PDE's in complex and symplectic geometry

复辛几何中的偏微分方程

• 批准号: 0804099
• 负责人: Benjamin Weinkove
• 金额: $ 13.63万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2008-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0804099&HistoricalAwards=false
• 关键词: PDE; complex; symplectic; geometry

The PI plans to investigate projects in four areas, related by the theme of nonlinear PDE's and their interaction and applications to complex and symplectic geometry. In the first project, the PI will work on Donaldson's program of extending Yau's theorem in Kahler geometry to symplectic 4-manifolds. The ultimate goal is to tackle some fundamental questions in symplectic and almost complex geometry, such as: given an almost complex 4-manifold, when does there exist a compatible symplectic form? The second project deals with the constant scalar curvature equation on Kahler manifolds. It is believed that the existence of a solution to this nonlinear PDE should be equivalent to the algebraic stability of the manifold. The PI will examine this via the natural energy functionals associated to this problem, building on his work with Phong, Song and Sturm. The third project concerns the Kahler-Ricci and Calabi flows and aims to elucidate the relationship between the convergence properties of these parabolic flows and conditions of stability and positive curvature. The final project also involves parabolic PDE's: the geometric flows of Donaldson which arise naturally from considerations of moment maps and diffeomorphism groups. The PI will investigate the behavior of these flows, on Kahler and hyperkahler manifolds, and possible applications to the study of canonical metrics and the space of symplectic forms.The fundamental laws of physics are described, in the language of mathematics, by differential equations. Understanding the behavior of solutions to differential equations (such as Einstein's equations) is key to unravelling the mystery of the geometry and structure of the universe. This proposal concerns equations which arise naturally in the study of geometry and are related to, and inspired by, the physical laws. There is a complex interaction between such differential equations, which are described locally, and the underlying geometry, which is global. This project aims to investigate this rich interplay between local and global and in the process touch on some basic problems in mathematics: what are the natural structures in geometry, and how do we find and describe them?

PI计划调查四个领域的项目，与非线性偏微分方程组及其在复杂和辛几何中的相互作用和应用有关。在第一个项目中，PI将致力于Donaldson的计划，将Kahler几何中的Yau定理推广到辛四维流形。本课程的最终目的是解决辛几何和几乎复杂的几何中的一些基本问题，例如：给定一个几乎复杂的4-流形，什么时候存在相容的辛形式？第二个项目研究Kahler流形上的常数量曲率方程。这类非线性偏微分方程解的存在性应等价于流形的代数稳定性。PI将以他与Phong、Song和Sturm的工作为基础，通过与这个问题相关的自然能量泛函来检验这一点。第三个项目涉及Kahler-Ricci流和Calabi流，目的是阐明这些抛物线流的收敛性质与稳定性和正曲率条件之间的关系。最后的项目还涉及抛物型偏微分方程组：Donaldson的几何流，它自然地产生于对矩映射和微分同胚群的考虑。PI将研究这些流在Kahler流形和Hyperkahler流形上的行为，以及在正则度量和辛形式空间的研究中的可能应用。了解微分方程解的行为(如爱因斯坦方程)是解开宇宙几何和结构之谜的关键。这一建议涉及在几何学研究中自然产生的方程，这些方程与物理定律有关，并受到物理定律的启发。在这种局部描述的微分方程式和基本几何结构之间存在复杂的相互作用，后者是全局的。这个项目旨在研究局部和全球之间的这种丰富的相互作用，并在这个过程中触及一些基本的数学问题：几何中的自然结构是什么，以及我们如何发现和描述它们？