Studies in Algebraic Combinatorics

这位研究者将继续他在计数和代数组合学方面的研究。他认为，有很多机会加强组合学和其他数学分支之间的千丝万缕的联系。他将研究如何将全正性理论的最新突破应用于有关立方多面体中的面的计数和无爪图中的稳定集的计数等公开问题。研究人员还将继续研究凸多面体的组合性质，这些性质已经出现在统计推断和半单李代数的表示理论等领域。特别是，他将进一步研究一些与Kostant配分函数有关的凸多面体。此外，他还将研究卡克、康采维奇、瓦尔琴科等人工作中出现的一些杂项问题。组合数学领域最早系统地研究于20世纪60年代的S，直到最近才达到高度成熟的水平。这是一个与许多其他学科有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，以及最近没有数学的生物学。研究人员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和其他数学分支之间的联系，并将导致开发出可供科学家在数学之外使用的工具。

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.