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.

代数几何形状发现对科学和工程中产生的问题的应用增加了。这在代数几何，暹罗双年期会议以及《应用代数和几何形状的暹罗杂志》中的工业和应用数学学会（SIAM）活动组建立中得到了认可。在这个项目中，PI将进行研究和培训活动，以扩展并加深代数几何形状在包括数学物理在内的应用中的作用。他还将在他的机构，国内和国际上从事数学外展活动。该项目将为学生提供研究培训机会。在此项目中，PI将在该项目中将组合代数几何形状的方法应用于数学物理学的光谱理论问题。 离散的周期性操作员的频谱是算术复合品种中真正的代数性超曲面，并且该领域的许多开放性问题都适合代数几何形状的想法。 PI还将致力于对Galois组和列举几何形状中的逆Galois问题进行分类。为此，PI将开发计算和理论上的工具，用于利用和研究应用程序中的Galois理论。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估的评估来支持的。