RUI: Characterization Problems, Outer Models, and Forcing

该研究者将继续他在枚举和代数组合学方面的研究。他觉得有很多机会可以加强组合学和其他数学分支之间的无数联系。他将研究如何将总正性理论的最新突破应用于诸如立方多面体中的计数面和无爪图中的稳定集计数等开放问题。在统计推断和半单李代数的表示理论等领域中出现的凸多面体的组合性质，研究者也将继续他的研究。特别地，他将进一步研究一些与Kostant分割函数有关的凸多面体。此外，他还将研究卡茨、康茨维奇、瓦琴科等人在工作中出现的一些杂项问题。组合学领域在20世纪60年代首次被系统地研究，直到最近才达到高度成熟。这是数学的一个领域，它与许多其他学科有着密切的联系，从数学中的代数、几何和统计学，到计算机科学、高能物理、化学，以及最近的生物学。研究者的主要兴趣是发展组合学和其他数学领域之间的联系。他将研究一系列问题，这些问题将继续发展组合学与其他数学分支之间的联系，并将导致数学以外的科学家可以使用的工具的发展。

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.