Modern research mathematics is moving steadily in the direction of more collaboration. The stereotypical mathematician may be a solo thinker, waiting for inspiration to strike, but in fact pairs and small teams of researchers are becoming the norm. The Research in Pairs (RiP) program at the Oberwolfach Institute in Germany is one of several established models that seek to support collaboration by supporting joint stays.

At Mathcamp, we have implemented an Oberwolfach-style Research in Pairs program in a novel setting: a nationally recognized five-week summer program for mathematically talented high school students!

Created with the support of a grant from the National Science Foundation, the Research in Pairs program at Mathcamp hosts by pairs of mathematical collaborators. These visitors work on research during their stay, as well as teaching a companion course, with the goal of making the production of new mathematics visible to our students. We encourage mathematical interactions between the campers and the research visitors, both inside and outside of class.

Over 100 high school students will be exposed to research in progress every summer through this new program. A smaller group of students will be systematically introduced to brand-new mathematics in the companion course.

Research in Pairs at Mathcamp (RIPM) provides a unique mentoring experience for the visiting researchers. More importantly, students at Mathcamp gain an early window into the practice of doing mathematics research. The pipeline to careers in the mathematical sciences loses talented students partly because so few can imagine themselves as creative, contributing mathematicians. RIPM introduces talented teenagers to role models who can both show them what research looks like and discuss its daily rewards and challenges.

Summer 2015:

- Mohamed Omar (Harvey Mudd College) and Matthew Stamps (Yale-NUS College). They taught a course called "Counting the Faces of Cut-Up Spaces". Their research project was entitled "The Matroid of a Tensor Product of Graphs."
- Adam Larios (University of Nebraska-Lincoln) and Jared Whitehead (Brigham Young University). They taught a course called "The Hidden Dance of Partial Differential Equations". Their research project was entitled "Universal Bounds on the Attractor of the Boussinesq Equations of Ocean Flows".

Summer 2014:

- Allison Henrich (Seattle University) and Sam Nelson (Claremont McKenna College). They taught a course called "Knots, Games, Coloring, and Algebra."
*Sam Nelson writes:*My research partner Allison Henrich and I spent a week at Canada/USA Mathcamp in July 2014 as a Research in Pairs pair. We are both topologists who study algebraic and combinatorial invariants of knots, including a generalization of knots known as virtual knot. While there, we made progress on our project defining axiomatic knot theory and started a new project on forbidden twisted virtual knot theory.

We team-taught a week course on knot theory with interactive components as well as lecture components; the students seemed very excited by Allison's colloquium talk on knot games, after which we were frequently joined at mealtime by students and faculty alike for discussion. Our research-in-pairs panel discussion was well-attended and featured a lively discussion of what the life of a researcher is like. My mathcamp experience ended on a high note with a camper telling me she is starting her freshman this fall at fellow Claremont consortium member Pomona College where she might see me in classes.

- Mohamed Omar (Harvey Mudd College) and Matthew Stamps (Royal Institute of Technology, Sweden). They taught a course called "Graph Coloring." Their research project was entitled "The Geometry of Metric Graphs".
*Mohamed Omar writes:*During our visit to the Canada/USA Mathcamp, Matthew Stamps and I worked on the problem of generalizing the space of phylogenic trees to include all metric graphs on a given number of taxa (i.e. vertices). We made some pivotal discoveries at Mathcamp, most vividly that some of the assumptions we had made about the topology of this space were dead wrong.

We had some very fruitful interaction with the Mathcamp students. Many evenings were spent in the lounge publicly displaying the work we were doing. We had several students ask us about the content of our work and about the practice of mathematical research. We had the opportunity to field questions from students about the life of a mathematician in an informal seminar. Matt and I played many strategic board games with campers, and most notably ran an exciting trivia night that more than half the camp attended!

Summer 2013:

- Sam Payne (Yale) and David Jensen (SUNY Stonybrook). They taught a course called "Chip Firing and Sandpiles".

Summer 2012:

- Ben Schmidt (Michigan State University) and Craig Sutton (Dartmouth). They taught a course called "Constructible Numbers". Research topics: differential geometry, length spectra.
- Joe Rabinoff (Harvard) and Kirsten Wickelgren (Harvard). They taught a course called "Unique Factorization and Fermat's Last Theorem". Research topics: algebraic geometry, algebraic topology, number theory
- Rachel Epstein (Harvard) and Karen Lange (Wellesley College). They taught a course called "Randomness". Research topics: mathematical logic.
- Chris Hoffman and Matthew Kahle (Ohio State). They taught a course called "The Probabilistic Method". Research topics: probability in geometry and topology.

Summer 2011:

- Sarah Koch (Harvard) and Jim Belk (Bard). They taught a course called "Complex Dynamics". Research topics: complex dynamics, polynomials.
*Sarah Koch writes:*At Mathcamp, we worked on a problem that lies in the intersection of topology, dynamics, geometry, group theory and combinatorics. It arises from Thurston's Topological Characterization of Rational Maps, a fundamental result that is the cornerstone of the field of Complex Dynamics. This result provides criteria under which a "floppy" topological object has a natural rigid geometry (an undercurrent of much of Thurston's work). Loosely speaking, this theorem provides a purely topological criterion under which a

*critically finite*ramified cover*f*:*S*^{2}→*S*^{2}is*equivalent*to a rational map*F*:*P*^{1}→*P*^{1}.When we were at Mathcamp in 2011, we proved that if (

*S*^{2},*P*_{f}) → (*S*^{2},*P*_{f}) is an obstructed Thurston map such that |*P*_{f}|=4, then there are precisely an integer's worth of (classes of) Thurston maps with the same combinatorics as f. We have a conjecture about the general case. During the week of our visit, we taught a course in Complex Dynamics. We had about 10-15 students on a given day. We used a lot of pictures in our presentations; this is a major advantage enjoyed by people who study Complex Dynamics. It is possible to use these beautiful pictures to explain the basic ideas of the subject. The students in our course seemed to be very interested in the material we spent some time going over the concepts that we used to draw the pictures, and we spent some time exploring the pictures we saw. They really liked zooming into the Mandelbrot Set a thrilling experience!I gave a colloquium about matings of polynomials, and it was a lot of fun! I showed some movies on the computer and a lot of pictures. The audience was asking a lot of questions, and many people seemed very interested in what they were looking at. As a speaker, it is very exciting to have such an enthusiastic audience.

Summer 2010 (our inaugural year):

- Noah Snyder (Columbia) and Emily Peters (MIT). They taught a course called "Planar Algebras". Research topics: operator algebras, quantum groups.

*Noah Snyder writes:*"During our week at Mathcamp we worked on a longterm research project to classify subfactors of index less than 5, and we taught a joint class on the Temperley-Lieb algebra in which the students worked through a proof of a version of the Jones index theorem. That week we were able to make progress which led to a substantial breakthrough in our research program. Furthermore, our class introduced our students to the topic we were researching, concluding with an explanation of the main results of our recent papers.

Both our class and our research concerned the study of subfactors. Factors are certain operator algebras which one can think of as noncommutative analogues of fields. Thus the theory of inclusions of factors is a noncommutative analogue of Galois theory, with planar algebras playing the role of groups. The theory of subfactors attracted great attention starting in the 80's due to Jones's celebrated index theorem, and relationships to knot theory, quantum groups, statistical mechanics, and quantum field theory. The index theorem gives a list of all possible indices less than 4, and thus is a key step towards classifying subfactors of index less than 4. Using the modern language of planar algebras it's possible to state and prove the main content of the index theorem in language accessible to advanced high school students."

We are looking for collaborators who meet the following criteria:

- Aspects of your work are accessible to high school students (with a little poetic license if necessary);
- You're comfortable with high school students and willing to try "fishbowl research" - working in public and letting people peek in at the process;
- You're interested in giving a colloquium and a short course (3-5 days at one hour per day) on some math that is suitable for the students and loosely related to your joint work;
- You're willing to seek out students and actively try to engage them in your interests.

If this sounds like you, get in touch!

Contact us for more information. This program was supported by NSF grant DMS-1242617 in 2012 - 2015.