### Recent Submissions

• #### 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, ...
• #### 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 ...
• #### 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 ...
• #### 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 ...
• #### 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 ...

• #### 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, ...
• #### Cornered Asymptotically Hyperbolic Metrics ﻿

This thesis considers asymptotically hyperbolic manifolds that have a finite boundary in addition to the usual infinite boundary – cornered asymptotically hyperbolic manifolds. A theorem of Cartan-Hadamard type near infinity ...
• #### Problems in Algebraic Vision ﻿

This thesis studies several fundamental mathematical problems that arise from computer vision using techniques in algebraic geometry and optimization. Chapters 2 and 3 consider the fundamental question of the existence of ...
• #### On f-vectors of polytopes and matroids ﻿

The f-vector of a simplicial complex is a fundamental invariant that counts the number of faces in each dimension. A natural question in the theory of simplicial complexes is to understand the relationship between the ...
• #### Some Inverse Problems in Analysis and Geometry ﻿

The aim of a typical inverse problem is to recover the interior properties of a medium by making measurements only on the boundary. These types of problems are motivated by geophysics, medical imaging and quantum mechanics ...
• #### On T-Semisimplicity of Iwasawa Modules and Some Computations with Z3-Extensions ﻿

For certain Zp-extensions of abelian number fields, we study the Iwasawa module associated to the ideal class groups. We show that generic Zp-extensions of abelian number fields are T-semisimple. We also construct the ...
• #### Formal group laws and hypergraph colorings ﻿

This thesis demonstrates a connection between formal group laws and chromatic symmetric functions of hypergraphs, two seemingly unrelated topics in the theory of symmetric functions. A formal group law is a symmetric ...