Spatial Systems Laboratory
(The picture shows three graphs, called “snake graphs”, obtained by starting with a single cell with 4 vertices and 4 edges and then adjoining new cells, with each new cell being either to the right of the last cell or above it. These snakes have 89, 493, and 191 perfect matchings, respectively. Snakes are a useful tool for the combinatorial study of Markoff numbers. This picture was created by Rui Viana and Hal Canary.)
Summary
The University's of Wisconsin's Spatial Systems Laboratory (that's SSL for short -- pronounced “sizzle”) is a gathering of undergraduates, graduate students, and UW faculty engaged in exploring mathematical systems drawn from or inspired by the real world. These models are simple enough for us to simulate and prove theorems about, but rich enough in phenomena that we hope that our explorations will generate insights that may be of interest to people outside of mathematics.
The featured project for Fall 2003 and Spring 2004 is a team research effort focusing on combinatorial models related to non-linear algebraic recurrences. In some cases, the combinatorial models have already been described in explicit terms, and our goal is to use the algebra to draw conclusions about the model. But, just as often, we have an algebraic system that seems to be “about” some kind of combinatorial model, except that we don't know what that model is, and must reverse-engineer it from the algebra. If you have Maple, you can click here to see an example (mws.after) of the kind of algebra we're hoping to understand combinatorially. One experiment we tried (suggested by Emilie Hogan) can be found here.
A different sort of experiment (related to formal iteration of Newton's method for iterative solution of polynomial equations, applied to quadratic equations) led to the work-sheet that can be found here.
We now have drafts of the two main articles that were produced this year (the titles are still tentative): Newton's Method as a Formal Recurrence by Hal Canary, Carl Edquist, Samuel Lachterman, and Brendan Younger (now available in the arXiv as math.CO/0407115) and A New Family of Somos-Like Recurrences by Paul Heideman and Emilie Hogan [see the arXiv version or the published version]. We also have a draft of a write-up of one aspect of the work the group did last semester: On Perfect Matchings of the 2-by-2-by-n Grid"
by Paul Heideman.How it works
The faculty leader this semester is Prof. James Propp, and the graduate intern is Stephen Griffeth.
You can read the minutes of all the SSL meetings that were held during 2003-2004.
SSL undergraduate participants meet for 4 hours a week, and are expected to invest roughly 6 additional hours per week outside of group meetings. They are paid on an hourly basis. Based on my experience with running similar research efforts in the past, team-work (and more particularly the sort of brainstorming that occurs when all group-members are present) will play a pivotal role in the success of the research. Consequently, regular attendance will be required.
A course in algebraic combinatorics, Math 491, was offered in conjunction with SSL during Fall 2003. (It was possible to take Math 491 without participating in SSL, and vice versa, but many students found them to be complementary and mutually reinforcing, as Math 491 provided tools that were helpful to students tackling research problems in the Spatial Systems Laboratory.)
For web-pages describing earlier work by undergraduates on related topics, check out the web-site of the Boston area Tilings Research Group and of Research Experiences in Algebraic Combinatorics at Harvard.
The Past
You can find out who has been involved in SSL in past semesters, and what they've done, by visiting our vault for Spring 2000 and our vault for Spring 2001.
People
The Spatial Systems Laboratory during 2003-2004 consisted of the following people:
| Undergraduate participants: | ||
|---|---|---|
| Carl Edquist | ccedquist@wisc.edu | |
| Paul Heideman | ppheideman@students.wisc.edu | |
| Emilie Hogan | eahogan@wisc.edu | |
| Sam Lachterman | slachterman@fuse.net | |
| Abigail Scott | abigailscott@students.wisc.edu | |
| Brendan Younger | bcyounger@students.wisc.edu | |
| Between degrees: | ||
| Hal Canary | hal[e-snail]ups.physics.wisc.edu | |
| Graduate students: | ||
| Stephen Griffeth | griffeth@math.wisc.edu | |
| Martin Hock | mhock@cs.wisc.edu | |
| Postdocs and faculty: | ||
| John Vano | jvano@math.wisc.edu | |
| Melania Alvarez-Adem | melania@math.wisc.edu | |
| Jim Propp | propp@math.wisc.edu | (608) 263-5148 |
Thanks
SSL is sponsored by the National Science Foundation through their VIGRE (Vertical Integration of Graduate Research and Education) program. Additional support comes from the National Security Agency through a grant from its Mathematical Sciences Program.
This page maintained by Jim Propp
Last updated January 28, 2004.