This dissertation is devoted to the study of inverse spectral problem of Hankel operators. It is well-known that spectral characteristics of a Hankel operator does …
This thesis consists of four chapters on the author's research in algebraic dynamics. The first chapter introduces the main results. The second chapter is an …
In the field of arithmetic dynamics, we study number theoretic aspects of discrete dynamical systems induced by rational maps on projective spaces. Among the rational …
We extend the results presented by Antos et al. [1] for nonparametric estimation methods using complete datasets to nonparametric regression estimators that use right censored …
Domain exchange transformations (DETs) are dynamical systems of piecewise translations defined on piecewise smooth Jordan domains. DETs are two-dimensional generalizations of interval exchange transformations which …
In this thesis, we study the elliptic boundary value problems on irregular domains. We first obtain the $W^{2,p}$ and $C^2$ regularity theories for second-order, non-divergence …
Chapter 1 presents joint work with Nikolaos Kapouleas and is concerned with constructions of new closed, embedded minimal surfaces in the round three sphere using …
We prove a weak factorization result on birational maps of Deligne-Mumford stacks, and deduce the following: Let $U \subset X$ be an open embedding of …
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 existence and regularity of harmonic maps between 2-dimensional simplicial complexes. This work begins by defining metrics on these simplicial complexes and describing their …
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, …
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 …
The main subject of this dissertation is the study of certain moduli spaces intimately related to the enumerative geometry of complex algebraic varieties and orbifolds. …
In a celebrated paper published in 1983, R. Mañé, P. Sad, and D. Sullivan proved a result about holomorphic families of injections called the λ-Lemma …
In this thesis, we investigate the stability properties of a variety of PDE models concerning fluid dynamics, including charged fluids (plasmas). The models which we …
We construct homotopical categories of equivariant spectra and the canonical degree 1 functors between them. We construct a universal degree 2 functor on spectra and …
This thesis explores transversality between algebraic varieties and linear subspaces in the setting of finite fields. The main contributions are effective Bertini-type results for smooth …