Papers & Results

Below are some write-ups and pre-prints that have come out of my research.  Click the titles for the pdfs.

Continuous Families

 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)
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Heisenberg Geometry

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.

Limits of 2d geometries

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.

Pseudomodular groups

 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)

COMPLEX HYPERBOLIC AND REAL PROJECTIVE

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 "self-dual real projective geometry".
​(IN PREPARATION)

Talks

2018:
The Heisenberg Plane, Graduate Student Topology Conference

2017
What it feels like to get right-multiplied 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

Conferences

2018:

​Graduate Student Topology, Conference, 

2017:
Lee Mosher Birthday Conference, Princeton