RUI: Characterization Problems, Outer Models, and Forcing

点击翻译按钮获取中文摘要

ABSTRACTThe investigator will continue his research in enumerative andalgebraic combinatorics. He feels that there are many opportunitiesfor strengthening the myriad connections between combinatorics andother branches of mathematics. He will investigate how recentbreakthroughs in the theory of total positivity can be applied to openproblems concerning such topics as counting faces in cubical polytopesand counting stable sets in clawfree graphs. The investigator willalso continue his research on the combinatorial properties of convexpolytopes that have arisen in such areas as statistical inference andthe representation theory of semisimple Lie algebras. In particular,he will investigate further some convex polytopes related to Kostant'spartition function. He will in addition pursue a number ofmiscellaneous problems arising in the work of Kac, Kontsevich,Varchenko, and others.The field of combinatorics was first systematically investigated inthe 1960's and only recently has reached a high level of maturity. Itis an area of mathematics that has close connections with many othersubjects, ranging from algebra, geometry, and statistics withinmathematics, andcomputer science, high-energy physics, chemistry, andmost recently biology without. The investigator's primary interest isthe development of connections between combinatorics and other areasof mathematics. He will investigate a number of problems that wouldcontinue the development of the connections between combinatorics andother branches of mathematics and that would lead to the developmentof tools that could be used by scientists outside of mathematics.