Geometric Partial Differential Equations and Algebraic Geometry

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Geometric partial differential equations describe the fundamental laws of the universe, from gravity to electromagnetism to fluid flow and the equations of motion of string theory. It is a fundamental fact that these equations do not always admit solutions. Understanding when solutions do and do not exist provides deep insights into the nature of our universe; for example, we can narrow down the possible shapes of the universe, or the kinds of behaviors charged particles in space might exhibit. This project aims to understand these problems, and their connections to underlying algebraic structures in a variety of settings connected with high-energy physics, and string theory. The PI plans to investigate several problems studying the relationship between existence and regularity problems for geometric PDEs on complex manifolds and algebraic geometry. The three main settings in which this will be carried out are the existence problem for the deformed Hermitian-Yang-Mills equation, which describes BPS D-Branes on the B-side of mirror symmetry, the Kahler-Ricci flow and connections to the minimal model program, and the existence of canonical destabilizers in the Yau-Tian-Donaldson conjecture for Fano manifolds. Each of these directions involves relating estimates for elliptic/parabolic PDEs and algebraic geometry through techniques involving birational geometry, geometric invariant theory and Riemannian convergence theory.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.