# Research

## Rectifiability

Given a set (or metric space) how can I tell if and how well the set resembles $\mathbb{R}^{d}$? 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 $\mathbb{R}^{d}$. 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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To appear in Rev. Mat. Iberoam.
• , with D. Dabrowski.
Submitted
• , with M. Hyde.
Submitted
• , with M. Villa.
To appear in Anal. PDE.
• , with X. Tolsa and T. Toro.
Trans. Amer. Math. Soc. 373 (2020), no. 11
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Publ. Mat. 62 (2018), 161--176
• , with R. Schul..
Math. Annal. 370 (3), 1389-1476
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 in the above link 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.
• , with G. David and T. Toro.
Math. Zeit., 286(3), 861-891
• , with M. Mourgoglou.
Anal. PDE 9 (2016), no. 1, 99--109.s
• , with G. David and T. Toro.
Math. Ann. 364 (2016), no. 1-2, 151--224.
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Rev. Mat. Iberoam. 32 (2016), no. 2, 589--648.
• , with X. Tolsa.
GAFA 25 (2015), no. 5, 1371--1412.
• , with M. Badger and T. Toro.
• , with R. Schul.
Proc. Amer. Math. Soc. 142 (2014), no. 4, pp. 1351-1357.
• , with R. Schul.
Geom. Funct. Anal. 22 (2012), no. 5, pp. 1062-1123.
• , with R. Schul.
Proc. Lond. Math. Soc. (3) 105 (2012), no. 2, 367--392.

## 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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• , with M. Mourgoglou and X. Tolsa.
Trans. Amer. Math. Soc. 373 (2020), no. 6, 4359-4388
• , with J. Garnett, M. Mourgoglou, and X. Tolsa.
To appear in IMRN.
• , with S. Hofmann, J.M. Martell, M. Mourgoglou, and X. Tolsa.
Invent. Math. 222 (2020), no. 3, 881-993
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Potential Anal. 53 (2020), no. 3, 1025-1041
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IMRN (2019) np. 3, 6717-6771.
• , with M. Akman and M. Mourgoglou.
• , with M. Mourgoglou.
Anal. PDE 12 (2019), no. 8, 1891-1941
• , with M. Mourgoglou, X. Tolsa, A. Volberg.
Amer. J. Math. 141 (2019), no. 5, 1259-1279
• , with M. Mourgoglou.
Rev. Mat. Iberoam. 34 (2018), no. 1, 305-330.
• , with M. Mourgoglou and X. Tolsa.
Comm. Pure Appl. Math. vol. 70 no. 11 p. 2121--2163.
• , with M. Mourgoglou and X. Tolsa.
IMRN Vol. 2017, No. 12, pp. 3751-3773.
• , with M. Mourgoglou and X. Tolsa.
Anal. PDE (2017) 10 no. 3, 559-588
• , with S. Hofmann, J.M. Martell, K. Nystr\"om, and T. Toro.
J. Eur. Math. Soc. (2017) 19, no. 4, 967-981
• , with S. Hofmann, S. Mayboroda, J.M. Martell, M. Mourgoglou, X. Tolsa, and A. Volberg.
Geom. Funct. Anal. (2016) 26: 703.
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Potential Analysis, 45 (2016) no. 3, 403-433.

## General GMT

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 specific areas.

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Ann. Acad. Sci. Fenn. Math. 44 (2019), no. 2, 889-901
• , with J. Hickman and S. Li.
PAMS v. 146, no. 10, October 2018, p. 4331--4337
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Ark. Mat. 53 (2015), no. 1, 1--36.

## Miscellaneous

• , with M. Hall and R. Strichartz.
Trans. Amer. Math. Soc. 360 (2008), no. 4, 2089--2130.
• , with J. Bedrossian.
Trans. Amer. Math. Soc. 367 (2015), no. 5, pp. 3095-3118.