Studies in Algebraic Combinatorics

调查员将继续他的研究在枚举和代数组合。 他认为，有许多机会加强组合数学和其他数学分支之间的无数联系。 他将研究如何在总积极性理论的最新突破，可以应用到开放的问题，如在立方多面体计数面和计数无爪图稳定集等主题。 调查员还将继续他的研究，在统计推断和半单李代数的表示理论等领域出现的凸多面体的组合性质。 特别是，他将进一步调查一些凸多面体有关的Kostant的分区功能。 此外，他还将继续研究卡茨、孔采维奇、瓦尔琴科等人的工作中出现的一些杂项问题。组合数学领域在20世纪60年代首次进行了系统的研究，直到最近才达到高度成熟。 它是一个数学领域，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，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数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数学和其他数学分支之间的联系，并将导致开发可供数学之外的科学家使用的工具。

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.