### Recent Submissions

• #### On the Duflot filtration for equivariant cohomology rings ﻿

We study the Fp-cohomology 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 p-toral 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 non-isomorphic 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 α-logarithmic measures on the symmetric group, including the well- ...
• #### Inverse Problems for Linear and Non-linear 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 formal-local methods, we prove that a separated and normal Deligne-Mumford 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 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) ...
• #### 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 one-dimensional 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 first-order 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 g2-number of various classes of spheres and manifolds ﻿

For a $(d-1)$-dimensional simplicial complex $\Delta$, we let $f_i=f_i(\Delta)$ be the number of $i$-dimensional faces of $\Delta$ for $-1\leq i\leq d-1$. One classic problem in geometric combinatorics is the following: ...
• #### Structure and complexity in non-convex and non-smooth 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 ...
• #### Spectral analysis in bipartite biregular graphs and community detection ﻿

This thesis concerns to spectral gap of random regular graphs and consists of two main con- tributions. First, we prove that almost all bipartite biregular graphs are almost Ramanujan by providing a tight upper bound for ...
• #### Strichartz estimates for the wave equation on Riemannian manifolds of bounded curvature ﻿

Wave packet methods have proven to be a useful tool for the study of dispersive effects of the wave equation with coefficients of limited differentiability. In this thesis, we use scaled wave packet methods to prove ...
• #### Novel uses of the Mallows model in coloring and matching ﻿

A natural model of a highly ordered random ranking is the Mallows model. Disorder is measured by the number of inversions; these are pairs of elements whose order is reversed. The Mallows model assigns to each ranking of ...
• #### 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 ...