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Some of the material in is restricted to members of the community. By logging in, you may be able to gain additional access to certain collections or items. If you have questions about access or logging in, please use the form on the Contact Page.
This dissertation presents studies of effective methods in two main areas of algebraic geometry: intersection theory and characteristic classes, and combinatorial algebraic geometry. We begin in chapter 2 by giving an effective algorithm...
In this thesis we present an algorithm that finds closed form solutions for homogeneous linear recurrence equations. The key idea is transforming an input operator Linp to an operator Lg with known solutions. The main problem of this...
We present three algorithms to reduce homogeneous linear differential equations to their simplest form. Factoring a differential operator reduces a differential equation L(y)=0 to equations of minimal order, but this is not the only...
We investigate some algebro-geometric aspects of several families of elliptic fibrations relevant for F-theory model building along with some physical applications. In particular, we compute topological invariants of elliptic fibrations...
Let $N$ be a positive integer. We first discuss a method for computing intersection numbers between subspaces of $S_{2}(Gamma_{0}(N), C)$. Then we present a new method for computing a basis of q-expansions for $S_{2}(Gamma_{0}(N), Q)$, ...
To solve globally bounded order $3$ linear differential equations with rational function coefficients, this thesis introduces a partial $_3F_2$-solver (Section~\ref{3F2 type solution}) and $F_1$-solver (Chapter~\ref{F1 solver}), where $...
This thesis studies the minimal Mahler measure of the primitive integral elements of a field K. The Mahler measure of a polynomial with integer coefficients is the product of the moduli of all of the all roots outside the unit circle and...
This dissertation finds some partial results in support of two positivity conjectures regarding the Chern-Schwartz-MacPherson (CSM) classes of graph hypersurfaces (conjectured by Aluffi and Marcolli) and Schubert varieties (conjectured...
The goal of this thesis is to develop rigorous foundations for octonionic structures for their potential use in theoretical physics. I begin by defining the octonions and exploring some of their algebraic properties. In particular, they...
This paper presents an analysis of the capital needs, needed return on capital, and optimum reinsurance retention for insurance companies, all in the context where claims are either paid out or known with certainty within or soon after...
A linear differential equation with rational function coefficients has a Bessel type solution when it is solvable in terms of Bessel functions, change of variables, algebraic operations and exponential integrals. For second order...
This thesis addresses the role of topology and geometry in quantum gravity. A major topic will be how inequivalent differentiable structures (exotic smoothness) can play a physically significant role in both semiclassical gravity and...
We work over an algebraically closed ground field of characteristic zero. The group of PGL(4) acts naturally on the projective space P^N parameterizing surfaces of a given degree d in P^3. The orbit of a surface under this action is the...
The Shafarevich-Tate group of an elliptic curve is an important invariant of the curve whose conjectural finiteness can sometimes be used to determine the rank of the curve. The second part of the Birch and Swinnerton-Dyer (BSD)...
This thesis presents three algorithms each of which returns a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gt-transformations). The first and third algorithm...
In this thesis we develop a few algorithms that are useful for factoring linear recurrence operators. We approach the factorization problem from three directions. First, considering reduction modulo a prime leads to the study of...
The study of periods arose in number theory and algebraic geometry, periods are interesting transcendental numbers like multiple zeta values, on the other hand periods are integrals of algebraic differential forms over domains described...
This thesis introduces two new algorithms to find hypergeometric solutions of second order regular singular differential operators with rational function or polynomial coefficients. Algorithm 3.2.1 searches for solutions of type: exp(∫ r...
Characteristic Classes and Local Invariants of Determinantal Varieties and a Formula for Equivariant Chern-Schwartz-MacPherson Classes of Hypersurfaces
Determinantal varieties parametrize spaces of matrices of given ranks. The main results of this dissertation are computations of intersection-theoretic invariants of determinantal varieties. We focus on the Chern-Mather and Chern...
Let L be a linear ordinary differential equation with coefficients in C(x). This thesis presents algorithms to solve L in closed form. The key part of this thesis is 2-descent method, which is used to reduce L to an equation that is...
This thesis consists of three independent projects in the field of algebraic geometry. The first of these is the focus of Chapter 2. There we define a family of integer sequences we refer to as the family of secant indices of projective...
This thesis presents an algorithm for factoring polynomials over the rationals which follows the approach of the van Hoeij algorithm. The key theoretical novelty in our approach is that it is et up in a way that will make it possible to...
In this thesis we generalize results from classical Coxeter systems to mixed-sign Coxeter systems which are denoted by a triple (W, S, B)consisting of a reflection group W, a distinguished set of generators Sfor the group for W, and a...
The thesis work we present here focuses on solving a conjecture raised by Aluffi about Chern-Schwartz-MacPherson classes. Let $X$ be a nonsingular variety defined over an algebraically closed field $k$ of characteristic $0$, $D$ a...
Let L be a second order linear differential equation with rational function coefficients. We want to find a solution (if that exists) of L in terms of 2F1-hypergeometric function. This thesis presents two algorithms to find such solution...
Clifford analysis is seen as the higher dimensional analogue of complex analysis. This includes a rich study of Clifford algebras and, in particular, monogenic functions, or Clifford-valued functions that lie in the kernel of the Cauchy...
This dissertation studies certain intersection numbers of exceptional divisions arising from blowing up subspaces of lattices associated to graphs. These permit the computation of the Segre class of a scheme associated to the graph...
This dissertation covers several topics around the idea of the Schwartz-MacPherson Chern classes, which were independently constructed by M.H. Schwartz around 1965 and R. MacPherson in the early 1970's. First we review a more recent...
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