Partial Differential Equations on Complex and Symplectic Manifolds

PI 提出的研究重点是与复辛流形几何相关的几个基本问​​题，这些问题可以使用非线性偏微分方程来研究。在第一个项目中，PI 将研究唐纳森最近的一个猜想，该猜想旨在以他与 Weinkove 和丘的合作为基础，将卡勒几何中的丘定理扩展到辛四流形。如果得到证实，这个猜想将提供一个强大的新工具来构造紧辛四流形上的辛形式，并将在辛拓扑中具有惊人的应用。第二个项目涉及紧致卡拉比-丘流形的几何形状，特别是卡拉比-丘流形上的里奇平坦卡勒度量在其上同调类接近卡勒锥体边界时退化的方式。弦理论家也结合镜像对称性对这些简并进行了研究。 PI 提议继续研究这些退化，并研究接近大型复杂结构极限的五次三重族的 Ricci 平坦度量。第三个项目属于紧凑卡勒流形的规范度量领域，例如卡勒-爱因斯坦或恒定标量曲率卡勒度量。人们认为这种规范度量的存在应该等价于流形的代数稳定性。 PI 将使用与这些问题相关的两个自然演化方程（Kahler-Ricci 流和 Calabi 流）来研究这一问题，目的是通过使用自然能量泛函将流的极限行为与代数稳定性联系起来。最终项目还涉及规范卡勒度量，更具体地说，涉及已知稳定的上同调类中具有充足规范束的复杂表面上常标量曲率卡勒度量的存在问题。我们将考虑的大多数问题，例如爱因斯坦方程，最初是由寻找自然基本定律模型的物理学家发现的。最近，与所提出的研究密切相关的几何方面已经在高能物理学中得到了应用，并被用来加深我们对宇宙和基本粒子的理解。 PI 研究的几何思想围绕着寻找几何空间的最佳形状（具有最大可能对称性的形状）的问题，并了解在不存在这种最佳形状的空间中可能形成的奇点。这些问题的任何进展不仅会阐明数学中的一些基本问题，而且还将在物理学和其他科学中得到应用。

The PI proposed research focuses on several basic problems related to the geometry of complex and symplectic manifolds, which can be studied using nonlinear PDEs. In the first project the PI will study a recent conjecture of Donaldson that aims at extending Yau's theorem in Kahler geometry to symplectic four-manifolds, building on his work with Weinkove and Yau. If proved, this conjecture would provide a powerful new tool to construct symplectic forms on compact symplectic four-manifolds, and would have striking applications to symplectic topology. The second project regards the geometry of compact Calabi-Yau manifolds, and specifically the way in which Ricci-flat Kahler metrics on a Calabi-Yau manifold can degenerate when their cohomology class approaches the boundary of the Kahler cone. These degenerations have also been studied by string theorists in connection with mirror symmetry. The PI proposes to continue his study of these degenerations, as well as investigating Ricci-flat metrics on a family of quintic threefolds near a large complex structure limit. The third project falls in the area of canonical metrics on compact Kahler manifolds, such as Kahler-Einstein or constant scalar curvature Kahler metrics. It is believed that the existence of such canonical metrics should be equivalent to the algebraic stability of the manifold. The PI will study this using two natural evolution equations associated to these problems, the Kahler-Ricci flow and the Calabi flow, with the aim of connecting the limiting behaviour of the flows to algebraic stability through the use of natural energy functionals. The final project also involves canonical Kahler metrics, and more specifically the problem of existence of constant scalar curvature Kahler metrics on complex surfaces with ample canonical bundle in cohomology classes that are known to be stable.Most of the problems that we will consider, for example the Einstein equations, were originally discovered by physicists who were searching for models of the fundamental laws of nature. More recently, geometric aspects closely related to the proposed research have found applications in high energy physics, and are being used to deepen our understanding of the Universe and of elementary particles. The geometric ideas of the PI's research revolve around the problem of finding the optimal shape of a geometric space, the one with the largest possible symmetry, and understanding the possible singularities that form in spaces where such an optimal shape does not exist. Any progress on these questions will not only shed some light on some basic problems in mathematics, but will also have applications in physics and other sciences.