Markov Chain Monte Carlo (George Cobb together with Yung-Pin Chen of Lewis and Clark College).
George Cobb together with Yung-Pin Chen of Lewis and Clark College directed a group of nine students: Valentin Burlacu (Amherst College), Rebecca Horowitz (Hampshire College), Lin Lin (Mount Holyoke College), Ana Mocanu (Amherst College), Jane Ni (Mount Holyoke College), Timothy Teravainen (New College, Florida), Man Yu Yum (Mount Holyoke College). Five of these students were supported with local funds and the remaining four with NSF funds. (A fifth student was accepted and offered NSF support, but withdrew the day before the program started.)
Seven of these students worked in the area of discrete Markov chain Monte Carlo, using computer simulation to estimate the number of binary matrices with given row and column totals. (This problem is one that arises out of ecology, among other applications, and has been an active area of research by Diaconis and Liu.) The areas of mathematics involved include the theory of Markov chains, elementary graph theory, basic group theory, and spectral theory for finite matrices. Two students investigated combinatorial formulas for counting the size of specialized families of such matrices. Results from the first group were presented at MathFest in August 2001; results from both groups were presented at a poster session in San Diego at the annual MAA meeting in January, 2002. A paper based on the work of the first group has been accepted for publication in The American Mathematical Monthly; a paper based on the work of the other two students has been submitted to Discrete Mathematics. These papers are available on the Web at Summer 2001.