Differential Geometry and Partial Differential Equations

本课题是微分几何、广义相对论和偏微分方程的交叉学科。其中一个主题是研究正曲率流形。具体而言，他计划利用里奇流和最小曲面技术进一步研究正各向同性曲率流形。他还希望在掐度略低于1/4的点掐度条件下证明球体定理。在相对论中，他计划研究静态物质解的几何，希望证明在这种解中物质体不能被凸集分开。他还计划研究一般的彭罗斯不等式；即具有非零秒基本形式的黑洞初始集的猜想不等式。他还将研究一系列关于满足自由边界条件的最小子流形的问题以及与特征值问题的联系。最后他计划继续他的最小拉格朗日和特殊拉格朗日子流形的研究。他将试图证明一个关于Calabi-Yau流形的积分同调子群的不变性的猜想，该流形是由最小拉格朗日循环在周围Calabi-Yau结构变形时产生的。从曲率特性的角度来理解空间是数学和科学的一个基本概念。自然定律经常提供曲率特性的信息，我们必须从这些信息中推断出空间的信息。一个典型的例子是广义相对论，它的运动方程是用曲率来描述的，从曲率我们必须确定宇宙的具体性质。这个项目处理一系列这类问题，比如黑洞是如何从平滑的初始数据中形成的。此外，几何在成像和计算机图形学等应用领域也很重要。该项目的研究将推进构成此类应用基础的核心几何思想。

This project lies at the interface between Differential Geometry, General Relativity, and Partial Differential Equations. One theme will be the study of manifolds of positive curvature. Specifically the proposer plans to further his study of manifolds of positive isotropic curvature (PIC) using Ricci flow and minimal surface techniques. He also hopes to prove sphere theorems under pointwise pinching conditions with pinching slightly below 1/4. In relativity he plans to study the geometry of static matter solutions hoping to show that matter bodies cannot be separated by convex sets in such solutions. He also plans to study the general Penrose inequality; that is, the conjectured inequality for black hole initial sets with nonzero second fundamental form. He will also pursue a range of questions concerning minimal submanifolds satisfying free boundary conditions and connections to eigenvalue problems. Finally he plans to continue his study of minimal lagrangian and special lagrangian submanifolds of Kahler-Einstein manifolds. He will attempt to prove a conjecture concerning the invariance of the subgroup of the integral homology of a Calabi-Yau manifold which is generated by minimal lagrangian cycles when one deforms the ambient Calabi-Yau structure.Understanding spaces in terms of their curvature properties is a fundamental idea in mathematics and science. The laws of nature often provide information about curvature properties, and from those we must deduce information about the spaces. A prime example of this is General Relativity where the equations of motion are described by curvature, and from those we must determine concrete properties of the universe. This project deals with a range of questions of this type such as the way in which black holes form from smooth initial data. Furthermore, geometry is important in application areas such as imaging and computer graphics. The research of this project will advance the core geometric ideas which form the basis of such applications.