Polygon iterations, which can be thought of as discrete dynamical systems on the space of polygons, provide an abundance of interesting discrete dynamical systems in …
We consider the one-dimensional nonlinear Schroedinger equation (NLS) with focusing point nonlinearity, which replaces the pure power nonlinearity in the standard NLS by inserting a …
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 study the boundary of the moduli space of Higgs bundles using analytic methods such as harmonic maps and partial differential equations to give new …
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 …
The present work brings together the fields of random maps and Loewner evolution by constructing explicit embeddings of critical Galton-Watson trees in the upper half-plane …
This thesis studies three problems stemming from different areas of combinatorics. The first problem is the study of the length of the longest path in …