Combinatorial Algebraic Geometry for Spectral Theory and Galois Groups

代数几何在科学和工程中出现的问题中有越来越多的应用。这在工业与应用数学学会（SIAM）代数几何活动小组的建立、SIAM两年一次的会议以及SIAM应用代数和几何杂志中得到了认可。在这个项目中，PI将进行研究和培训活动，以扩展和深化代数几何在应用中的作用，包括数学物理。他还将在国内和国际上参与他所在机构的数学推广活动。本项目将为学生提供研究训练的机会。更详细地说，在这个项目中，PI将应用组合代数几何中的方法来解决数学物理中的光谱理论问题。离散周期算子的谱是算术环变中的一个真正的代数超曲面，这一领域的许多开放问题都适用于代数几何的思想。PI还将致力于伽罗瓦群的分类和枚举几何中的伽罗瓦逆问题。为此，PI将开发计算和理论工具，用于开发和研究伽罗瓦理论的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.