Broadly, my academic interests were in Riemannian geometry and geometric analysis. More specifically,
my background and training was in spectral geometry,
where one is interested in the interplay between the geometry of a manifold and the spectrum of some
geometrically motivated differential operator (e.g. the Laplace-Beltrami operator).
In my PhD thesis I investigated the singularities of the wave trace on lens spaces and flat
manifolds (the so-called Poisson relation).
After getting my PhD in Mathematics at Dartmouth College, I spent a year at Université Laval,
where I was a CRM Postdoctoral Fellow in the Department of Mathematics. At Laval I worked with
Alexandre Girouard on questions related to
the Steklov spectrum. After Laval, I did another post-doc at the University of Michigan. At
Michigan, I branched out to investigate questions motivated by mathematical physics, fluid mechanics
and numerical analysis.
Semi-classical incarnations of the Inverse Spectral Method: The inverse spectral method
has been used to study,
among other things, the continuous limit of the Toda lattice and the dispersionless
limit of the KdV equation.
Alejandro Uribe and I investigated an analog of the inverse spectral method for a class
of "tri-diagonal"
Toeplitz operators obtained from quantizing the 2-sphere. The work was inspired by this
article. Our preliminary
results can be found here.
Point vortex dynamics: Point vortices provide a useful toy model for helping to
understand vorticity in
fluid flows. They also have many connections to other areas of mathematics, e.g.
dynamical systems,
Hamiltonian dynamics and potential theory.
Jörn Zimmerling and I mentored three undergraduates in a
Lab of
Geometry project
where we investigated the dynamics of the vortices whose motion is confined to a curved
surface (like a
sphere or a torus). The video to the right is a simulation of three planar vortices in
an equilateral triangle
configuration where the vortices are known to steadily rotate around the center of the
triangle.