Combinatorial Algebraic Geometry for Spectral Theory and Galois Groups

Texas A&M University, College Station TX

Investigators

Abstract

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.