Enumeration Problems in Algebraic Geometry and Representation Theory

代数几何和表示论中的枚举问题

• 批准号: 1802289
• 负责人: Christopher Manon
• 金额: $ 0.96万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802289&HistoricalAwards=false
• 关键词: Enumeration; Problems; Algebraic; Geometry; Representation

This research project concerns both algebraic geometry, which seeks to characterize solutions to algebraic equations with geometry, and representation theory, which is a systematic investigation of symmetry. Despite their abstract nature, the answers to many questions in both of these subjects boil down to being able to compute certain numbers. The purpose of this research project is to use new techniques from algebraic geometry and commutative algebra to recast these numbers as combinatorial quantities, making them easier to understand with a computer. By utilizing conceptual connections to other scientific fields, this research will also advance the understanding of several questions in mathematical physics and mathematical biology. Undergraduate students are involved directly as collaborators in the project, providing them with training in advanced mathematical topics and the use of 3D printing techniques in mathematical research.The algebraic geometry of moduli spaces of principal bundles and branching varieties naturally produces two interesting enumeration problems: counting branching multiplicities of a map of reductive groups, and finding the dimension of the spaces of conformal blocks from the Wess-Zumino-Novikov-Witten model of conformal field theory. This research aims to further understanding of these quantities using the theory of Newton-Okounkov bodies and the quickly evolving field of Berkovich geometry. These theories will be used to provide new polyhedral descriptions of conformal blocks and branching multiplicities, as well as further the understanding of the topology and symplectic geometry of the spaces under consideration.

该研究项目涉及代数几何形状，该几何形状试图表征用几何形状来代数方程的解决方案，也是代表理论，这是对对称性的系统研究。尽管具有抽象的性质，但这两个主题中许多问题的答案归结为能够计算某些数字。该研究项目的目的是使用代数几何形状和交换代数中的新技术将这些数字作为组合数量重新销售，从而使它们更易于使用计算机理解。通过利用与其他科学领域的概念联系，这项研究还将提高人们对数学物理学和数学生物学中几个问题的理解。本科生直接作为合作者参与了该项目，为他们提供了高级数学主题的培训以及在数学研究中使用3D打印技术。主要的捆绑包和分支品种模量空间的代数几何形状自然地产生了两个有趣的枚举问题：计数群体的综合型，并从综合型组成的繁殖力，从而构成了一组型号的繁殖，并发现了型号的繁殖型，并建立了综合型，并发现了精力的繁殖。 Wess-Zumino-Novikov - 纽约文理论模型。这项研究旨在使用牛顿 - 科恩科夫身体的理论和伯科维奇几何学的迅速发展的领域进一步了解这些数量。这些理论将用于提供共形块和分支多重性的新的多面描述，并进一步了解所考虑的空间的拓扑结构和符号几何形状。