Modern research mathematics is moving steadily in the direction of more collaboration. The stereotypical mathematician may be a lone shut-in 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!
Supported by a grant from the National Science Foundation, we host visits to Mathcamp 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.
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 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!
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 : S2 → S2 is equivalent to a rational map F : P1 → P1.
When we were at Mathcamp in 2011, we proved that if (S2, Pf ) → (S2, Pf ) is an obstructed Thurston map such that | Pf |=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 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:
If this sounds like you, get in touch!
Contact us for more information. This program is supported by NSF grant DMS-1242617.