1 Results from Prior NSF

During the past five years, the PI has received funding from individual grants from the National Science Foundation and the National Security Agency, as follows: DMS 9500936 (Random Tilings, 1995-1998); DMS 9803249 (Research on Tilings, 1999-2003). As part of these grants, the PI has also received REU Supplements. The PI received additional support from the VIGRE grants awarded by NSF to the University of Wisconsin and Harvard University. He also received funding from the National Security Agency, the most recent of which was MDA904-00-1-0060. During the funded period (from 1995 to the present), the PI wrote or co-wrote seventeen published articles on the funded topics. Much of the PI’s work (funded through the Statistics and Probability subdivision within the Division of Mathematical Sciences) was probabilistic in nature, originating from the PI’s desire to understand the behavior of random tilings. The articles “Exact sampling with coupled Markov chains and applications to statistical mechanics” (with David Wilson, 1996), “Generating random elements of a finite distributive lattice” (1997), “How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph” (with David Wilson, 1998), and “Coupling from the past: a user’s guide” (with David Wilson, 1998) developed and promulgated a general approach to the random generation of combinatorial objects such as tilings, which initiated a (still ongoing) flurry of activity by many researchers in the field of Markov Chain Monte Carlo (the new fashion of “exact sampling” or “perfect sampling”, spearheaded by David Wilson; see http://dbwilson.com/exact/ for a comprehensive bibliography). The PI also contributed to combinatorial probability theory by co-organizing (with David Aldous) a workshop on discrete probability and co-editing a volume of workshop proceedings (“Microsurveys in Discrete Probability”, 1998). Having helped develop new methods of random sampling, the PI applied these methods to the empirical study of random tilings, obtaining many conjectures about the ways in which the shape of the boundary of a region can influence the behavior of a random tiling of that region using tiles of a specified kind. Working with other researchers, the PI was able to prove many of these conjectures, in such articles as “Local statistics for random domino tilings of the Aztec diamond” (with Henry Cohn and Noam Elkies, 1996), “The shape of a typical boxed plane partition” (with Henry Cohn and Michael Larsen, 1998), and “A variational principle for domino tilings” (with