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.

几何部分微分方程描述了宇宙的基本定律，从重力到电磁学再到流体流以及弦理论运动方程。 这些方程并不总是承认解决方案是一个基本事实。 了解解决方案何时确实存在和不存在，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，或者可能显示出空间中充电颗粒的行为。 该项目旨在了解这些问题，以及它们与与高能量物理学和弦理论相关的各种环境中的基础代数结构的联系。 PI计划研究研究复杂歧管和代数几何形状上的几何PDE之间存在关系与规律性问题之间关系的几个问题。 将执行此操作的三个主要环境是变形的Hermitian-Yang-Mills方程的存在问题，该方程描述了镜像对称性的B侧BPS D-Branes，Kahler-Ricci流以及与最小模型程序的连接以及在Yau-Donaldian denodian denodian contian contian contian denodian-dandur for的范围内的连接。 这些方向中的每一个都涉及通过涉及Birational几何形状，几何不变理论和Riemannian融合理论的技术来关联椭圆/抛物线PDE的估计以及代数几何形状。这项奖项反映了NSF的法规使命，并被认为是通过基金会的知识优点和广泛的范围来评估的，并值得通过评估来进行评估。