The work in this thesis concerns two problems in arithmetic dynamics: forward orbit problems over finite fields, and inverse image problems over local fields. We …
We study the boundary of the moduli space of Higgs bundles using analytic methods such as harmonic maps and partial differential equations to give new …
We study existence and regularity of harmonic maps between 2-dimensional simplicial complexes. This work begins by defining metrics on these simplicial complexes and describing their …
We study the roles of domain and target curvatures in harmonic maps into metric spaces with upper curvature bounds. We begin computing the domain and …
While the study of algebraic curves and their moduli has been a celebrated subject in algebraic and arithmetic geometry, generalizations of many results that hold …
Broadly, topology is the study of shapes. In this thesis we specifically study surfaces through graphs which are defined by characteristics of the surface. The …
This thesis is devoted to understanding incompressible surfaces properly embedded in a hyperbolic four-punctured sphere bundle. The ideas are drawn from previous works of Floyd, …
The focus of this dissertation is on the invertibility of certain topological summary statistics for metric objects. The first set of results concern persistence diagrams …
Arc diagrams are simple, combinatorial objects associated to surfaces with boundary. They consist of homotopy classes of disjoint curves, and can be thought of as …
This thesis presents two general constructions of minimal surfaces by PDE gluing methods carreid out in collaboration with my advisor Nicolaos Kapouleas. In the first …
Convex polyhedra is an attractive and fundamental subject in which geometry is nicely joined with combinatorics and topology. My thesis studies convex polyhedra from two …
We study the numerical invariants attached to families of Abelian fourfolds constructed by Mumford with extra codimension 2 Hodge cycles that are generically not endowed …
A $(p,q,r)$-complex hyperbolic triangle group is a group generated by complex reflections across complex geodesics meeting at angles $\pi/p, \pi/q, \pi/r$. In this thesis, we …
This thesis focuses on weighted blow-ups. Weighted blow-ups are an important class of birational transformations, which has, for stacks, a similar role to the one …
Hironaka showed in his 1964 groundbreaking work that singularities of algebraic varieties admit a resolution in characteristic zero. Over the years, the proof of Hironaka’s …
Planar target patterns are radially symmetric time-periodic structures that connect a core region with a spatially periodic traveling wave in the far field. These patterns …
This thesis deals with sharp weighted estimates for classical operators in harmonic analysis. In the first chapter we study sharp weighted estimates in terms of …
