Geometric Partial Differential Equations and Algebraic Geometry

几何偏微分方程描述了宇宙的基本定律，从引力到电磁到流体流动，以及弦理论的运动方程。 一个基本事实是，这些方程并不总是有解。 了解什么时候存在和不存在解决方案，可以深入了解我们宇宙的本质;例如，我们可以缩小宇宙的可能形状，或者空间中带电粒子可能表现出的行为类型。 这个项目旨在了解这些问题，以及它们与高能物理学和弦理论相关的各种设置中的潜在代数结构的联系。PI计划研究几个问题，研究复杂流形上的几何偏微分方程和代数几何的存在性和正则性问题之间的关系。 这将进行的三个主要设置是变形的厄米-杨-米尔斯方程的存在性问题，它描述了BPS D-膜的镜像对称的B侧，卡勒-里奇流和连接到最小模型程序，并存在规范的不稳定因素在姚田-唐纳森猜想的法诺流形。 每个方向都涉及到椭圆/抛物偏微分方程和代数几何的相关估计，通过涉及双有理几何，几何不变理论和黎曼收敛理论的技术。该奖项反映了NSF的法定使命，并已被认为是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的支持。

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.