Math 491, offered in conjunction with the NSF-funded Spatial Systems Laboratory (SSL), will bring students to the point of being able to conduct supervised original research in low-dimensional combinatorics, using algebraic and bijective techniques. Methods taught will include recurrence relations (linear and non-linear), transfer matrices, and generating functions; special topics will include triangulations, tilings, Markoff numbers, and Somos sequences.
This course will not require much in the way of prior background in combinatorics (beyond the level of, say, the use of binomial coefficients in counting combinations), but students who take the course will find that a background in discrete mathematics or graph theory, and in the writing of proofs, will be helpful.
There will be an emphasis on discovery and the use of computers (specifically Maple). Prior knowledge of Maple is not required. For more information, see http://www.math.wisc.edu/~propp/491/.
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 a class of mathematical models lying at the interface between algebra and combinatorics. Some of these models have arisen from chemistry and physics, but our main interest will be in mathematical issues. Computers will play a large part in the research.
Participants in the Spatial Systems Laboratory receive pay (rather than academic credit).
For more information, see http://www.math.wisc.edu/~propp/SSL/.