The distribution of values of arithmetic functions in residue classes is a problem of significant interest in elementary, analytic and combinatorial number theory. Much w[...]

In this expository thesis, the focus will be on bounds for point-plane incidences. We begin by stating and proving two results about point-line incidences in R^2. We then[...]

In this dissertation we discuss two of our recent papers. In their 1984 paper, Lasota, Li, and Yorke presented an argument that if $S:[0,1]\to [0,1]$ is piecewise $C^2$ w[...]

This dissertation studies the representation theory of classical Lie superalgebras from a categorical point of view. Given a classical Lie superalgebra $\fg = \fg_{\bar 0[...]

This dissertation is about two different aspects of symplectic geometry. The first part is about the Hofer-Zehnder conjecture. Arnold conjecture says that the number of $[...]

In this paper, we explore properties of the ring of polynomials P(R, R) over a finite ring R. We examine the kernel of the evaluation map E to find #P(R,R), first for whe[...]

This thesis discusses Christoffel words, a subset of the free monoid on two letters, and algorithms which can produce them. It covers Christoffel morphisms, standard fact[...]

In this work,we use splines of arbitrary degree and arbitrary smoothness to .nd approximations of the 3D Navier-Stokes equations in velocity-pressure formulation. We star[...]

A discrete tensegrity framework can be thought of as a graph in Euclidean n-spacewhere each edge is of one of three types: an edge with a fixed length (bar) or an edgewit[...]

In this work, we generalize Bertrams work on rank two vector bundles on a smooth irreducible projective curve to an irreducible singular curve C with only one node as sin[...]

AbstractLetH(x) =Pnx`(n) n6 2x. Motivated by a conjecture of P. Erdos, Y.-K. Lau developed a new method and proved #f1nT:H(n)H(n+ 1)<0g T: We consider arithmetic [...]

We study properties of spherical Bernstein-Bezier splines. Algorithms for practical implementation of the global splines are presented for a homogeneous case as well as a[...]

Boe, Kujawa and Nakano recently investigated relative cohomology for classical Lie superalgebras and developed a theory of support varieties. The dimensions of these sup[...]

Every transfer of information is subject to some probability of error. Codingtheory addresses the problems of detecting and correcting the errors which occurin these tran[...]

In general little is known about the structure of H(), the higher right-derived functors of the induction from a Borel subgroup of a simple reductive algebraic group (not[...]

The tokamak has been the preferred design for fusion test reactors since the 70's. While the tokamak holds promise to be one of the most efficient in the field of fusion [...]

We describe work towards a compact moduli space of elliptic K3 surfaces with marked divisor given by a small multiple of the sum of rational curves in an ample class of s[...]

The ring of algebraic cycles modulo algebraic equivalence on an abelian variety is an interesting and mysterious object. When the abelian variety is the Jacobian of a sm[...]

For $ngeq 0$, we study a formalism for approximating rational homotopy types of differential graded algebras with cohomological $n$-equivalences, algebra morphisms induci[...]

The Riemann Mapping Theorem guarantees that the upper half plane is conformallyequivalent to the interior domain determined by any polygon. Schwarz-Christoffel transforma[...]

In this paper we will introduce the reader to projective geometry and what it means for two figures to be projectively equivalent. This leads to a discussion on the proj[...]

This thesis is organized as an introduction to the Nash problem on arc spaces and asurvey of all (known to the author) results, which give a positive or negative answer t[...]