The investigator will continue his research in enumerative and algebraic combinatorics. He feels that there are many opportunities for strengthening the myriad connections between combinatorics and other branches of mathematics. He will investigate how recent breakthroughs in the theory of total positivity can be applied to open problems concerning such topics as counting faces in cubical polytopes and counting stable sets in clawfree graphs. The investigator will also continue his research on the combinatorial properties of convex polytopes that have arisen in such areas as statistical inference and the representation theory of semisimple Lie algebras. In particular, he will investigate further some convex polytopes related to Kostant's partition function. He will in addition pursue a number of miscellaneous problems arising in the work of Kac, Kontsevich, Varchenko, and others.
The field of combinatorics was first systematically investigated in the 1960's and only recently has reached a high level of maturity. It is an area of mathematics that has close connections with many other subjects, ranging from algebra, geometry, and statistics within mathematics, andcomputer science, high-energy physics, chemistry, and most recently biology without. The investigator's primary interest is the development of connections between combinatorics and other areas of mathematics. He will investigate a number of problems that would continue the development of the connections between combinatorics and other branches of mathematics and that would lead to the development of tools that could be used by scientists outside of mathematics.
