CLE Moore Instructor/NSF Postdoc
Department of Mathematics
Office: 2-174
Email: maxe is my username.


I am a CLE Moore Instructor and an NSF Postdoc at MIT broadly interested in applications of harmonic analysis and geometric measure theory to PDE. At MIT my sponsoring scientist is Professor David Jerison and most of my research here has been on problems related to free boundary regularity. I spent the Spring of 2017 at MSRI as a postdoc in the Harmonic Analysis program, where my mentor was Professor Svitlana Mayboroda.

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). While in graduate school I spent several months at the University of Washington, working with Professor Tatiana Toro

My CV is available pdf.


(Log-)Epiperimetric Inequality and Regularity over Smooth Cones for Almost Area-Minimizing Currents

with Luca Spolaor and Bozhidar Velichkov
Submitted. 2018. ArXiv. 19 pages.
[Abstract ±]

Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional

with Luca Spolaor and Bozhidar Velichkov
Submitted. 2018. ArXiv. 36 pages.
[Abstract ±]

Reifenberg Flatness and Oscillation of the unit Normal Vector

with Simon Bortz
Submitted. 2017. ArXiv. 25 pages.
[Abstract ±]

Free Boundary Regularity for Almost-Minimizers

with Guy David and Tatiana Toro
Submitted. 2017. ArXiv. 70 pages.
[Abstract ±]

Quantitative Stratification for Some Free-Boundary Problems

with Nick Edelen
Transactions of the AMS. to appear. ArXiv. 30 pages.
[Abstract ±]

Parabolic NTA Domains in R2

Communications in PDE Vol. 42, 2017, pp 1524-1536. Published Version (behind paywall). ArXiv.
[Abstract ±]

A Free Boundary Problem for the Parabolic Poisson Kernel

Advances in Math. Vol. 314, 2017, pp 835-947. Published Version (behind paywall). ArXiv.
[Abstract ±]

Structure of Sets which are Well Approximated by Zero Sets of Harmonic Polynomials

with Matthew Badger and Tatiana Toro
Analysis & PDE. Vol. 10, 2017, pp 1455-1495. Published Version (behind paywall). ArXiv.
[Abstract ±]

A Two-Phase Free Boundary Problem For Harmonic Measure

Ann. Sci. de l'ENS. Vol. 49, 2016, pp 859-905. Published Version (behind paywall). ArXiv.
[Abstract ±]

Older Papers (pre-graduate school)

Selected Talks

Parabolic NTA domains in R2. (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/Miscellaneous Organizing

I am not teaching the 2017-2018 Academic year. However, if you are an MIT undergraduate interested in doing a reading/research project and want to learn some cool analysis, please feel free to contact me.

Together with Murat Akman, I am co-organizing the AMS Special Section on PDE in Rough Domains at Northeastern University, Boston, MA April 21-22, 2018.

Application Materials

While applying to the NSF Postdoc, I found it very helpful to read several previous proposals which other mathematicians made available on their websites. In that spirit, here is my NSF Summary and NSF Description from my sucessful NSF Postdoc Application in the Fall of 2016. I also applied unsucessfully for the NSF postdoc in the fall of 2015, but seem to have lost those materials.