# Math

## Research Interests

I'm a topologist and a geometer. Within topology, my interests are focused in low-dimensional topology, and in particular contact topology and knot theory. Within geometry, I'm interested in mathematical relativity, Kerr black holes, spinor formalism, and PDEs.

## Papers

### An Odd Analog of Plamenevskaya's Invariant of Transverse Knots

Plamenevskaya defined an invariant of transverse links as a distinguished class in the even Khovanov homology of a link. I define an analog of Plamenevskaya’s invariant in the odd Khovanov homology of Ozsváth, Rasmussen, and Szabó. The analog is also an invariant of transverse links and has similar properties to Plamenevskaya’s invariant. I also show that this invariant can be identified with an equivalent invariant in the reduced odd Khovanov homology. Additionally, I demonstrate computations of the invariant on various transverse knot pairs with the same topological knot type and self-linking number.

## Unpublished Work

### Spinor Formalism for Spacetime Initial-Data Sets

General relativity expresses gravity as the curvature of a spacetime's timelike vectors in a in a spacelike direction. Manifestly, this is in the form of a fully non-linear PDE in the metric of a spacetime. Coordinates in spacetimes are traditionally expressed as four real numbers: three spacelike coordinates, and one timelike coordinate. There is an alternative formulation of a spacetime called the spinor formulation where the four real-number coordinates are replaced with two complex numbers. My work involves making rigorous the definitions from physics for the initial-value formulation of spinor spacetimes. This makes it possible to analyze the differential properties of the objects necessary for proving Kerr stability in this framework.

### Matrix Representation of nonKerrness

Astrophysicists believe that all sufficiently isolated black holes become maximally symmetric in the asymptotic future. Such a maximally symmetric black hole is called a Kerr black hole and can be uniquely specified by its mass and angular momentum. Fundamental to proving this conjecture is establishing that Kerr black holes are stable: that if their symmetry is perturbed, they will naturally return to equilibrium. In its spinor formulation, the Killing vectors of Kerr spacetimes simplify so that it is possible to define an invariant measuring a spacetime's deviation from Kerr as the solution to a linear elliptic PDE. This work calculates the matrix components of this PDE so that—given initial data—its nonKerrness invariant can be analysed and evolved numerically.

I defined orbigraphs: local quotients of $$k$$-regular graphs as a graph-theoretic analog to local quotients of Riemannian manifolds, or Riemannian Orbifolds. In my work, I defined analogs to good and bad orbifolds and used the spectrum of an orbigraph's adjacency matrix to study these objects. This work was supervised by Liz Stanhope and was continued by a number of her students after me, culminating in this paper.