I am a CLE Moore Instructor (and, as of July, 2017, an NSF Postdoc) at MIT broadly interested in applications of harmonic analysis and geometric measure theory to PDE. At MIT, I am working mostly with Professor David Jerison and on problems related to free boundary regularity. I am spending the Spring of 2017 at MSRI as a postdoc in the Harmonic Analysis program.

Up until June of 2016, I was a graduate student at the University of Chicago and my advisor was Professor Carlos Kenig. You can find my thesis, on free boundary problems for harmonic
and caloric measure, here (and you can see a picture of me defending said thesis to the right).

My CV is available
pdf.

##
Papers

with

Guy David and

Tatiana Toro
*Submitted.* 2017.

ArXiv. 70 pages.

[Abstract ±]
In this paper we study the free boundary regularity for almost-minimizers of
the functional $$ J(u)=\int_{\mathcal O} |\nabla u(x)|^2
+q^2_+(x)\chi_{\{u>0\}}(x) +q^2_-(x)\chi_{\{u<0\}}(x)\ dx $$ where
$q_\pm \in L^\infty(\mathcal O)$. Almost-minimizers satisfy a variational
inequality but not a PDE or a monotonicity formula the way minimizers do (see
[AC], [ACF], [CJK], [W]). Nevertheless we succeed in proving that, under a
non-degeneracy assumption on $q_\pm$, the free boundary is uniformly
rectifiable. Furthermore, when $q_-\equiv 0$, and $q_+$ is Hölder continuous
we show that the free boundary is almost-everywhere given as the graph of a
$C^{1,\alpha}$ function (thus extending the results of [AC] to
almost-minimizers).
with

Nick Edelen
*Submitted.* 2017.

ArXiv. 30 pages.

[Abstract ±]
In this paper we prove the rectifiability of and measure bounds on the
singular set of the free boundary for minimizers of a functional first considered by Alt-Caffarelli
(Crelle's Journal '81). Our main tools are the Quantitative Stratification and Rectifiable-Reifenberg
framework of Naber-Valtorta (Annals '17), which allow us to do a type of ``effective
dimension-reduction." The arguments are sufficiently robust that they apply to a broad
class of related free boundary problems as well.
*Submitted* 2016.

ArXiv. 12 pages.

[Abstract ±]
We show that each connected component of the boundary of a parabolic NTA domain in **R**^{2} is given by a graph.
We then apply this observation to classify blowup solutions in **R**^{2} to a free boundary problem for caloric measure
first considered by Hofmann, Lewis and Nyström (Duke, '04).
*Advances in Math.* to appear.

ArXiv. 91 pages.

[Abstract ±]
We study parabolic chord arc domains, introduced by Hofmann, Lewis and Nyström (Duke, '04), and prove a free boundary regularity result below the continuous threshold.

More precisely, we show that a Reifenberg flat, parabolic chord arc domain whose Poisson kernel has logarithm in VMO must in fact be a vanishing chord arc domain (i.e. satisfies a vanishing Carleson measure condition).
This generalizes, to the parabolic setting, a result of Kenig and Toro (Ann. ENS '03) and answers in the affirmative a question left open in the aforementioned paper of Hofmann et al.
A key step in this proof is a classification of ``flat" blowups for the parabolic problem.
with

Matthew Badger and

Tatiana Toro
*Submitted.* 2015.

ArXiv. 41 pages.

[Abstract ±]
The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure.
In order to understand the fine structure of these free boundaries a detailed study of the singular points of these zero sets is required.
In this paper we study how "degree k points" sit inside zero sets of harmonic polynomials in **R**^{n} of degree d (for all n ≥ 2 and 1 ≤ k ≤ d) and inside
sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials.
We obtain a general structure theorem for the latter type of sets, including sharp Hausdorff and Minkowski dimension estimates on the singular set
of "degree k points" (k ≥ 2) without proving uniqueness of blowups or aid of PDE methods such as monotonicity formulas.
In addition, we show that in the presence of a certain topological separation condition, the sharp dimension estimates improve and depend on the parity of k.
An application is given to the two-phase free boundary regularity problem for harmonic measure below the continuous threshold introduced by
Kenig and Toro.
* Ann. Sci. de l'ENS.* **Vol. 49**, 2016, pp 859-905.

Published Version (behind paywall). ArXiv.

[Abstract ±]
We study a 2-phase free boundary problem for harmonic measure first considered by Kenig and Toro and prove a sharp Hölder regularity result.

The central difficulty is that there is no *a priori* non-degeneracy in the free boundary condition.
Thus we must establish non-degeneracy by means of monotonicity formulae.

###
Older Papers (pre-graduate school)

##
Selected Talks

- Parabolic NTA domains in
**R**^{2}. (AMS Sectional Meeting Talk)
- AMS Sectional Meeting, Special Session on Geometric Aspects of Harmonic Analysis, September 2016.
- The structure of the singular set of a two-phase free boundary problem for harmonic measure. (AMS Sectional Meeting Talk)
- AMS Sectional Meeting, Special Session on Geometric Measure Theory and Its Applications, March 2016.
- A Two-Phase Problem for Harmonic Measure (Seminar Talk)
- Seattle Rainwater Seminar. University of Washington, Seattle. Seattle, WA. March 2015.

##
Teaching

I am not teaching in the Spring of 2017 (I'm at MSRI), and I won't be teaching the 2017-2018 Academic year. However, I am available to supervise MIT undergraduates interested in doing reading/research during the 2017-2018 Academic year. If you want to learn some cool analysis, please feel free to contact me.