# Max Engelstein

Assistant Professor
Department of Mathematics
University of Minnesota-Twin Cities
Office: Vincent Hall 230

I am an Assistant Professor (Tenure Track) at the University of Minnesota, Twin-Cities broadly interested in the interplay between harmonic analysis, geometric measure theory, the calculus of variations and PDE.

Previously, I was a CLE Moore Instructor and an NSF Postdoc at MIT, where my sponsoring scientist is Professor David Jerison. I spent the Spring of 2017 at MSRI as a postdoc in the Harmonic Analysis program, where my mentor was Professor Svitlana Mayboroda. Before that, 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.

## Papers

### Rectifiability and almost everywhere uniqueness of the blow-up for the vectorial Bernoulli free boundaries

with Guido De Philippis Luca Spolaor and Bozhidar Velichkov
Preprint ArXiv. 10 pages.
[Abstract ±]

### Quantitative Stability for Minimizing Yamabe Metrics

with Robin Neumayer and Luca Spolaor
Submitted ArXiv. 22 pages.
[Abstract ±]

### Non-uniqueness of Bubbling for Wave Maps

with Dana Mendelson
Submitted ArXiv. 26 pages.
[Abstract ±]

### Regularity for almost-minimizers of variable coefficient Bernoulli-type functionals

with Guy David, Mariana Smit Vega Garcia and Tatiana Toro
Math. Zeitschrift, to appear. ArXiv. 41 pages.
[Abstract ±]

### Two Phase Free Boundary Problem for Poisson Kernels

with Simon Bortz, Max Goering, Tatiana Toro and Zihui Zhao
Indiana University Math J., to appear. 2019. ArXiv. 42 pages.
[Abstract ±]

### The Riemannian Quantitative Isoperimetric Inequality

with Otis Chodosh and Luca Spolaor
Journal of the European Math Soc., to appear. 2019. ArXiv. 28 pages.
[Abstract ±]

### Minimizers for the thin one-phase free boundary problem.

with Aapo Kauranen, Martí Prats, Georgios Sakellaris and Yannick Sire
Comm. Pure. Appl. Math., Vol 74, 2021, pp 1801-2022. ArXiv. 43 pages.
[Abstract ±]

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

with Guy David and Svitlana Mayboroda
Duke Math J., Vol 170, 2021, pp 455-501. 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
Revisita Matematica Ib., 36 , 2020, pp 1375-1408. 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
Duke Math J., Vol 169, 2020, pp 1541-1601. ArXiv. 36 pages.
[Abstract ±]

### Reifenberg Flatness and Oscillation of the unit Normal Vector

with Simon Bortz
Not For Publication. Superseded by BEGTZ above. 2017. ArXiv. 25 pages.
[Abstract ±]

### Free Boundary Regularity for Almost-Minimizers

with Guy David and Tatiana Toro
Advances in Math. 350, 2019, pp 1109-1192. 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 ±]

## Recent 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.

## Teaching/Organizing

In the Academic Year 2021-2022, I am teaching Math 2373: CSE Linear Algebra and ODE. More details will be linked to here when they are available.

I am a co-organizer of the UMN PDE seminar which meets at Wednesday at 3:35pm during the academic year. If you are interested in attending or giving a talk please contact me.

I am currently taking on graduate students. If you are a graduate student at UMN interested in the math that I do, feel free to email me.

In the Summer of 2021, I led the MathILy-EST REU on Discrete Free Boundary Problems. We are currently writing up our results. If you are a first or second year undergraduate student interested in doing an REU, I recommend checking out MathILy-EST.

In the Academic Year of 2020-2021, I taught 8601-8602, the first graduate sequence in Real Analysis. I have some (informal) notes from that class that I am happy to share on request.

Together with Mark Allen, Spencer Becker-Kahn and Mariana Smit Vega Garcia, I co-organized 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 August 2021--All Travel Tentative due to COVID]

CBMS Conference: Analysis, Geometry, and PDEs in a Lower-Dimensional World at Florida State University, December 2021
Harmonic Analysis and PDE", May 2022 at Yonsei University in Seoul, South Korea.
Special sessions on Elliptic PDEs and Geometric Variational Problems and on Analysis of Nonlinear PDEs and Applications
AIMS Conference on Dynamical Systems, Differential Equations and Applications, Rome, June 2022.

## 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.

I also have my application materials from my tenure track job search in the Fall of 2018 and my NSF grant application in the Fall of 2019. I feel a little too shy to share them on my website right now but if you are currently applying and want to see my materials please ask (even if you don't know me!)

Support/Disclaimer: I'm partially supported by the NSF (DMS 1703306 and DMS 2000288) 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, the Simons Foundation or my employer, the University of Minnesota.