P  A  P  E  R  S 

The transition from $\mathbb{C}\mathsf{P}^1$ to $(\mathbb{R}\oplus\mathbb{R})\mathsf{P}^1$.

Families of Geometries,
​Algebras & Transitions

 ABSTRACT: This paper proposes a general framework for studying transitional behavior in geometric topology.  A theory of families of geometries is developed, robust enough to contain all currently known transitional geometries arising from conjugacy limits.  This is used to construct a theory of Klein geometries over $\mathbb{R}$-algebras and the relationship between transitioning algebraic structure and transitions of the corresponding geometries is investigated in detail.  This generalizes the construction in the paper below.

A Transition of
​Complex Hyperbolic Space

ABSTRACT: By degenerating the algebraic structure of $\mathbb{C}$, we construct a transition of geometries from complex hyperbolic space to a new geometry built out of $\mathbb{R}\mathsf{P}^n$ and its dual.  This transition provides a geometric context for considering the flexing of hyperbolic orbifolds, as defined by Cooper, Long & Thistlethwaite.  As an application, we connect the convex projective and complex hyperbolic deformations of triangle groups via this transition.

The developing image of a Heisenberg torus

The Heisenberg plane

ABSTRACT: 
This paper studies the geometry given by the projective action of the Heisenberg group on the plane. The closed orbifolds admitting Heisenberg structures are those with vanishing Euler characteristic and singularities of order at most two, and the corresponding deformation spaces are computed. Heisenberg geometry is of interest as a transitional geometry between any two of the constant-curvature geometries $\mathbb{S}^2,\mathbb{E}^2,\mathbb{H}^2$, and regenerations of Heisenberg tori into these geometries are completely described. 
​
ARXIV: https://arxiv.org/abs/1805.04256

O T H E R  W O R K

Transition from $\mathbb{H}^2$ to $\mathbb{S}^2$ through $\mathbb{E}^2$ within $\mathbb{R}\mathsf{P}^2$.

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)

T  A  L  K  S

2018:
History of manifold classification problems, ongoing lecture series, UCSB
Conformal embeddings of Flat tori in R^3, UCSB
What would the sky look like in a finite volume universe? UCSB Allosphere
Inside a hyperbolic 3-manifold, UCSB Allosphere
Heisenberg Wallpaper, Topology Student's Workshop
Cayley Graphs, Tilings and Symmetries in dimension 3, UCSB Allosphere
The Heisenberg Plane, Graduate Student Topology Conference


2017
What it feels like to get right-multiplied by a quaternion, UCSB Allosphere
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