Geometric Partial Differential Equations and Algebraic Geometry

几何偏微分方程和代数几何

• 批准号: 1810924
• 负责人: Tristan Collins
• 金额: $ 15.83万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-15 至 2019-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1810924&HistoricalAwards=false
• 关键词: Geometric; Partial; Differential; Equations; Algebraic

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.

几何偏微分方程描述了宇宙的基本定律，从重力到电磁学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。 了解解决方案何时存在或不存在可以让我们深入了解宇宙的本质；例如，我们可以缩小宇宙可能的形状，或者空间中带电粒子可能表现出的行为类型。 该项目旨在了解这些问题，以及它们与高能物理和弦理论相关的各种设置中的基础代数结构的联系。 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的存在性和正则性问题之间的关系。 执行此操作的三个主要设置是变形 Hermitian-Yang-Mills 方程的存在问题，该方程描述镜像对称 B 侧上的 BPS D-Branes、Kahler-Ricci 流以及与最小值的连接模型程序，以及 Fano 流形的 Yau-Tian-Donaldson 猜想中规范失稳子的存在。 这些方向中的每一个都涉及通过涉及双有理几何、几何不变量理论和黎曼收敛理论的技术对椭圆/抛物线偏微分方程和代数几何进行相关估计。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和知识进行评估，被认为值得支持。更广泛的影响审查标准。