Browsing Mathematics by Title
Now showing items 120 of 124

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 nonisomorphic 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 firstorder 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 ... 
Analytic and geometric aspects of the elliptic measure on nonsmooth domains
Harmonic/elliptic measure arises naturally in probability and in the study of boundary value problems for elliptic operators. It has attracted the attention of many mathematicians to study the relationship between the ... 
Arithmetic Properties of the Derived Category for CalabiYau Varieties
This thesis develops a theory of arithmetic FourierMukai transforms in order to obtain results about equivalences between the derived category of CalabiYau varieties over nonalgebraically 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 StableLike Processes
We establish the boundary Harnack principle for certain classes of symmetric stablelike processes in $\mathbf{R}^d$ on arbitrary open sets as well as censored stablelike 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 ... 
Brownian particles interacting with a Newtonian Barrier: Skorohod maps and their use in solving a PDE with free boundary, strong approximation, and hydrodynamic limits.
In this thesis, we pioneer the use of Skorohod maps in establishing the hydrodynamic behavior of an interacting particle system. This technique has the benefit of using stochastic methods to show both existence and uniqueness ... 
The C*algebra of a finite T_0 topological space
We are concerned with the following motivating question: how can one extend the classical GelfandNaimark theorem to the simplest nonHausdorff topological spaces? Our model space is a finite $T_0$ topological space, or ... 
Classification of connected Hopf algebras up to primecube 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 ... 
Classification of Line Modules and Finite Dimensional Simple Modules over a Deformation of the Polynomial Ring in Three Variables
Let $\Bbbk$ be a field and $A$ the noncommutative $\Bbbk$algebra generated by $x_1, x_2, x_3$ subject to the relations $$ q x_ix_j  q^{1} x_jx_i \; = \; x_k $$ as $(i,j,k)$ ranges over all cyclic permutations of ... 
Combinatorial Laguerre Series
(20140224)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 ThreeDimensional 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 SwinnertonDyer critical subgroup of elliptic ...