Papers & ResultsBelow are some writeups and preprints that have come out of my research. Click the titles for the pdfs.

ABSTRACT: A continuous family of subsets of $X$ parameterized by $Y$ is given by a map $Y\to\mathfrak{C}_X$ into the Hausdorff hyperspace of $X$. Here we propose a generalization of fibrations (to a situation where the fibers need not have constant homotopy type) as an abstract notion of continuity in this sense. (IN PREPARATION)

ABSTRACT: The Heisenberg group acts transitively on the plane, and the resulting geometry arises as a nontrivial conjugacy limit of all three constant curvature geometries within projective geometry. This paper studies Heisenberg geometry, in particular Heisenberg structures on orbifolds and regenerations of these structures as constant curvature conemanifold structures.

ABSTRACT: Here we classify the conjugacy limits of the isometry groups of the constant curvature geometries as subgroups of $PGL(3,\mathbb{R})$. I wrote this to teach myself the material, hopefully it proves useful to other learners.

ABSTRACT: A group $\Gamma<\mathsf{PSL}(2,\mathbb{Q})$ is pseudomodular if $\Gamma$ is not comenusrable with $\mathsf{PSL}(2,\mathbb{Z})$ but the cusp set of $\Gamma$ is still the extended rationals $\mathbb{Q}\cup\{\infty\}$. Here we extend a technique of Lu, Tan and Vo to construct infinite families of pseudomodular groups. (TO BE POSTED SOON)

ABSTRACT: When $\Gamma\to SO(2,1)$ is a smooth point of $\mathsf{Hom}(\Gamma,SL(3,\mathbb{R})$, it is also a smooth point of $\mathsf{Hom}(\Gamma,SU(2,1))$ and the dimension of the Zariski tangent spaces the hyperbolic representation are equal. Here we provide a geometric explanation for this fact via a transition from complex hyperbolic geometry to "selfdual real projective geometry".
(IN PREPARATION) 
Talks2018:
The Heisenberg Plane, Graduate Student Topology Conference 2017 What it feels like to get rightmultiplied by a quaternion, UCSB Ways to Die in Hyperbolic Space, UCSB The many faces of the Trefoil Knot Complement, UCSB Daily Life in Constant Curvature Geometries, UCSB 2016 Pseudomodular Groups, UCSB What is a manifold and what are they good for?, UCSB What does Homology Measure? Seifert Fibered spaces and Tesselations, UCSB 2015 Covers of the Modular Surface, UCSB Flows on the space of Planar Lattices, UCSB Visualizing spaces of Isometries, UCSB 2014 Polytopes on the Integer Lattice, UCSB 
Conferences2018:
Graduate Student Topology, Conference, 2017: Lee Mosher Birthday Conference, Princeton 