Combinatorial Algebraic Geometry for Spectral Theory and Galois Groups

代数几何已发现越来越多的应用于科学和工程中出现的问题。这是公认的建立社会工业和应用数学（SIAM）活动组在代数几何，SIAM两年一次的会议，并在SIAM杂志上应用代数和几何。在这个项目中，PI将开展研究和培训活动，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光谱理论问题。 离散周期算子的谱是算术复曲面簇中的真实的代数超曲面，这一领域的许多未解决的问题都服从于代数几何的思想。PI还将致力于分类伽罗瓦群和计数几何中的逆伽罗瓦问题。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，并被认为值得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来支持。

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.