varieties. Parshin's theory is a generalization of the classical one-dimensional residue theory. The main difference between the Parshin's definition and the one-dimensional case is that in higher dimensions one computes the residue not at a point but at a complete flag of irreducible subvarieties. Parshin, Beilinson, and Lomadze also proved the Reciprocity Law for residues: if one fixes all elements of the flag, except for one, and consider all possible choices of the missing element, then only finitely many of these choices give non-zero residues, and the sum of these residues is zero. Parshin's constructions are completely algebraic. In fact, they work in very general settings, not only over complex numbers. However, in the complex case one would expect a more geometric variant of the theory. In my thesis I study Parshin residues from the geometric point of view. In particular, the residue is expressed in terms of the integral over a smooth cycle. Parshin-Lomadze Reciprocity Law for residues in the complex case is proved via a homological relation on these cycles. The thesis consists of two parts. In the first part the theory of Leray coboundary operators for stratified spaces is developed. These operators are used to construct the cycle and prove the homological relation. In the second part resolution of singularities techniques are applied to study the local geometry near a complete flag of subvarieties.

In the early 70's Parshin introduced his notion of the multidimensional residues of meromorphic top-forms on algebraic varieties. Parshin's theory is a generalization of the classical one-dimensional residue theory. The main difference between the Parshin's definition and the one-dimensional case is that in higher dimensions one computes the residue not at a point but at a complete flag of irreducible subvarieties. Parshin, Beilinson, and Lomadze also proved the Reciprocity Law for residues: if one fixes all elements of the flag, except for one, and consider all possible choices of the missing element, then only finitely many of these choices give non-zero residues, and the sum of these residues is zero. Parshin's constructions are completely algebraic. In fact, they work in very general settings, not only over complex numbers. However, in the complex case one would expect a more geometric variant of the theory. In my thesis I study Parshin residues from the geometric point of view. In particular, the residue is expressed in terms of the integral over a smooth cycle. Parshin-Lomadze Reciprocity Law for residues in the complex case is proved via a homological relation on these cycles. The thesis consists of two parts. In the first part the theory of Leray coboundary operators for stratified spaces is developed. These operators are used to construct the cycle and prove the homological relation. In the second part resolution of singularities techniques are applied to study the local geometry near a complete flag of subvarieties.

Iterative CT reconstruction algorithms are superior to the standard convolution backpropagation (CBP) methods when reconstructing from a small number of views (hence less radiation), but are computationally costly. To reduce the execution time, this work implements and tests a parallel approach to iterative algorithms using a cluster of workstations, which is a low cost system found in many offices and non-academic sites. A previous implementation showed little speedup because of the significant cost of inter-processor communication. In this thesis, several data partitioning methods are examined, including some image tiling methods that exploit the spatial locality demonstrated by local CT. Using these methods, computation can proceed locally, without the need for inter-processor communication during every iteration. A relative speedup of up to 17 times is obtained using 25 processors, demonstrating that good performance can be obtained running computationally intensive CT reconstruction algorithms on distributed memory hardware. / October 2007

This is the integral text of my thesis. The first part is an expanded version of "Riemann-Roch theorems for Deligne-Mumford stacks", where I deal with Artin stacks over general bases. In the second part, I prove some Riemann-Roch statment for D-modules on Deligne-Mumford stacks, and I also consider the problem of algebraization of analytic stacks.

Cryptography is defined to be the practice and studying of hiding information and is used in applications present today; examples include the security of ATM cards and computer passwords ([34]). In order to transform information to make it unreadable, one needs a series of algorithms. Many of these algorithms are based on elliptic curves because they require fewer bits. To use such algorithms, one must find the rational points on an elliptic curve. The study of Algebraic Number Theory, and in particular, rare objects known as power bases, help determine what these rational points are. With such broad applications, studying power bases is an interesting topic with many research opportunities, one of which is given below. There are many similarities between Cyclic and Dihedral fields of prime degree; more specifically, the structure of their field discriminants is comparable. Since the existence of power bases (i.e. monogenicity) is based upon finding solutions to the index form equation - an equation dependant on field discriminants - does this imply monogenic properties of such fields are also analogous? For instance, in [14], Marie-Nicole Gras has shown there is only one monogenic cyclic field of degree 5. Is there a similar result for dihedral fields of degree 5? The purpose of this thesis is to show that there exist infinitely many monogenic dihedral quintic fields and hence, not just one or finitely many. We do so by using a well- known family of quintic polynomials with Galois group D₅. Thus, the main theorem given in this thesis will confirm that monogenic properties between cyclic and dihedral quintic fields are not always correlative.

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This thesis explores two infinite families of self-complementary arc-transitive graphs: the familiar Paley graphs and the newly discovered Peisert graphs. After studying both families, we examine a result of Peisert which proves the Paley and Peisert graphs are the only self-complementary arc transitive graphs other than one exceptional graph. Then we consider other families of graphs which share many properties with the Paley and Peisert graphs. In particular, we construct an infinite family of self-complementary strongly regular graphs from affine planes. We also investigate the pseudo-Paley graphs of Weng, Qiu, Wang, and Xiang. Finally, we prove a lower bound on the number of maximal cliques of certain pseudo-Paley graphs, thereby distinguishing them from Paley graphs of the same order.

One of the fundamental problems in Algebraic Geometry is to study solutions to certain systems of polynomial equations in several variables, or in other words, find rational points on a given variety which is defined by equations. In this paper, we discuss the existence of del Pezzo surface of degree 1 and 2 with a unique rational point over any finite field [Special characters omitted.] , and we will give a lower bound on the number of rational points to each q. Furthermore, we will give explicit equations of del Pezzo surfaces with a unique rational point. Also we will discuss the rationality property of the del Pezzo surfaces especially in lower degrees.

This thesis explores two infinite families of self-complementary arc-transitive graphs: the familiar Paley graphs and the newly discovered Peisert graphs. After studying both families, we examine a result of Peisert which proves the Paley and Peisert graphs are the only self-complementary arc transitive graphs other than one exceptional graph. Then we consider other families of graphs which share many properties with the Paley and Peisert graphs. In particular, we construct an infinite family of self-complementary strongly regular graphs from affine planes. We also investigate the pseudo-Paley graphs of Weng, Qiu, Wang, and Xiang. Finally, we prove a lower bound on the number of maximal cliques of certain pseudo-Paley graphs, thereby distinguishing them from Paley graphs of the same order.

It is not hard to see that the number of equiangular lines in a complex space of dimension $d$ is at most $d^{2}$. A set of $d^{2}$ equiangular lines in a $d$-dimensional complex space is of significant importance in Quantum Computing as it corresponds to a measurement for which its statistics determine completely the quantum state on which the measurement is carried out. The existence of $d^{2}$ equiangular lines in a $d$-dimensional complex space is only known for a few values of $d$, although physicists conjecture that they do exist for any value of $d$. The main results in this thesis are: \begin{enumerate} \item Abelian covers of complete graphs that have certain parameters can be used to construct sets of $d^2$ equiangular lines in $d$-dimen\-sion\-al space; \item we exhibit infinitely many parameter sets that satisfy all the known necessary conditions for the existence of such a cover; and \item we find the decompose of the space into irreducible modules over the Terwilliger algebra of covers of complete graphs. \end{enumerate} A few techniques are known for constructing covers of complete graphs, none of which can be used to construct covers that lead to sets of $d^{2}$ equiangular lines in $d$-dimensional complex spaces. The third main result is developed in the hope of assisting such construction.