Geometric Partial Differential Equations and Algebraic Geometry

几何偏微分方程描述了宇宙的基本定律，从重力到电磁再到流体流动和弦理论的运动方程。这是一个基本的事实，这些方程并不总是允许解。了解解决方案何时存在，何时不存在，可以提供对我们宇宙本质的深刻洞察；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。这个项目的目的是了解这些问题，以及它们与高能物理和弦理论相关的各种环境中基本代数结构的联系。PI计划研究几个问题，研究复流形和代数几何上几何偏微分方程解的存在性和正则性问题之间的关系。进行这项工作的三个主要背景是：描述镜像对称B侧BPS D膜的变形Hermitian-Yang-Mills方程的存在性问题，Kahler-Ricci流及其与最小模型程序的联系，以及Fano流形的Yau-Tian-Donaldson猜想中正则失稳子的存在。这些方向中的每一个都涉及通过涉及二次几何、几何不变理论和黎曼收敛理论的技术来关联椭圆/抛物型偏微分方程和代数几何的估计。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.