## Search

Now showing items 1-10 of 11

#### Maximum Norm Regularity of Implicit Difference Methods for Parabolic Equations

(2011)

We prove maximum norm regularity properties of L-stable finite differencemethods for
linear-second order parabolic equations with coefficientsindependent of time, valid
for large time steps. These results are almostsharp; ...

#### Persistent Cohomology Operations

(2011)

The work presented in this dissertation includes the study of cohomology and cohomological
operations within the framework of Persistence. Although Persistence was originally
defined for homology, recent research has developed ...

#### Estimating the Intrinsic Dimension of High-Dimensional Data Sets: A Multiscale, Geometric Approach

(2011)

This work deals with the problem of estimating the intrinsic dimension of noisy, high-dimensional
point clouds. A general class of sets which are locally well-approximated by <italic>k</italic>
dimensional planes but which ...

#### A Spectral Deferred Correction Method for Solving Cardiac Models

(2011)

Many numerical approaches exist to solving models of electrical activity in the heart.
These models consist of a system of stiff nonlinear ordinary differential equations
for the voltage and other variables governing channels, ...

#### The Graph Cases of the Riemannian Positive Mass and Penrose Inequalities in All Dimensions

(2011)

We consider complete asymptotically flat Riemannian manifolds that are the graphs
of smooth functions over $\mathbb R^n$. By recognizing the scalar curvature of such
manifolds as a divergence, we express the ADM mass as ...

#### Dimensionality Reduction and Learning on Networks

(2011)

Machine learning is a powerful branch of mathematics and statistics that allows the
automation of tasks that would otherwise require humans a long time to perform. Two
particular fields of machine learning that have been ...

#### Multi-Variable Period Polynomials Associated to Cusp Forms

(2011)

My research centers on the cohomology of arithmetic varieties. More speciﬁcally,
I am interested in applying analytical, as well as topological methods to gain better
insight into the cohomology of certain locally ...

#### The Geometry of Gravitational Lensing: Magnification Relations, Observables, and Kerr Black Holes

(2011)

Gravitational lensing is the study of the bending of light by gravity. In such a scenario,
light rays from a background star are deflected as they pass by a foreground galaxy
(the lens). If the lens is massive enough, then ...

#### Scalar curvature rigidity theorems for the upper hemisphere

(2011)

In this dissertation we study scalar curvature rigidity phenomena for the upper hemisphere,
and subsets thereof. In particular, we are interested in Min-Oo's conjecture that
there exist no metrics on the upper hemisphere ...

#### Stochastic Microlensing: Mathematical Theory and Applications

(2011)

Stochastic microlensing is a central tool in probing dark matter on galactic scales.
From first principles, we initiate the development of a mathematical theory of stochastic
microlensing. We first construct a natural ...