Geometric Analysis on Complex Manifolds

主要研究人员的研究涉及复流形的研究，复流形是使用复数定义的高维曲面空间。这类空间的最简单的例子称为黎曼曲面，它是具有一个复数维(因此也就是两个实数维)的复流形，并且包括熟悉的球面和圆环面。例如，高维复杂流形包括作为弦理论的基本工具的Calabi-Yau流形。复流形是数学中普遍存在的对象，在物理和工程中有着广泛的应用。拟议的研究项目将扩大我们使用分析技术，特别是偏微分方程组(PDE)的高维复杂流形的几何知识。这些项目位于几个数学学科的交叉点，如微分、代数和辛几何、复分析和偏微分方程组，所有这些领域的技术都需要攻击它们。这些问题的研究进展不仅对数学中的一些基本问题有一定的帮助，而且在物理和其他科学中也有应用。主要研究人员建议使用几何分析和非线性偏微分方程组的方法来研究复杂流形的几何问题。第一个项目是关于分析在复流形上构造流的应用，它被用来研究紧Kahler流形上的(1，1)上同调类的几何。在第二个项目中，主要研究人员将开发新的分析技术来构造非Kahler复流形上的特殊度量，通过求解(n-1，n-1)形式的Monge-Ampere方程，建立在主要研究人员与SzekelyHidi和Weinkove的早期工作的基础上，该工作最终解决了Gauduchon猜想。第三个项目是关于了解Ricci-Flat Calabi-Yau流形的折叠极限。这与镜面对称理论密切相关，镜面对称理论的灵感来自于物理方面的考虑。第四个项目是以Donaldson的程序为中心，将Kahler几何中Calabi猜想的Yau解推广到四个辛流形及其在辛拓扑中的应用。

The principal investigator's research concerns the study of complex manifolds, which are higher-dimensional curved spaces that are defined using the complex numbers. The simplest examples of such spaces are called Riemann surfaces, which are complex manifolds with one complex dimension (and therefore two real dimensions), and include the familiar surfaces of the sphere and of a donut. Higher-dimensional complex manifolds include for example Calabi-Yau manifolds, which are a fundamental tool in string theory. Complex manifolds are ubiquitous objects in mathematics, and have wide-ranging applications in physics and engineering. The proposed research projects will expand our knowledge of the geometry of higher-dimensional complex manifolds using analytic techniques, and in particular partial differential equations (PDEs). These projects lie at the intersection of several mathematical disciplines, such as differential, algebraic and symplectic geometry, complex analysis and PDEs, and techniques from all these fields are necessary to attack them. 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.The principal investigator proposes to use techniques from geometric analysis and nonlinear partial differential equations to investigate problems about the geometry of complex manifolds. The first project is about applications of analysis to the construction of currents on complex manifolds, which are used to study the geometry of (1,1) cohomology classes on compact Kahler manifolds. In the second project the principal investigator will develop new analytic techniques to construct special metrics on non-Kahler complex manifolds, by solving Monge-Ampere equations for (n-1,n-1) forms, building upon earlier work of the principal investigator with Szekelyhidi and Weinkove which culminated in the solution of Gauduchon's conjecture. The third project is about understanding collapsed limits of Ricci-flat Calabi-Yau manifolds. This is closely related to the theory of mirror symmetry, which was inspired by physical considerations. The fourth project is centered on Donaldson's program to extend Yau's solution of the Calabi Conjecture in Kahler geometry to symplectic four manifolds, and to its applications to symplectic topology.