CAREER: Differential Equations, Algebraic Geometry, and String Theory

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The structure of the universe is encoded by mathematics. For example, the large scale behavior of the universe is determined by Einstein's theory of general relativity, while the small scale behavior is dictated by quantum mechanics. A prominent goal of theoretical physics has been to find a mathematical description of the universe which unites these large and small scale descriptions. This work has led to the discovery of string theory, and a vast array of beautiful and intricate mathematical structures and equations. The equations of string theory are difficult and complicated, and are expected to connect radically different branches of mathematics and physics. Understanding these equations, and the structure they contain, is the primary goal of these education and research projects. This research agenda aims to understand mathematical descriptions of physical phenomena in string theories ranging from particle formation and decay to the emergence of gravity in "holographic" string theories. The education plans aim to provide opportunities for undergraduate and graduate students to engage in learning and research projects related to the research program. Planned activities include sponsored research projects for undergraduates, a bi-annual weekend conference for graduate students, and a two week summer program, including a week long summer school and graduate students and advanced undergraduates vertically integrated with existing summer research programs.These projects aim to investigate several problems in geometry which are inspired by string theory, and which involve techniques from differential equations, differential geometry, and algebraic geometry. The first project involves studying two equations arising in mirror symmetry: the deformed Hermitian-Yang-Mills equation, and the special Lagrangian equation. The goal is to develop a theory that relates the study of these equations to structures in algebraic geometry, using an infinite dimensional version of classical geometric invariant theory. These equations present exciting new mathematical challenges ranging from the analysis of fully nonlinear systems to understanding the algebraic structure encoded in certain singular geometric objects, in analogy with connections between multiplier ideals and singular, positively curved metric on holomorphic line bundles. As a result, this project will involve developing new ideas in partial differential equations, as well as differential and algebraic geometry. These ideas are to be tested in a variety of simplified models, which will lead to the resolution of several open problems in Kahler geometry. Another line of work will study the phenomenon of emergent gravity in string theory, and in particular the AdS/CFT conjecture, and connection with combinatorial facets of AdS/CFT.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.