Geometric Analysis on Complex Manifolds

首席研究员的研究涉及复杂流形的研究，它是用复数定义的高维弯曲空间。这种空间最简单的例子被称为黎曼曲面，它是具有一个复维（因此是两个实维）的复杂流形，包括球体和甜甜圈的熟悉表面。高维复流形包括例如Calabi-Yau流形，它是弦理论的基本工具。复杂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。提出的研究项目将扩展我们使用解析技术，特别是偏微分方程（PDEs）的高维复杂流形几何知识。这些项目位于几个数学学科的交叉点，例如微分几何、代数几何和辛几何、复杂分析和偏微分方程，所有这些领域的技术都是解决这些问题所必需的。在这些问题上取得进展，不仅会对数学中的一些基本问题有所启发，而且还会在物理学和其他科学中得到应用。主要研究者建议使用几何分析和非线性偏微分方程的技术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电流构造中的应用，用于研究紧卡勒流形上（1,1）上同调类的几何。在第二个项目中，首席研究员将开发新的分析技术，通过求解（n-1,n-1）形式的蒙日-安培方程，在非kahler复流形上构建特殊度量，这是在首席研究员与Szekelyhidi和Weinkove的早期工作的基础上完成的，该工作最终解决了Gauduchon猜想。第三个项目是关于理解Ricci-flat Calabi-Yau流形的坍缩极限。这与镜像对称理论密切相关，镜像对称理论的灵感来自于物理方面的考虑。第四个项目集中在Donaldson的程序上，将Yau在Kahler几何中的Calabi猜想的解扩展到辛四流形，以及它在辛拓扑中的应用。

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.