Papers:

*Properties of Discrete Free Boundary Problems on Arbitrary Graphs*by Helen Dai, Max Engelstein, Tess Harvey, Annemily Hoganson, Zoe Markman, River Newman, and Hugo Sanchez; in preparation.

Papers:

*Routing by matching on convex pieces of grid graphs*by Hannah Alpert, RJ Barnes, Seth Bell, Annika Mauro, Na'ama Nevo, Nataya Tucker, and Hanna Yang; in preparation.

Papers:

*Counting locally flat-foldable origami configurations via 3-coloring graphs*by Alvin Chiu, William Hoganson, Thomas C. Hull, and Sylvia Wu; preprint.*Connectivity in origami flip graphs for flat-foldable vertices*by Thomas C. Hull, Manuel Morales, Sarah Nash, and Natalya Ter-Saakov; in preparation.*Maximal-size origami flip graphs of flat-foldable vertices: properties and algorithms*by Thomas C. Hull, Manuel Morales, Sarah Nash, and Natalya Ter-Saakov; in preparation.

**2021 Research Description:**
In 2021 the MathILy-EST topic was **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.

**2021 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.

**2020 Research Description:**
In 2020 the research topic of MathILy-EST was **sorting vertex labels on convex pieces of grids**, under the direction of Dr. Hannah Alpert.

Ignoring the purple pentagon for a moment, we see in Figure 1 a graph with seven vertices and eight edges, along with two ways to number the vertices \(1\) through \(7\). We refer to these numberings as *configurations*, and we refer to the set of all \(7!\) configurations as a *configuration space*. We say that two configurations are *adjacent*, or one step apart, if they differ by swapping a set of disjoint pairs of numbers, with each pair sharing an edge of the graph; the configurations in Figure 1 are adjacent by swapping along the three teal edges.

Figure 1: Two configurations have distance \(1\) if we can get from one to the other by selecting a set of disjoint edges and swapping the two numbers on each edge.

The *distance* between two configurations is the minimum number of steps needed to get from one to the other, and the *diameter* of the configuration space is the maximum distance between any two configurations.

The diameters of these configuration spaces have been studied for rectangular grids and for some other classes of graphs, but in this REU we will select graphs according to their geometric, rather than graph-theoretic, properties. In Figure 1 the graph is constructed by intersecting an infinite rectangular grid with the purple pentagon; we can consider all configuration spaces on graphs that are intersections of a rectangular grid with a polygon \(P\), focusing on the case where the grid boxes are very small relative to \(P\). *When \(P\) is an arbitrary convex polygon in \(\mathbb{R}^2\), does the diameter of the configuration space behave roughly the same as when \(P\) is a rectangle aligned with the grid?*

The geometric motivation is a continuous version of the problem: the configuration space of \(n\) disks of radius \(r\) in the polygon \(P \subseteq \mathbb{R}^2\) is the set of ways to arrange the disks, labeled \(1\) through \(n\), disjointly inside \(P\). In the REU, we will study discrete configuration spaces with the hope of finding properties that are also true of continuous configuration spaces.

**2020 Research Mentor:** Dr. Hannah Alpert is an NSF Postdoctoral Fellow in mathematics at the University of British Columbia in Vancouver, Canada. During her undergraduate years she participated in REUs every summer, and she is an author on six published papers on work she completed before entering graduate school. She first started working on questions related to the REU topic in 2014, with multiple related collaborations still ongoing, and her research interests lie more broadly within geometry and topology.

**2019 Research Description:**
In 2019 the research topic of MathILy-EST was **the combinatorics of flat origami**, an area that combines discrete mathematics with geometry, under the direction of Dr. Thomas Hull.
Origami, otherwise known as paper folding, is modeled by a *crease pattern* \((P,C)\) where \(P\) is a compact region of \(\mathbb{R}^2\) (the paper) and \(C\) is a straight-line planar graph drawn on \(P\). A *flat origami* with crease pattern \((P,C)\) is a function \(f:P\to\mathbb{R}^2\) that is an isometry on every face of \(C\), continuous, and non-differentiable on the edges and vertices of \(C\).

Intuitively, flat origamis are folded paper models that can be pressed in a book without crumpling or adding new creases; see Figure 1 for some examples. We model how the faces of \(C\) get folded above or below neighboring faces via a *mountain-valley (MV) assignment* on the creases. Formally, a MV assignment is a function \(\mu:E(C)\to\{-1,1\}\), where \(E(C)\) is the edge set of \(C\) and \(-1\) denotes mountain (convex) crease while \(1\) denotes a valley (concave) crease.

Figure 1: (a) Mountain and valley creases, with the classic crane crease pattern. (b) The Miura-ori crease pattern, with MV assignment, and the folded result.

A MV assignment is called *valid* if it can be realized in a flat origami \(f\) without causing any self-intersections of the paper. (For example, a crease pattern where all the creases are mountains would not be possible to fold flat, and thus is invalid.)

The major open problem in this area, which has applications in materials science and engineering, is the following: *Given a flat-foldable crease pattern \(C\), how many different valid MV assignments exist on \(C\)?* A recent advance has been the discovery that for some flat-foldable crease patterns this question can be translated into a graph-coloring problem. Developing a general theory of the link between valid MV-assignments and graph colorings (or other combinatorial structures) is the main thrust of this REU. Many smaller open problems exist, and will be explored, as we work towards this goal.

**2019 Research Mentor:** Dr. Thomas Hull is an Associate Professor of Mathematics at Western New England University in Springfield, MA. He has been researching the mathematics of paper folding since 1990 and is considered one of the world experts on the topic. His book *Project Origami* explores the use of origami in math education at the college level, and he is currently finishing *Origametry: Mathematical methods in paper folding*, a research-level monograph on origami mathematics forthcoming from Cambridge University Press.

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