Now showing items 1-20 of 178

• #### A Complete Implementation of Graphlopedia: A Fingerprint Database for Graphs ﻿

The OEIS is a widely used and incredibly powerful tool to catalogue user-submitted sequences of interest to the mathematical community, and to associate with them a list of citations where these sequences can be found. In ...
• #### A Functorial Approach to Algebraic Vision ﻿

We study multiview moduli problems that arise in computer vision. We show that these moduli spaces are always smooth and irreducible, in both the calibrated and uncalibrated cases, for any number of views. We also show ...
• #### 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 ...
• #### An Extremal Property of the Square Lattice ﻿

\nI{Motivated} by a 2019 result of Faulhuber-Steinerberger \cite{extremal} on the hexagonal lattice $\Lambda$, we demonstrate that the square lattice $\Z^2$ exhibits the same local extremal property as $\Lambda$, where ...
• #### Analytic and geometric aspects of the elliptic measure on non-smooth 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 ...
• #### Applications of the Cyclic Sieving Phenomenon to Words, Branching Rules, and Tableaux ﻿

Since Reiner-Stanton-White defined the cyclic sieving phenomenon associated to a finite cyclic group action and a polynomial, many compelling examples of cyclic sieving have been found. In this thesis, we focus on what we ...
• #### Approximation Algorithms for Scheduling and Fair Allocations ﻿

In this thesis, we will have discussions on two main topics, max-min allocation and schedulingjobs with precedent constraints on machines with communication delays. New approximation algorithms are given in Chapter 2, 4 ...
• #### Arithmetic of Totally Split Modular Jacobians and Enumeration of Isogeny Classes of Prime Level Simple Modular Abelian Varieties ﻿

In this thesis, we aim to give algorithms for computing two key invariants of the modular Jacobians $J_0(N)$. We first give methods for computing the rational torsion order of rank-0 Jacobians $J_0(N)$ that are isogenous ...
• #### 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 ...
• #### Birational Functors in the Derived Category ﻿

In this thesis, we study a class of derived equivalences that naturally induce birational maps. We give several equivalent criteria for a birational correspondence to exist, and prove the correspondence induces a $K$-equivalece, ...
• #### 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 ...
• #### Brownian Motion, Quasiconformal Mappings and the Beltrami Equation ﻿

Consider a Jordan domain $\Omega$ in the plane with $3$ distinct points marked on its boundary. These $3$ points split $\partial \Omega$ into $3$ arcs. For each $z \in \Omega$, we can assign it the harmonic coordinates by ...