PDE's in complex and symplectic geometry

PI计划调查四个领域的项目，这些领域与非线性偏微分方程及其在复杂几何和辛几何中的相互作用和应用的主题有关。在第一个项目中，PI将致力于Donaldson将Kahler几何中的Yau定理扩展到辛4流形的计划。最终目标是解决辛几何和几乎复杂几何中的一些基本问题，例如：给定一个几乎复杂的4流形，何时存在兼容的辛形式？第二个项目处理Kahler流形上的常数标量曲率方程。认为该非线性偏微分方程解的存在性等价于流形的代数稳定性。PI将在他与Phong， Song和Sturm合作的基础上，通过与这个问题相关的自然能量函数来研究这个问题。第三个项目涉及Kahler-Ricci和Calabi流，旨在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线PDE: Donaldson的几何流，它自然地产生于矩映射和微分同构群的考虑。PI将研究这些流在Kahler和超Kahler流形上的行为，以及在规范度量和辛形式空间研究中的可能应用。物理学的基本定律是用数学语言用微分方程来描述的。理解微分方程（如爱因斯坦方程）解的行为是解开宇宙几何和结构之谜的关键。这一建议涉及几何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。这些微分方程是局部描述的，而底层几何是全局描述的，两者之间存在复杂的相互作用。该项目旨在研究局部和全局之间丰富的相互作用，并在此过程中触及数学中的一些基本问题：几何中的自然结构是什么，我们如何发现和描述它们？

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?