PDE's in complex and symplectic geometry

PI计划调查四个领域的项目，与非线性偏微分方程的主题及其相互作用和复杂和辛几何的应用有关。 在第一个项目中，PI将致力于唐纳森的计划，将Kahler几何中的Yau定理扩展到辛4-流形。 最终目标是解决辛几何和几乎复几何中的一些基本问题，例如：给定一个几乎复4-流形，何时存在相容的辛形式？ 第二个项目研究Kahler流形上的常数量曲率方程。 我们认为这个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将通过与这个问题相关的自然能量泛函来研究这个问题，建立在他与Phong，Song和Sturm的工作基础上。 第三个项目涉及Kahler-Ricci和Calabi流，旨在阐明这些抛物流的收敛性质与稳定性和正曲率条件之间的关系。 最后一个项目还涉及抛物型偏微分方程的：自然产生的考虑矩映射和自同构群的唐纳森的几何流。 PI将研究这些流在Kahler和hyperkahler流形上的行为，以及在正则度量和辛形式空间研究中的可能应用。物理学的基本定律用数学的语言通过微分方程来描述。 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。 这个建议涉及的方程自然出现在几何研究和有关，并受到启发，物理定律。 在局部描述的这种微分方程和全局的基本几何之间存在复杂的相互作用。 这个项目旨在研究局部和全局之间的这种丰富的相互作用，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，我们如何找到和描述它们？

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?