Now showing items 1-20 of 105

• #### A Survey of Tverberg Type Problems ﻿

Tverberg's theorem, which celebrates its fiftieth anniversary this year, is a central result in the fields of discrete geometry and topological combinatorics. Proved in 1966, it was a major step in solving questions whether, ...
• #### 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 ...
• #### Algorithms for convex optimization with applications to data science ﻿

Convex optimization is more popular than ever, with extensive applications in statistics, machine learning, and engineering. Nesterov introduced optimal first-order methods for large scale convex optimization in the 1980s, ...
• #### Algorithms in Discrepancy Theory and Lattices ﻿

This thesis deals with algorithmic problems in discrepancy theory and lattices, and is based on two projects I worked on while at the University of Washington in Seattle. A brief overview is provided in Chapter 1 (Introduction). ...
• #### 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 ...
• #### Bin packing, number balancing, and rescaling linear programs ﻿

This thesis deals with several important algorithmic questions using techniques from diverse areas including discrepancy theory, machine learning and lattice theory. In Chapter 2, we construct an improved approximation ...
• #### Bispectral Operator Algebras ﻿

This dissertation is an amalgamation of various results on the structure of bispectral differential operator algebras, ie. algebras of differential operators with possibly noncommutative coefficients in a variable $x$ ...
• #### 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 ...