My work focuses on the connections between geometric measure theory, harmonic and complex analysis, and related fields such as elliptic partial differential equations. In addition to the brief descriptions below, all my preprints are available on arXiv
, and my CV is here
Preprints and Publications
- Improved Hölder continuity of quasiconformal maps, to appear in Ann. Acad. Sci. Fenn. Math. Quasiconformal maps have certain Hölder continuity properties, among other deep results in their regularity theory. These characteristics are related to some deep questions - for example, it is an open problem to find a (general) way of detecting when a solution to a Beltrami equation is bilipschitz. This is also related to the connectivity of the manifold of chord-arc domains. This work involves studying the Hölder continuity exponent from certain local averages of the Beltrami coefficient, as well as looking at the local geometric distortion of a map through isoperimetric inequalities.
- Geometric bounds for Favard length, published in Proc. Amer. Math. Soc. This work involves Favard length and Buffon needle problems in the plane, developing new techniques and estimates that apply to some classes of self-similar sets that arise naturally in fractal geometry. These can be used to give bounds for average projection lengths of sets. This gives an interesting notion of the size of a set (as well as a quantitative measure of rectifiability) that follows classical work of Marstrand and Besicovitch.
- Stretching and rotation sets of quasiconformal mappings, published in Ann. Acad. Sci. Fenn. Math. Quasiconformal maps have many useful geometric distortion properties; in this work I studied the size (in the sense of Hausdorff measure) of the sets where a given quasiconformal map can have prescribed stretching and rotation. The analysis relies on a Cantor set construction (following ideas of my advisor Ignacio Uriarte-Tuero and the work of Lauri Hitruhin) to show that these distortion sets can be very large. The notions of size involve both those in geometric measure theory (gauged Hausdorff measure) and potential theory (Riesz capacities).
- Commutators of the Hilbert transform along a parabola (slides given at ORAM 2019); joint with Zihua Guo, Ji Li, and Brett Wick. Lp bounds for commutators are frequently used to characterize BMO spaces, going back to classical work of Coifman, Rochberg, and Weiss. For the Hilbert and Riesz transforms (among many other singular integral operators), commutators against functions are bounded precisely when the function has bounded mean oscillation. In this work, we study what happens in the case that the operator is the Hilbert transform along a curve. This involves substantial new difficulties, because this operator is singular in a much different way than a typical Calderón-Zygmund operator.