Problems in Complex Geometry, Partial Differential Equations, and Mathematical Physics

复杂几何、偏微分方程和数学物理问题

• 批准号: 2203273
• 负责人: Duong Phong
• 金额: $ 39.81万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203273&HistoricalAwards=false
• 关键词: Problems; Complex; Geometry; Partial; Differential

The search for a unified theory of all forces of nature has led through string theories to new equations requiring a challenging and unfamiliar geometry. This research is devoted to the development of a theory of geometric partial differential equations which can address such equations. Such a theory is needed progress is to be made in this area of interface between partial differential equations, geometry, and high energy physics, and arrive some day at an understanding of physical laws at their most fundamental level. The training of both undergraduate and graduate students is an integral component of the project. It is particularly important for the next generation of scholars to be introduced to these difficult issues as soon as possible.A characteristic feature of these new equations is a curvature condition combined with a cohomological constraint. Thus they can be viewed as defining “canonical metrics”. The PI will strive to develop geometric flow methods to find such metrics, and to consider flows of both tensor and spinor fields. These geometric flows are interesting in their own right, from the point of view of the theory of partial differential equations. Of particular interest is to determine the manifestations of supersymmetry and duality, which are expected in particular to link aspects of complex geometry to aspects of symplectic geometry. A hybrid cohomology theory will also be developed with possible applications to superstring perturbation 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.

对自然界所有力的统一理论的探索已经将弦理论引入了新的方程，这需要一个具有挑战性和不熟悉的几何。本研究致力于发展一种能解决这类方程的几何偏微分方程理论。在偏微分方程、几何和高能物理的交界领域，需要取得进展，并有一天在最基本的层面上理解物理定律。本科生和研究生的培训是该项目的一个组成部分。让下一代学者尽快了解这些棘手的问题尤为重要。这些新方程的一个特征是曲率条件与上同调约束的结合。因此，它们可以被视为定义“规范指标”。PI将努力发展几何流动方法来找到这样的度量，并考虑张量场和旋量场的流动。从偏微分方程理论的角度来看，这些几何流本身就很有趣。我们特别感兴趣的是确定超对称和对偶的表现形式，它们特别被期望将复几何的各个方面与辛几何的各个方面联系起来。一个混合上同调理论也将发展，并可能应用于超弦微扰理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。