Broadly, I am interested in the intersection of geometry and analysis. In particular, I study analysis and geometry in the setting of metric spaces, where there may be no linear or smooth structure. This field has applications to many other areas of pure mathematics, as well as to computer science and imaging technology.
My thesis (pdf) focused on the existence theory for quasisymmetric mappings. Quasisymmetry is a metric version of conformality. Extending work of Bonk and Kleiner, I give a metric version of the classical uniformization theorem for simply connected Riemann surfaces. I also show that a wide class of (possibly non-smooth) metric surfaces have a quasisymmetric structure if they satisfy a necessary geometric condition called linear local contractibility. These theorems have applications to complex analysis in the plane, hyperbolic geometry, and bi-Lipschitz parameterization problems. Much of the material in my thesis is reproduced in my first two papers.
The classical uniformization theorem states that any simply connected Riemann surface is conformally equivalent to the disk, the plane, or the sphere, each equipped with a standard conformal structure. We give a similar uniformization for Ahlfors 2-regular, linearly locally connected metric planes; instead of conformal equivalence, we are concerned with quasisymmetric equivalence.
We show that a locally Ahlfors 2-regular and locally linearly locally contractible metric surface is locally quasisymmetrically equivalent to the disk. We also discuss an application of this result to the problem of characterizing surfaces in some Euclidean space that are locally bi-Lipschitz equivalent to an open ball in the plane.
A theorem of Balogh, Koskela, and Rogovin states that in Ahlfors Q-regular metric spaces which support a p-Poincaré inequality, 1 ≤ p ≤ Q, an exceptional set of σ-finite (Q-p)-dimensional Hausdorff measure can be taken in the definition of a quasiconformal mapping while retaining Sobolev regularity analogous to that of the Euclidean setting. Through examples, we show that the assumption of a Poincaré inequality cannot be removed.
We show that for any length-compact metric space Y and any 1 < q ≤ n, there is a continuous surjection in a suitably defined Sobolev-Lorentz space W1,n,q ([0,1]n, Y). On the other hand, we show that mappings in the space W1,n,1 ([0,1]n, Y) satisfy condition (N). This implies that the target Y can be at most n-dimensional.