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Local and Global Convergence for ConvexComposite Optimization
Convexcomposite optimization seeks to minimize f(x):=h(c(x)) over x in R^n, where h is closed, proper, and convex, and c is smooth. Such problems include nonlinear programming, minimax optimization, estimation of nonlinear ... 
Homological algebra of StanleyReisner rings and modules
Associated to each simplicial complex $\Delta$ and each field $\field$ is the StanleyReisner ring $\field[\Delta]$. The answers to a multitude of questions related to simplicial complexes have historically been found ... 
Major Index Statistics: Cyclic Sieving, Branching Rules, and Asymptotics
Major index statistics have been studied for over a century in many guises and appear throughout algebraic combinatorics. We pursue major index statistics from two complementary perspectives: algebraic and asymptotic. We ... 
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 ... 
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 ... 
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 ... 
On the Duflot filtration for equivariant cohomology rings
We study the Fpcohomology rings of the classifying space of a compact Lie group G using methods from equivariant cohomology. Building on ideas of Duflot and Symonds we study a “rank filtration” on the ptoral equivariant ... 
Two Inverse Problems Arising in Medical Imaging
In this thesis, we discuss two inverse problems arising in medical imaging. The first problem is about a hybrid imaging method using coupled boundary measurements, which combines electrical impedance tomography (EIT) with ... 
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 ... 
Twistor Spaces for Supersingular K3 Surfaces
We develop a theory of twistor spaces for supersingular K3 surfaces, extending Artin's analogy between supersingular K3 surfaces and complex analytic K3 surfaces. Our twistor spaces are families of twisted supersingular ... 
Random Permutations and Simplicial Complexes
We study the asymptotic behavior of distributions on two different combinatorial objects, permutations and simplicial complexes. First we study strong αlogarithmic measures on the symmetric group, including the well ... 
Inverse Problems for Linear and Nonlinear Elliptic Equations
An inverse problem is a mathematical framework that is used to obtain information about a physical object or system from observed measurements. A typical inverse problem is to recover the coefficients of a partial differential ... 
Some Theorems on the Resolution Property and the Brauer map
Using formallocal methods, we prove that a separated and normal DeligneMumford surface must satisfy the resolution property, this includes the first class of separated algebraic spaces which are not schemes. Our analysis ... 
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) ... 
The Inverse Problem of Thermoacoustic Tomography in Attenuating Media
Thermoacoustic tomography is a developing medical imaging technique that combines the propagation of electromagnetic and ultrasound waves with the purpose of producing a high contrast and high resolution internal image of ... 
Topics in Continuum Theory
Continuum Theory is the study of compact, connected, metric spaces. These spaces arise naturally in the study of topological groups, compact manifolds, and in particular the topology and dynamics of onedimensional and ... 
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, ... 
The geometry of uniform measures
Uniform measures have played a fundamental role in geometric measure theory since they naturally appear as tangent objects. They were first studied in the groundbreaking work of Preiss where he proved that a Radon measure ... 
On the g2number of various classes of spheres and manifolds
For a $(d1)$dimensional simplicial complex $\Delta$, we let $f_i=f_i(\Delta)$ be the number of $i$dimensional faces of $\Delta$ for $1\leq i\leq d1$. One classic problem in geometric combinatorics is the following: ... 
Structure and complexity in nonconvex and nonsmooth optimization
Complexity theory drives much of modern optimization, allowing a fair comparison between competing numerical methods. The subject broadly seeks to both develop efficient algorithms and establish limitations on efficiencies ...