My main research interests lie in the area of 2-step nilpotent Lie groups with left-invariant pseudo-Riemannian metrics. These are manifolds with rich algebraic structures together with geometric structures that respect the algebra. My interests are primarily in the two extreme cases. On one side, I've defined a class of spaces called modified \(H\)-type groups for which the connection between the two structures is arguably strongest. In these spaces the Lie bracket and metric tensor are linked to one another by a quadratic form.

The opposite extreme case are groups with degenerate centers. In these groups the metric behaves in such a way that the natural algebraic components contain totally null subspaces. This scenario can only occur in the pseudo-Riemannian setting, and not with Riemannian metrics. There is a lot still to learn about the geometric consequences imposed by degenerate centers. The extension of Riemannian results to pseudo-Riemannian metrics with degenerate centers can frequently become somewhat complicated.

My PhD work was focused on Ehresmann connections, sometimes called general connections. A general connection represents the least amount of information needed to add to a fiber bundle, \(\pi :E \to M\), in order for it to "have a geometry." Ehresmann originally defined connections to be horizontal distributions on the bundle that possess a property we now call horizontal path lifting, or HPL. Naïvely, the HPL property ensures that all fibers of the bundle are indeed "connected" by the horizontal distribution. In my dissertation work, I provide a complete classification of horizontal distributions that possess the HPL property.

A group of colleagues and friends work in the area of numerical conformal mapping. A conformal map is a function that maps \(\mathbb{C} \to \mathbb{C}\) biholomorphically. Every analytic function (on \(\mathbb{C}^1\)) with non-vanishing derivative is conformal. Geometrically, conformal maps preserve angles, but not necessarily lengths. Numerical methods can be used to map domains with complicated boundaries to "nicer" domains—usually circle domains—conformally. Problems involving objects that are invariant under conformal maps, such as the Laplacian, can then be solved in the nicer domain, then the solution can be sent back to the original domain.

My contribution to the group has been to study the alternating projection method originally introduced by Proznak, and later studied by Wegmann. I am currently working to get this method coded in Python.