The Teichmüller space of a surface that admits a hyperbolic structure can be equipped with the Weil-Petersson metric. The Weil-Persson Ending Lamination is a candidate …
Call a diagram D of spaces ``absolute'' if, for all enriched homotopy functors F, the induced map F(holim(D)) -> holim(FD) is an equivalence of spaces. …
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 …
We present a series of C^2-Morse functions on the Deligne-Mumford compactification M_{g,n} bar of the moduli space of genus g Riemann/hyperbolic surfaces with n punctures. …
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, …
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 …
Let $S=S_{g}$ be a surface of genus $g>1$, and $\Teich(S)$ be the Teichm\"uller space endowed with the Weil-Petersson metric and $\Mod(S)$ be the mapping class …
This thesis will discuss generalizations of Selberg’s 3/16 theorem to hyperbolic surfaces of infinite area. These Selberg type theorems are deeply related to orbit counting …
We investigate two different topics in discrete mathematics: the geometry of piecewise-linear surfaces and the long-term behavior of discrete dynamical systems. A regular polygon surface …
The focus of this thesis is the mapping class group of surfaces, which consists of isotopy classes of orientation-preserving homeomorphisms of the surface. This dissertation …
We study combinatorial and probabilistic properties of tilings of the plane by $m\times1$ horizontal rectangles and $n\times1$ vertical rectangles also called tilings by bars. In …
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 …
The central objects of this thesis are Veech surfaces: Riemann surfaces with exceptionally symmetric flat structures that generalize flat tori. They play a crucial role …
