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上得到越来越多的应用。工业和应用数学学会(SIAM)在代数几何活动小组的设立、SIAM两年一度的会议以及SIAM的《应用代数和几何》杂志都承认了这一点。在这个项目中，PI将继续开展研究和培训活动，以扩大和深化代数几何在应用中的作用，包括数学物理。他还将在他的机构、国内和国际上开展数学推广活动。这个项目将为学生提供研究培训的机会。更详细地说，在这个项目中，PI将应用组合代数几何的方法来解决数学物理中的谱论问题。离散周期算子的谱是算术环面簇中的实代数超曲面，这一领域的许多未决问题都服从于代数几何的思想。PI还将致力于对计数几何中的伽罗瓦群和伽罗瓦逆问题进行分类。为此，PI将开发工具，包括计算和理论，在应用中开发和研究伽罗瓦理论。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.