**Who:** Six early college students who are deeply but perhaps informally prepared
for mathematics research.

**What:** Eight weeks of mathematical research on discrete free boundary problems. More details.

**When:** June 13 - August 8, 2021
(8 weeks).

**Where:** Alongside the participants in the
MathILy summer program.

**Why:** Because students who are prepared for research should have
the opportunity to do it.

**The application process:** Consists of personal information, information about your mathematical
studies, a multi-question personal statement, and two recommendations (instructions will be sent to recommenders when you submit the aforementioned information), all of which must be received by April 2, 2021. More details.

**Stipend:** $3800, plus on-campus housing and meals if MathILy-EST is in person, or a meal allowance if MathILy-EST is online.

**Note about SARS-CoV-2/COVID-19:** MathILy-EST *will* take place in 2021---it has not yet been determined whether we will meet in person or synchronously online.

**Research Description:**
In 2021 the MathILy-EST topic will be **discrete free boundary problems**, under the direction of Dr. Max Engelstein.

Free boundary problems are a class of equations that often arise when some physically natural energy is minimized: every time you pull plastic wrap tight over a heaping plate of leftovers you solve a free boundary problem. Here, the edge of the plate is a fixed boundary, and the "free boundary" is where the plastic wrap comes detached from the pile of leftovers. See Figure 1.

This summer we will look at discrete versions of "Bernoulli" free boundary problems. We'll label each point of a square grid (more fancily, \(\mathbb Z^2\)) with a height, or more formally, let \(f: \mathbb Z^2 \rightarrow \mathbb R\). We model the energy by counting the grid points with positive height and adding the average squared difference in height between each pair of adjacent grid points. That gives us \[ E(f) = \#\{v\mid f(v) > 0\} + \frac{1}{4}\sum (f(v)-f(w))^2,\] (with 1/4 because each vertex has four neighborhoods). This energy function was originally proposed to model water cavitation, where the change in pressure from some quick motion causes bubbles.

Figure 1: A plate of leftovers, with plastic wrap, fixed boundary, and free boundary shown.

Hugely simplifying, the first term represents the kinetic energy of the water flow and the second term takes into account the effects of pressure; the "free boundary" consists of points where \(f > 0\) that are adjacent to points where \(f = 0\) (this originally represented the interface between water and air). But we won't think much about water this summer---indeed, minimizers of a continuous version of \(E\) also have cool connections to random walks and eigenvalue problems; part of this project will be to see if these connections still exist in the discrete world!

A lot is known about the free boundary of critical points of the continuous version of \( E\) in the plane (see Figure 2 for some examples). But for the discrete problem there is still a lot to explore---what qualitative properties do these minimizers possess? Is the free boundary always connected or can it have lots of different components? Can the minimizing function alternate between positive and zero or does it switch precisely once? How do all these properties depend on the shape of the underlying grid (what if it is triangular or hexagonal as opposed to square)?

Figure 2: Critical points of continuous \( E \) are themselves functions. Here we have three examples of critical-point functions and their free boundaries.

**Research Mentor:** Dr. Max Engelstein is an assistant professor of mathematics at the University of Minnesota in Minneapolis. As an undergraduate he participated in two REUs both of which are related to the current topic. Since then he has supervised undergraduate research projects as a PhD student at U Chicago and a postdoc at MIT. One of his main research interests is free boundary problems and he has published seven papers on free boundary problems related to the current REU topic. His interests lie at the intersection of two of the oldest subjects in mathematics: harmonic analysis and the calculus of variations.

**Goals and Expectations:**
During the summer, each of the participants will be expected to...

- Meet each weekday (and some weekends) to collaborate with other participants on research projects.
- Record progress, ideas, experiments, conjectures, and proofs each week for submission to the MathILy-EST director.
- Meet regularly (at least twice per week) with the MathILy-EST director, both individually and as groups, to assess progress and receive advice on how to proceed.
- Attend the weekly Ever-EST Seminar to learn how to traverse the mathematical community's peaks and valleys.
- Attend the Daily Gather, an hour-long interactive mathematics lecture or activity that takes place each weekday afternoon.

Blackboards, whiteboards, glass... we have it all.

**Prerequisites:**
Applicants must be undergraduates in good standing. Preference will be given to first-year
college students, with second-year and entering college students also considered for
participation. NSF funding requires that participants must be US citizens or permanent residents.

**Transitioning into Research:**
MathILy-EST participants will have little-to-no formal experience with
mathematics research. To ease participants into the work they will be doing, the program
will start with the research mentor providing an overview of the project that includes
pointers to literature and concrete examples that are associated with questions to
investigate. Participants will be assigned sets of readings that they then must present
to one another, as well as hands-on explorations so that they can familiarize themselves
with relevant examples and generate data as a start to working on open problems.

The weekly record of progress that participants are expected to keep will provide practice in writing formally and building in time for detailed feedback and revisions of writing. In addition, it will provide fodder for paper drafts once research has progressed to the point of publication, and regular presentations on research progress will provide participants with feedback on their research methods and presentation skills.

Professional development for MathILy-EST participants will include...

- Instruction in LaTeX, Sage,
*Mathematica*, and any specialized software relevant to the research project. - How to find and access relevant books and journal articles, both through the library and MathSciNet
- What a CV is, and how to draft one
- How to give a presentation and a conference talk
- How to construct a research paper
- What sort of careers one can go into with a degree in mathematics
- How to apply for and choose graduate schools

**MathILy and MathILy-EST:**
The MathILy-EST REU will overlap with the MathILy summer program. The
two programs share space---MathILy-EST participants will stay in the same dorms as MathILy students
and instructors, and have work space in the same building as the MathILy classrooms---which
will provide plenty of opportunities for social interactions, from casual mathematical
conversations in the public areas of the dorm room to afternoon frisbee,
board, and card games, to the annual Philadelphia trip.

**After the Program:**
At the end of MathILy-EST, the research mentor will help participants put together plans
for presenting their research at local and national conferences, and will check in
regularly during the following semesters to make sure they're keeping to a reasonable
timeline for preparing publications and conference presentations. In addition, MathILy-EST
participants will be invited to the {MathILy, MathILy-Er, MathILy-EST} Yearly Gather at the Joint Mathematics
Meetings, which will provide a chance to catch up with their fellow participants and to
network with participants from other years and programs.

**Practical stuff:** We'll give you all the details after you've applied and if you are admitted.

**If after reading this whole page you still have questions,**
please do contact us at .

*MathILy, MathILy-Er, and MathILy-EST are projects of the nonprofit organization Mathematical Staircase, Inc..*