Studies in Algebraic Combinatorics

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。研究者还将继续研究凸多面体的组合性质，这些性质已经在统计推断和半单李代数的表示理论等领域出现。特别地，他将进一步研究一些与Kostant配分函数相关的凸多面体。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。这是一个与许多其他学科有着密切联系的数学领域，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。他将研究一些问题，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。

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.