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 the interplay between harmonic analysis, geometric measure theory, the calculus of variations and 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 will be starting as an Assistant Professor (Tenure Track), at the University of Minnesota, Twin-Cities in the summer of 2019.

Past Positions: 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 I spent the Spring of 2017 at MSRI as a postdoc in the Harmonic Analysis program, where my mentor was Professor Svitlana Mayboroda.

Click to See My: [CV [Updated Mar. 13th 2019] |Research| Teaching/Organizing | Slides and Reports | Travel| Old Materials ]

Support/Disclaimer: I'm partially supported by an NSF Postdoctoral Fellowship and am an Affiliate Scientist with the Simons Collaboration on Waves. Of course, all of the opinions expressed here are my own and don't necessarily represent the views of either the NSF or the Simons Foundation.


Square functions, non-tangential limits and harmonic measure in co-dimensions larger than one

with Guy David and Svitlana Mayboroda
Submitted. 2018. ArXiv. 36 pages.
[Abstract ±]

Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data

with Matthew Badger and Tatiana Toro
Submitted. 2018. ArXiv. 34 pages.
[Abstract ±]

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

with Luca Spolaor and Bozhidar Velichkov
Geometry & Topology, Vol 23, 2019, pp 513-540. ArXiv.
[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
Preprint, improved version in Progress. 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. Vol 371, 2019, pp 2043-2072. ArXiv.
[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)

Slides and Reports (all mistakes mine, even--especially!--when discussing joint work)

Geometry and the Dirichlet Problem in Any Co-Dimension (An Introduction to Harmonic Measure for Physcists and Engineers)
Simon's Foundation Collaboration on the Localization of Waves, Workshop on the Spectral Properties of Disordered Systems, January 2019.
An Epiperimetric Approach to Isolated Singularities (Oberwolfach Extended Abstract)
Oberwolfach, Calculus of Variations Workshop, July-August 2018.
Regularized Distance and Harmonic Measure in Co-Dimension Greater than One (Seminar Talk)
PCMI, Harmonic Analysis Research Program Seminar, July 2018.
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.


In the Spring of 2019 I am the course administrator for 18.03 Differential Equations.

Together with Mark Allen, Spencer Becker-Kahn and Mariana Smit Vega Garcia, I am co-organizing the AMS Special Section on Interactions between Geometric Measure Theory, PDE, and Harmonic Analysis at the University of Hawaii, Manoa, HI March 22-24, 2019.

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

Upcoming Travel [Current March 13th, 2019]

Geometric and Harmonic Analysis (giving a minicourse)
University of Connecticut, CT March 29th-31st, 2019.
Research visit to University of Washington, Seattle. April 22nd-26th, 2019.
Geometric Analysis Seminar
Rutgers University, NJ April 30th, 2019.
Harmonic Analysis and PDE
University of Helsinki, Finland June 3rd-7th, 2019.

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.