Combinatorial Algebraic Geometry for Spectral Theory and Galois Groups

谱论和伽罗瓦群的组合代数几何

• 批准号: 2201005
• 负责人: Frank Sottile
• 金额: $ 30.38万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2201005&HistoricalAwards=false
• 关键词: Combinatorial; Algebraic; Geometry; Spectral; Theory

Algebraic geometry has found increasing applications to problems arising in the sciences and engineering. This was recognized in the establishment of the Society for Industrial and Applied Mathematics (SIAM) Activity Group in Algebraic Geometry, SIAM biennial conferences, and in the SIAM Journal on Applied Algebra and Geometry. In this project the PI will pursue research and training activities that will extend and deepen the role of algebraic geometry in applications, including mathematical physics. He will also engage in mathematical outreach activities at his institution, nationally, and internationally. This project will provide research training opportunities for students.In more detail, in this project the PI will apply methods from combinatorial algebraic geometry to problems in spectral theory from Mathematical Physics. The spectrum of a discrete periodic operator is a real algebraic hypersurface in an arithmetic toric variety, and many open questions in this area are amenable to ideas from algebraic geometry. The PI will also work towards classifying the Galois groups and the inverse Galois problem in enumerative geometry. To that end, the PI will develop tools, both computational and theoretical, for exploiting and studying Galois theory in applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

代数几何越来越多地应用于解决科学和工程中出现的问题。工业与应用数学学会 (SIAM) 代数几何活动小组的成立、SIAM 两年一次的会议以及 SIAM 应用代数和几何杂志都对此予以了认可。在该项目中，PI 将开展研究和培训活动，以扩展和深化代数几何在包括数学物理在内的应用中的作用。他还将在他的机构、国内和国际范围内从事数学推广活动。该项目将为学生提供研究培训机会。更详细地说，在该项目中，PI 将应用组合代数几何的方法来解决数学物理中的谱理论问题。 离散周期算子的谱是算术环面变体中的实代数超曲面，并且该领域的许多开放性问题都适用于代数几何的思想。 PI 还将致力于对计数几何中的伽罗瓦群和逆伽罗瓦问题进行分类。为此，PI 将开发计算和理论工具，用于在应用中开发和研究伽罗瓦理论。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。