Now showing items 1-20 of 82

• #### An Algorithmic Framework for High Dimensional Regression with Dependent Variables ﻿

(2014-02-24)
We present an exploration of the rich theoretical connections between several classes of regularized models, network flows, and recent results in submodular function theory. This work unifies key aspects of these problems ...
• #### Alternate Approaches to the Cup Product and Gerstenhaber Bracket on Hochschild Cohomology ﻿

The Hochschild cohomology $HH^\bullet(A)$ of an algebra $A$ is a derived invariant of the algebra which admits both a graded ring structure (called the cup product) and a compatible graded Lie algebra structure (called the ...
• #### Applications of Metric Embeddings in Solving Combinatorial Problems ﻿

(2013-11-14)
Metric embeddings constitute one of the fundamental tools for exploiting the underlying geometric structure of many combinatorial problems. In this dissertation we study some of the applications of metric embeddings in the ...
• #### Arithmetic Properties of the Derived Category for Calabi-Yau Varieties ﻿

This thesis develops a theory of arithmetic Fourier-Mukai transforms in order to obtain results about equivalences between the derived category of Calabi-Yau varieties over non-algebraically closed fields. We obtain answers ...
• #### Aspects of Markov Chains and Particle Systems ﻿

The thesis concerns asymptotic behavior of particle systems and the underlying Markov chains used to model various natural phenomena. The objective is to describe and analyze stochastic models involving spatial structure ...
• #### Boundary Harnack Principle for Stable-Like Processes ﻿

We establish the boundary Harnack principle for certain classes of symmetric stable-like processes in $\mathbf{R}^d$ on arbitrary open sets as well as censored stable-like processes on $\mathcal{C}^{1,1}$-domains. Using ...
• #### Brownian Motion on Spaces with Varying Dimension ﻿

In this thesis we introduce and study Brownian motion with or without drift on state spaces with varying dimension. Starting with a concrete such state space that is the plane with an infinite pole on it, we construct a ...
• #### The C*-algebra of a finite T_0 topological space ﻿

We are concerned with the following motivating question: how can one extend the classical Gelfand-Naimark theorem to the simplest non-Hausdorff topological spaces? Our model space is a finite $T_0$ topological space, or ...
• #### Classification of connected Hopf algebras up to prime-cube dimension ﻿

We classify all connected Hopf algebras up to p^3 dimension over an algebraically closed field of characteristic p>0 under the mild restriction such that in dimension p^3, we only work over odd primes p when the primitive ...
• #### Closing the Loop: Optimal Stimulation of Neuronal Networks via Adaptive Control Algorithms ﻿

The Caenorhabditis elegans (C. elegans) worm is a well-studied biological organism model. The nervous system of C. elegans is particularly appealing to study, since it is a tractable fully functional neuronal network for ...
• #### Combinatorial Laguerre Series ﻿

(2014-02-24)
We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials. We describe how such a series can be computed by ...
• #### Competing Brownian Particles ﻿

Consider a finite system of N Brownian particles on the real line. Rank them from bottom to top: the (currently) lowest particle has rank 1, the second lowest has rank 2, etc., up to the top particle, which has rank N. The ...
• #### Computational aspects of modular parametrizations of elliptic curves ﻿

\abstract{ We investigate computational problems related to modular parametrizations of elliptic curves defined over $\mathbb{Q}$. We develop algorithms to compute the Mazur Swinnerton-Dyer critical subgroup of elliptic ...
• #### Conformal welding of uniform random trees ﻿

A conformally balanced tree is an embedding of a given planar map into the plane with constraints on the harmonic measure of its edges such that the resulting set is unique up to scale and rotation. Bishop (2013) showed ...
• #### Convex Optimization over Probability Measures ﻿

The thesis studies convex optimization over the Banach space of regular Borel measures on a compact set. The focus is on problems where the variables are constrained to be probability measures. Applications include ...
• #### Deformation invariance of rational pairs ﻿

Rational pairs, recently introduced by Kollár and Kovács, generalize rational singularities to pairs (X,D). Here X is a normal variety and D is a reduced divisor on X. Integral to the definition of a rational pair is the ...
• #### Deformations of Categories of Coherent Sheaves and Fourier-Mukai Transforms ﻿

(2013-07-25)
In modern algebraic geometry, an algebraic variety is often studied by way of its category of coherent sheaves or derived category. Recent work by Toda has shown that infinitesimal deformations of the category of coherent ...
• #### Detailing the Work of Leading a Productive Mathematics Discussion: A Study of a Practice-Based Pedagogy of Elementary Teacher Education ﻿

Abstract Detailing the Work of Leading a Productive Mathematics Discussion: A Study of a Practice-Based Pedagogy of Elementary Teacher Education Adrian Foster Cunard Chair of the Supervisory Committee: Professor Elham ...
• #### Dual Equivalence Graphs and their Applications ﻿

In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's ...
• #### Eigenvalue fluctuations for random regular graphs ﻿

One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are <italic>universal</italic>. We probe the edges of universality by studying ...