Research

My broad research interests are in geometric measure theory, analysis of metric spaces, and harmonic analysis. My work deals with identifying geometric conditions that detect when an arbitrary metric space may be parametrized by Euclidean space and studying the properties of these spaces. Recently, this has focused on Lipchitz analysis and (uniform) rectifiability.

1. Carleson measures, quasisymmetric extensions, and rectifiability of quasispheres, with M. Badger and T. Toro, in preparation.
   • Summary: In a recent paper of Badger, Gill, Rohde, and Toro, it is shown that a square Dini condition controlling the weak-quasisymmetry constant of a hoemomorphism $f:\mathbb{R}^{d+1}\rightarrow \mathbb{R}^{d+1}$ in the neighborhood of the unit sphere $\mathbb{S}^{d}$ guarantees that the image of the sphere is locally bi-Lipschitz parametrizable. In this paper, we weaken the global Dini condition to a Carleson measure condition on the weak-quasisymmetry and show that the image still contains large bi-Lipschitz pieces uniformly. In particular, we show that if $\tilde{H}_{f}(x,r)$ is one minus the weak-quasisymmetry constant of $f|_{B(x,r)}$, and $\tilde{H}_{f}(x,r)^{2}\frac{d\mathcal{H}^{d}dr}{r}$ is a Carleson measure on $\mathbb{S}^{d}\times(0,1)$, then every ball $B(\xi,s)$ with $\xi\in f(\mathbb{S}^{d})$ and $r\in (0,\mbox{diam} f(\mathbb{S}^{d}))$ contains a $L(C,d)$-bi-lipschitz image of a subset of $\mathbb{R}^{d}$ $A$ with $\mathcal{H}^{d}(A)\gtrsim_{C,d}s^{d}$. A good part of the discussion is establishing an extension theorem for approximately affine quasisymmetric maps in the spirit of the work of Tukia and Väisälä. The appearance of an extension theorem is tied up with the problem of isolating subsets of $\mathbb{S}^{d}$ upon which the quasisymmetric map $f:\mathbb{R}^{d+1}\rightarrow\mathbb{R}^{d+1}$ is absolutely continuous.


2. Wasserstein distance and rectifiability of doubling measures, with G. David and T. Toro, in preparation.
   • Summary: In a paper by X. Tolsa, he gives a new characterization of Ahlfors-regular uniformly rectifiable measures in Euclidean space in terms of the so-called $\alpha$-numbers. These are generalizations of David-Semmes $\beta_{1}$-numbers (in turn, a generalization of P. Jones $\beta$-numbers). While the $\beta_{1}$-number measures the average deviation of a portion of the support of a measure from lying in a plane, the $\alpha$-number measures the $1$-Wasserstein distance between a piece of the measure and planar Hausdorff measure. In our paper, we use the $\alpha$-number and some variants to derive sufficient conditions for a measure that is merely doubling (instead of Ahlfors regular) to be rectifiable.


3. Hausdorff dimension of wiggly metric spaces , submitted.
   • Summary: We answer a question of Bishop and Tyson (BT01, p. 3635) and generalize a result of Bishop and Jones by showing that compact connected metric spaces satisfying a uniform non-flatness condition have dimension strictly greater than one quantitatively. This gives an alternate proof that the quasisymmetric image of the antenna set (and more general kinds of antenna sets) into any metric space has Hausdorff dimension greater than one.


4. A quantitative metric differentiation theorem, with R. Schul, to appear in the Proceedings of the American Mathematical Society.
   
• Summary: This short note points out a simple consequence from the results in Hard Sard. For f, a Lipschitz function from a Euclidean space into a metric space, we show that one can estimate how often the pullback of the metric under f is approximately a seminorm. This is a quantitative version of Kirchheim's metric differentiation result from 1994, which we give in the form of a Carleson packing condition.


5. Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps, with R. Schul, GAFA, 22 no. 5 (2012), p. 1062-1123
• Summary: We prove a global implicit function theorem. In particular we show that any Lipschitz map $f:\mathbb{R}^n\times \mathbb{R}^m\to\mathbb{R}^n$ (with $n$-dim. image) can be precomposed with a bi-Lipschitz map $\bar{g}:\mathbb{R}^n\times \mathbb{R}^m\to \mathbb{R}^n\times \mathbb{R}^m$ such that $f\circ \bar{g}$ will satisfy, when we restrict to a large portion of the domain $E\subset \mathbb{R}^n\times \mathbb{R}^m$, that $f\circ \bar{g}$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map $\bar{g}$ distorts $\mathbb{R}^{n+m}$ in a controlled manner so that the fibers of $f$ are straightened out. Furthermore, our results stay valid when the target space is replaced by any metric space. A main point is that our results are quantitative: the size of the set $E$ on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman's 1979 construction of a $C^1$ map from $[0,1]^3$ onto $[0,1]^2$ with rank $\leq 1$ everywhere.
  
  On route we prove an extension theorem which is of independent interest. We show that for any $D\geq n$, any Lipschitz function $f:[0,1]^n\to \mathbb{R}^D$ gives rise to a large (in an appropriate sense) subset $E\subset [0,1]^n$ such that $f|_E$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of $\mathbb{R}^n$. This extends results of P. Jones and G. David, from 1988. As a simple corollary, we show that $n$-dimensional Ahlfors-David regular spaces lying in $\mathbb{R}^{D}$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in $\mathbb{R}^{D}$. This was previously known only for $D\geq 2n+1$ by a result of G. David and S. Semmes.
  
  To prove this extension theorem, we employ an intermediate result which, roughly speaking, says that if a bi-Lipschitz map $f:E\subseteq \mathbb{R}^{n}\rightarrow \mathbb{R}^{D}$ is close enough to being affine on all cubes intersecting $E$, then it permits a bi-Lipschtiz extension to all of $\mathbb{R}^{n}$ with only linear distortion in the bi-Lipschitz constant.
  


6. Bounded Mean Oscillation and the uniqueness of active scalar equations , with J. Bedrossian, accepted to TAMS.
• Summary: We consider a number of uniqueness questions for several wide classes of active scalar equations, unifying and generalizing the techniques of several authors (Yudovich, Bertozzi, et al) As special cases of our results, we provide a significantly simplified proof to the known uniqueness result for the 2D Euler equations in $L^1 \cap BMO$ and provide a mild improvement to recent results of Rusin for the 2D inviscid surface quasi-geostrophic (SQG) equations, which are now to our knowledge, the best results known for this model. We also obtain what are (to our knowledge) the strongest known uniqueness results for the Patlak-Keller-Segel models. We obtain these results via technical refinements of energy methods which are well-known in the $L^2$ setting but are less well-known in the $\dot{H}^{-1}$ setting.
  En route, we prove the following useful interpolation inequality for uniform domains $\Omega$, and give a new proof of the case $\Omega=\mathbb{R}^{n}$: If $f\in L^{p_{0}}(\Omega)\cap BMO(\Omega)$, then $||f||_{L^p(\Omega)}\lesssim_{p_{0},d} p^{1-\frac{p_{0}}{p}}||f||_{BMO(\Omega)}^{1-\frac{p_{0}}{p}}||f||_{L^{p_{0}}(\Omega)}^{\frac{p_{0}}{p}}$ for all $p\in (p_{0},\infty)$.
  



7. How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set , with R. Schul , Proc. London Math. Soc. 105 no.2 (2012), p. 367-392.
• Summary: We show there is $M>1$ such that any rectifiable curve $\Gamma\subseteq\mathbb{R}^{n}$ may be contained in a $M$-quasiconvex set $\tilde{\Gamma}$, where $\mathcal{H}^{1}(\tilde{\Gamma})\lesssim_{n} \mathcal{H}^{1}(\Gamma)$. By $M$-quasiconvex, we mean that for any $x,y\in\tilde{\Gamma}$ there is a curve $\gamma\subseteq\tilde{\Gamma}$ connecting $x$ and $y$ of length no more than $M|x-y|$. This generalizes a result of Peter Jones (1990) that was originally shown in the plane using complex analysis.



8. Conformal energy, conformal Laplacian, and energy measures , with M. Hall and R. Strichartz , in Transactions Amercan Mathematical Society, 360 (2008), p. 2089-2130.
• Summary: On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy $\mathcal{E}_{\phi}$ and conformal Laplacian $\Delta_{\phi}$ for a given conformal factor $\phi$, based on the corresponding notions in Riemannian geometry in dimension $n\neq2$. We derive a differential equation that describes the dependence of the effective resistances of $\mathcal{E}_{\phi}$ on $\phi$. We show that the spectrum of $\Delta_{\phi}$ (Dirichlet or Neumann) has similar asymptotics compared to the spectrum of the standard Laplacian, and also has similar spectral gaps (provided the function $\phi$ does not vary too much). We illustrate these results with numerical approximations. We give a linear extension algorithm to compute the energy measures of harmonic functions (with respect to the standard energy), and as an application we show how to compute the $L^{p}$ dimensions of these measures for integer values of $p\geq2$. We derive analogous linear extension algorithms for energy measures on related fractals.