## Research

Rectifable Sets, Harmonic Measure, General Geometric Measure Theory, Miscellaneous### Rectifiable Sets

Given a set (or metric space) how can I tell if and how well the set resembles Rd? There are various degrees of resemblance you could demand, like being isometric. One variant that I consider is the property of being rectifiable: a set is d-rectifiable if it can be covered up to d-dimensional measure zero by Lipschitz images of Rd. One can think of these sets as measure theoretic versions of smooth manifords: a rectifiable set is one that is "locally" parametrized by Lipschitz chart maps. There are even different degrees of rectifiability (like uniform rectifiability). Much of my work concerns with classifying when a set is rectifiable and studying the properties of rectifiable sets. There is a particular emphasis on quantitative analysis, and this only became more important around the 90s in connection with boundedness of singular integrals and Painleve's probelm.

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**Epilogue:**The counterexample mentioned above means that we cannot characterize rectifiable measures using $\alpha$-numbers we define. However, Damian Dąbrowski has developed a full characterization using $\alpha$-numbers based on the $W_{2}$-Wasserstein distance.

We develop a quantity $\beta_{E}^{d}(Q)$ that measures the $d$-flatness of a set $E\subseteq \mathbb{R}^{n}$ inside a cube $Q$ so that, if $E$ is a Reifenberg flat surface,
\[ (\mbox{diam } E)^{d}+\sum_{Q\subseteq \mathbb{R}^{n} \atop Q\cap E\neq\emptyset} \beta_{E}(3Q)^{2}\ell(Q)^{d}\sim_{n} \mathscr{H}^{d}(E).\]
The result also holds for surfaces $E\subseteq \mathbb{R}^{n}$ satisfying Condition B, that is, every ball centered on $E$ has two large balls in different components of $E^{c}$. This gives a generalization of the (Euclidean) Analyst's Traveling Salesman Theorem of Peter Jones and Kate Okikiolu, which originally held only for curves.

** Note:** The published version has many annoying typos and small mistakes that make for hard reading. For this reason, I am keeping an up-to-date version here where I correct any mistakes that I happen to find or that people point out. If you have any questions about the paper or notice an error, let me know so I can sort it. Slides, Video.

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*$1$-rectifiable*, meaning there are countably many curves $\Gamma_{i}$ of finite length for which $\mu(\mathbb{R}^{d}\backslash \bigcup \Gamma_{i})=0$. In this note, we give a characterisation of when a doubling measure with connected support in a metric space has a $1$-rectifiable subset of positive measure and show this set coincides up to a set of $\mu$-measure zero with the set of $x\in \mbox{supp} \mu$ for which $\liminf_{r\rightarrow 0} \mu(B(x,r))/r>0$.

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**Epilogue:**This result was generalized by Edelen, Naber and Valtorta. They have been used in a vareity of contexts, like line defects in liquid quasicrystals, obstacle problems, calculus of variations, and singular integrals.

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**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.

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**Epilogue:**This result was the main part of my thesis. It was improved upon by Raanan Schul and Guy C. David in 2020, see here for video seminar about their result, and the video also starts by giving a synopsis of our result.

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### Harmonic Measure

For a domain $\Omega\subseteq \mathbb{R}^{n}$, its *harmonic measure with pole at $x$*, denoted $\omega_{\Omega}^{x}$ can be defined two ways: one is that, for $A\subseteq \partial\Omega$, $\omega_{\Omega}^{x}(A)$ is the probability that a Brownian motion starting at $x\in \Omega$ first hits $\partial \Omega$ in the set $A$. Alternatively (and assuming our domain is "nice" enough), for each continuous function $f$ one can solve the Dirichlet problem to find a harmonic function $u_{f}$ on $\Omega$ that agrees with $f$ on the boundary; for $x\in \Omega$, the map $f\mapsto u_{f}(x)$ is a bounded linear functional and the Riesz Representation Theorem supplies a measure $\omega_{\Omega}^{x}$ so that $u_{f}(x) = \int_{\Omega}fd\omega_{\Omega}^{x}$. This is quite a natural measure to give the boundary of a domain, and so it's natural to ask what the relationship between the geometry of the domain and the behavior of its harmonic measure is. My work in this area has mostly focused on the relationship between the reguarity of harmonic measure and the rectifiable structure of the boundary.

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- We extend the results of Kenig, Preiss, and Toro [KPT09] by showing mutual
absolute continuity of interior and exterior elliptic measures for
*any*domains implies the tangent measures are a.e. flat and the elliptic measures have dimension $n$. - We generalize the work of Kenig and Toro [KT06] and show that VMO equivalence of doubling interior and exterior elliptic measures for general domains implies the tangent measures are always elliptic polynomials.
- In a uniform domain that satisfies the capacity density condition and
whose boundary is locally finite and has a.e. positive lower $n$-Hausdorff
density, we show that if the elliptic measure is absolutely continuous with
respect to $n$-Hausdorff measure then the boundary is rectifiable. This
generalizes the work of Akman, Badger, Hofmann, and Martell [ABHM17].
Finally, we generalize one of the main results of [Bad11] by showing that if
$\omega$ is a Radon measure for which all tangent measures at a point are
harmonic polynomials vanishing at the origin, then they are all homogeneous
harmonic polynomials.

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**Epilogue:**Tolsa recently generalized the results of this paper by showing that harmonic measure for

*any*domain whose boundary has positive and finite $s$-dimensional measure is singular with respect to the $s$-dimensional surface measure on the boundary.

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### General Geometric Measure Theory

Most of my work falls into the broad category of geometric measure theory. Here I list a few one-off results in this area that aren't connected to my more specific subareas areas.

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### Miscellaneous

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