My research focuses on deformations, degenerations and transitions in geometric topology. In particular, I am interested in continuous families of homogeneous spaces which abruptly change isomorphism type. Through my PhD and currently, I work on constructing a unified language for working with such objects and producing new examples of geometric transitions. Some other things I am currently interested in are as follows:

  • The computation of compactifications of moduli spaces of groups / manifolds relevant to geometric topology, bringing in tools from the algebro-geometric compactifications of configuration spaces.
  • Apply Transitional Geometry to other domains of mathematics, from the study of representation varieties to dynamical systems arising in classical mechanics.
  • Study the intrinsic geometry of homogeneous spaces including the anisotropic Thurston geometries in dimension three and new spaces discovered along geometric transitions.
Phd Thesis: Families of Geometries

Transition Geometry

A Geometric Transition is a continuously varying family of homogeneous spaces which abruptly changes isomorphism type. The ideas of transitional geometry trace all the way back to the original work of Bolyai, who in his construction of hyperbolic geometry described a limiting procedure to recover the standard theorems and relations of Euclidean geometry/trigonometry. This study was further purused by Klein who constructed a model of this in projective space (in modern language, desribing the degeneration of hypebolic space as the unit sphere in $\mathbb{R}^{n,1}$ as the minkowski inner product degenerates), and has become important more recently to geometric topology in the study of degenerating geometric structures on manifolds.

Families of Geometries

Beginning wtih the work of Klein, the usual description of a transition of geometries is as a conjugacy limit inside of some ambient geometry, usually projective space. The choice of such an ambient space affects both the results (which geometries limit to which others), and the methods of proof (which geometric techniques can be used to show a particular transition is actually continuous.) This preprint recounts work from my thesis, constructing an alternative formalism for discussing geometric transitions intrinsically, without need for an ambient space.

A Transition of Complex Hyperbolic Space

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 and Thistlethwaite. As an application, we connect the convex projective and complex hyperbolic deformations of triangle groups via this transition.

Transitional Geometry and the Kepler Problem

Under an inverse square central force law, solutions to the two body problem have conic section trajectories, with the common center of mass at one focus. The geometry of these orbits depends on the total energy: bound (periodic) orbits for negative energy, hyperbolic trajectories for positive energy, with the limiting case following parabolic paths. This prediction of Kepler was first confirmed by Newton, and since has attracted a wide variety of proofs. In this short note we connect the changing geometry of orbits to a transition of geometries in the phase space of the system: with orbits of energy $E$ related to the geodsic flow on a space of constant curvature $-E$.

Degenerations of Lie Groups and Chabauty Compactification

The space of closed subsets of a compact metric space $X$ is itself naturally a compact metric space under the Hausdorff metric. With care, this construction generalizes to broader contexts; in particular the Lie subgroups of a given (not necessarily compact) Lie group $G$ are topologized by the Chabauty topology. This Chabauty space of $G$ allows us to reason topologically about the relation between different isomorphism types of subgroups of $G$, and is fundamental to the study of limits of subgeometries / subgroups in geometric topology. .

Riemannian Limits of the Product Geometries

Hyperbolic geometry degenrates to Euclidean within projective geometry, as first noticed by Klein. More generally, up to finite index all eight Thurston geometries can be constructed as subgeometries of $\mathbb{R}\mathsf{P}^3$ and it is natural to ask when does Thurston geometry X have a conjugacy limit containing Thurston geometry Y? In this preprint we fill in the final two open cases in this $8\times 8$ grid: showing that the product geometries $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$ do not limit to $\mathsf{Nil}$.

Chabauty Compactification of the Orthogonal Groups

The collection of orthogonal groups in $\mathsf{GL}(n,\mathbb{R})$ is a union of components (in fact, conjugacy classes) determined by signature. The conjugacy limits of the orthogonal groups lie in the Chabauty closure of this space, whose topology we describe here. In particular, we investigate in detail a slice of this space, the Chabauty compactification of the space of orthogonal groups with respect to diagonal quadratic forms. Perhaps surprisingly the compactification is a manifold, succintly described as the (maximal) de Concini Procesi wonderful compactification of the coordinate hyperplane arrangement in $\mathbb{R}\mathsf{P}^{n-1}$. The original conjugacy classes of orthogonal groups form top dimensional permutohedral cells in this manifold, and the lower dimensional faces / their combinatorics parameterize all limits / their relationships.

Geometric Structures

A manifold $M$ admits a geometric structure locally modeled on a homogeneous space $(G,X)$ if it can be built out of little chunks of $X$ glued together isometrically by maps in $G$. Classical examples are Euclidean and Hyperbolic structures, but more exotic strutctures such as Convex Real Projecti ve, Anti de Sitter, Half Pipe, and Heisenberg structures are also important in geometric topology.

The Heisenbgerg Plane

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.