Browsing Mathematics by Title
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On the g2number of various classes of spheres and manifolds
For a $(d1)$dimensional simplicial complex $\Delta$, we let $f_i=f_i(\Delta)$ be the number of $i$dimensional faces of $\Delta$ for $1\leq i\leq d1$. One classic problem in geometric combinatorics is the following: ... 
On the Geometry of Rectifiable Sets with Carleson and Poincaretype inequlaities
A central question in geometric measure theory is whether geometric properties of a set translate into analytical ones. In 1960, E. R. Reifenberg proved that if an $n$dimensional subset $M$ of $\mathbb{R}^{n+d}$ is well ... 
On the mod 2 general linear group homology of totally real number rings
(1997)We study the mod 2 homology of the general linear group of rings of integers in totally real number fields. In particular, for certain such rings R, we construct a space JKR and show that the mod 2 homology of JKR is a ... 
The OrnsteinUhlenbeck Process In Neural DecisionMaking: Mathematical Foundations And Simulations Suggesting The Adaptiveness Of Robustly Integrating Stochastic Neural Evidence
(20130225)This master's thesis reviews the concepts behind a stochastic process known as the OrnsteinUhlenbeck Process, and then uses that process as a way to investigate neural decision making. In particular, MATLAB simulations ... 
Path algebras and monomial algebras of finite GKdimension as noncommutative homogeneous coordinate rings
This thesis sets out to understand the categories QGr A where A is either a monomial algebra or a path algebra of finite GelfandKirillov dimension. The principle questions are: \begin{enumerate} \item What is the structure ... 
Permutation diagrams in symmetric function theory and Schubert calculus
A fundamental invariant of a permutation is its inversion set, or diagram. Natural machinery in the representation theory of symmetric groups produces a symmetric function from any finite subset of <bold>N</bold><super>2</super>, ... 
Polynomials in Multiview Geometry
(20130417)We study multiview geometry and some of its applications through the use of polynomials. A threedimensional world point gives rise to n ≥ 2 twodimensional projections in n given cameras. The object of focus in this ... 
The Positive Semidefinite Rank of Matrices and Polytopes
The positive semidefinite (psd) rank of a nonnegative <italic>p</italic> × <italic>q</italic> matrix <italic>M</italic> is defined to be the smallest integer <italic>k</italic> such that there exist <italic>k</italic> × ... 
Problems in Algebraic Vision
This thesis studies several fundamental mathematical problems that arise from computer vision using techniques in algebraic geometry and optimization. Chapters 2 and 3 consider the fundamental question of the existence of ... 
Problems in computational algebra and integer programming
(2007)This thesis is a compendium of several projects that span the gap between commutative algebra and geometric combinatorics: one in tropical geometry, one in computational algebra, and two in discrete geometry and integer ... 
Qualitative stability properties of matrices
(1985)A matrix A is sign stable if some matrix with the same sign pattern of positive, negative, and zero entries has all eigenvalues with negative real parts. It is potentially stable if it is not sign unstable. The system A x ... 
The Radiative Transfer Equation in Photoacoustic Imaging
(20130725)Photoacoustic tomography is a rapidly developing medical imaging technique that combines optical and ultrasound imaging to exploit the high contrast and high resolution of the respective individual modalities. Mathematically, ... 
Random recursion
We study the limiting behavior of three stochastic processes. Two are interacting particle systems, the frog model and coalescing random walk. We work out transience and recurrence properties on various graphs. The last ... 
The regularity of Loewner curves
The Loewner differential equation, a classical tool that has attracted recent attention due to SchrammLoewner evolution (SLE), provides a unique way of encoding a simple 2dimensional curve into a continuous 1dimensional ... 
Results on singularities of pairs
Singularities of algebraic varieties have been studied extensively, and recently also the properties of singularities of pairs have been investigated. This thesis presents several results on singularities of different kinds ... 
Selfshrinking Solutions to Mean Curvature Flow
We construct new examples of selfshrinking solutions to mean curvature flow. We first construct an immersed and nonembedded sphere selfshrinker. This result verifies numerical evidence dating back to the 1980's and shows ... 
Selmer groups for elliptic curves with isogenies of prime degree
(2003)The MordellWeil theorem states that the points of an elliptic curve defined over a number field form a finitely generated, abelian group. The rank of this group, generally referred to as the rank of the elliptic curve, ... 
Sheaves on support varieties and varieties of elementary subalgebras
We present several results about two closely related types of objects: the projectivized scheme $\PG$ of one parameter subgroups of an infinitesimal group scheme $G$ and the variety $\bE(\fg)$ of maximal elementary subalgebras ... 
Shimura Degrees for Elliptic Curves over Number Fields
A crowning achievement of Number theory in the 20th century is a theorem of Wiles which states that for an elliptic curve E over <bold>Q</bold> of conductor N, there is a nonconstant map from the modular curve of level N ... 
Smoothness of Loewner Slits
(20130225)In this dissertation, we show that the chordal Loewner differential equation with C^{beta} driving function generates a C^{beta + 1/2} slit for 1/2 < beta <= 2, except when beta = 3/2 the slit is only proved to be weakly C^{1,1}.