Combinatorics in geometry and representation theory

几何和表示论中的组合学

• 批准号: 1160726
• 负责人: Thomas Lam
• 金额: $ 25万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1160726&HistoricalAwards=false
• 关键词: Combinatorics; geometry; representation; theory

The proposed project studies combinatorial problems arising from representation theory and algebraic geometry. A particular focus is on problems arising from the study of loop groups, or affine Lie algebras. The PI with collaborators will study the Schubert calculus of flag varieties of affine Lie groups, and the relations to quantum cohomology. This includes understanding the geometry of Schubert varieties and the combinatorics of the polynomials representing Schubert varieties in (co)homology. Together with collaborators, the PI also plans to develop a theory of total positivity for loop groups, aiming to extend work of Lusztig for finite-dimensional reductive groups. Connections with the theory of electrical networks, and to graphs embedded in surfaces will be investigated. Another direction of research is the study of infinite reduced words in infinite Coxeter groups. The PI will study partial orders on infinite reduced words, and explore probabilistic aspects of random infinite reduced words.Combinatorics is the study of discrete structures. Such structures come up in varied contexts throughout science. One such structure that arises in the proposed work is an electrical network consisting of resistors. A basic problem is to understand the relationship between the electrical properties of the network and the connectivity properties of the network. These problems have applications to electrical impedance tomography, a technique in medical imaging. Another type of discrete structure studied in the proposed work are structures with a large number of symmetries, including for example tilings of high-dimensional space in a regular pattern. A basic problem with these structures is to classify them and analyze the large scale behavior. Such structures are important for example in coding theory.

拟议的项目研究组合问题所产生的表示理论和代数几何。 一个特别的重点是从研究循环群，或仿射李代数所产生的问题。 PI与合作者将研究仿射李群的旗簇的舒伯特演算，以及与量子上同调的关系。这包括理解舒伯特品种的几何形状和代表舒伯特品种（共）同源性的多项式的组合。 与合作者一起，PI还计划开发循环群的全正性理论，旨在扩展Lusztig的有限维约化群的工作。 与电网络理论的连接，并嵌入在表面的图形将被调查。 另一个研究方向是研究无限Coxeter群中的无限约化词。 PI将研究无限约简词的偏序，并探索随机无限约简词的概率方面。 这样的结构出现在整个科学的不同背景下。 在所提出的工作中出现的一种这样的结构是由电阻器组成的电网络。 一个基本的问题是理解网络的电性质和网络的连通性之间的关系。 这些问题应用于电阻抗断层成像，一种医学成像技术。 在拟议的工作中研究的另一种类型的离散结构是具有大量对称性的结构，包括例如规则模式中的高维空间的平铺。 这些结构的一个基本问题是对它们进行分类并分析大尺度行为。 这种结构在编码理论中很重要。