Now showing items 1-20 of 118

• #### 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, ...
• #### Abelian Varieties with Small Isogeny Class and Applications to Cryptography ﻿

An elliptic curve $E$ over a finite field $\FF_q$ is called isolated if it admits few efficiently computable $\FF_q$-isogenies from $E$ to a non-isomorphic curve. We present a variation on the CM method that constructs ...
• #### 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 ...
• #### 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 ...
• #### 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 ...
• #### Compact Moduli of Surfaces in Three-Dimensional Projective Space ﻿

The main goal of this paper is to construct a compactification of the moduli space of degree $d \ge 5$ hypersurfaces in $\mathbb{P}^3$, i.e. a parameter space whose interior points correspond to (equivalence classes of) ...
• #### 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 ...
• #### Connections Between Lanczos Iteration and Orthogonal Polynomials ﻿

(2010-01-10)
In this thesis we examine the connections between orthogonal polynomials and the Lanczos algorithm for tridiagonalizing a Hermitian matrix. The Lanczos algorithm provides an easy way to calculate and to estimate the ...
• #### Convergence and approximation for primal-dual methods in large-scale optimization ﻿

(1990)
Large-scale problems in convex optimization often can be reformulated in primal-dual (minimax) representations having special decomposition properties. Approximation of the resulting high-dimensional problems by restriction ...