Geometric PDEs and Algebraic Geometry

几何偏微分方程和代数几何

基本信息

批准号：
1856457
负责人：
Tristan Collins
金额：
$ 0.29万
依托单位：
Massachusetts Institute of Technology
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：
2018-07-01 至 2018-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1856457&HistoricalAwards=false
关键词：
Geometric PDEs Algebraic Geometry

项目摘要

Geometric partial differential equations are ubiquitous in the study of natural physical phenomena-- for example, Einstein's equations in general relativity, and Maxwell's equations in electrodynamics. In practice these equations do not always admit a solution. Understanding the properties of spaces in which these equations have solutions, and furthermore understanding precisely the obstructions to finding solutions sheds new light on the underlying structure of the universe. A thorough understanding of the obstructions to solving these natural equations can lead to new and exciting predictions in the fields of string theory and high energy physics.The PI proposes to investigate several problems concerning the geometry of Kahler manifolds using techniques from partial differential equations and algebraic geometry. The central theme of this research is to relate the convergence or singularity formation of several geometric heat flows, such as the Kahler-Ricci flow and the J-flow, to algebraic geometry. In the setting of the Kahler-Ricci flow it is essential to understand the algebro-geometric properties of the blow-up set for the flow at any finite time singularity. In the non-projective setting, this involves finding transcendental approaches to algebro-geometric problems-- for example, Kawamata's base point free theorem. In the setting of the J-flow, the convergence or singularity formation of the flow is expected to be determined by an algebro-geometric stability condition. The PI aims to investigate how this stability condition is related to the analytic behavior of the flow for large time both in specific examples, and in general.

几何偏微分方程在自然物理现象的研究中无处不在，例如广义相对论中的爱因斯坦方程和电动力学中的麦克斯韦方程。实际上，这些方程并不总是有解。了解这些方程有解的空间的性质，并进一步准确地理解寻找解的障碍，为了解宇宙的基本结构提供了新的线索。彻底理解解决这些自然方程的障碍可以在弦理论和高能物理领域带来新的、令人兴奋的预测。PI建议使用偏微分方程和代数技术来研究有关卡勒流形几何的几个问题几何学。这项研究的中心主题是将几种几何热流（例如卡勒-里奇流和 J 流）的收敛或奇点形成与代数几何联系起来。在卡勒-里奇流的设置中，必须了解任意有限时间奇点处流的爆炸集的代数几何性质。在非射影环境中，这涉及寻找代数几何问题的先验方法，例如川俣的无基点定理。在 J 流的设置中，流的收敛或奇点形成预计由代数几何稳定性条件确定。 PI 旨在研究这种稳定性条件如何与特定示例和一般情况下长时间流的分析行为相关。