Who: Six early college students who are deeply but perhaps informally prepared for mathematics research.
What: Eight weeks of mathematical research on the combinatorial geometry of origami (paper folding). More details.
When: June 17 - August 12 (8 weeks).
Where: Bryn Mawr College, alongside the participants in the MathILy summer program. More details.
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 17, 2019. More details.
Stipend: \$3800, plus on-campus housing and meals while at Bryn Mawr.

Research Description: In 2019 the research topic of MathILy-EST will be 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.

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.

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.

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.

Why Bryn Mawr is an awesome place to be in the summer:
The dorms are really nice, and have air conditioning. (So do the classrooms.) You're very likely to have a single room.
The campus is beautiful...really beautiful. And very safe!
The food is excellent. (Still, if you suddenly need pizza or snacks, there are several pizza places and a grocery store near campus.)
Laundry is free.
You're right near Haverford, Bryn Mawr, Swarthmore, UPenn, Drexel, Temple, and Villanova, all excellent institutions of higher education.
Campus is also right near a hospital, shopping center, etc.
Philadelphia, which has tons to offer---including art and science and medicine and natural history museums, a zoo, shopping malls, a farmers market, the Italian Market, the Declaration of Independence (yes, the real thing), the Liberty Bell, and about a zillion other things---is a short train ride away.

Practical stuff: We'll tell you how to get to campus (planes and trains and buses and cars all work well), what you will want to bring (there is a list), details of internet access, how dietary restrictions are accommodated, and more, after you've applied and if you are admitted.