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.

对所有自然力量的统一理论的搜索已导致弦理论到需要挑战和陌生几何形状的新方程式。这项研究致力于发展几何偏微分方程理论，该方程可以解决此类方程。在部分微分方程，几何形状和高能量物理学之间的界面领域中，需要取得这种进步，并有一天能够在其最基本的水平上了解物理定律。本科和研究生的培训是该项目不可或缺的组成部分。对于下一代学者来说，尽快将这些困难问题引入这些困难问题尤其重要。这些新方程的特征是曲率条件与共同体学约束相结合。可以将它们视为定义“规范指标”。 PI将努力开发几何流量方法以找到此类指标，并考虑张量和纺纱场的流动。从部分微分方程理论的角度来看，这些几何流本身就是有趣的。特别感兴趣的是确定超对称性和二元性的表现，这特别是将复杂几何形状的各个方面与象征几何形状的各个方面联系起来。杂种共同体学理论还将在超声扰动理论中开发出可能的应用。该奖项反映了NSF的法定使命，并使用基金会的知识分子优点和更广泛的影响审查标准，通过评估诚实地认为支持了支持。