Differential Geometry and Partial Differential Equations

这个项目位于微分几何，广义相对论和偏微分方程之间的接口。其中一个主题将是研究流形的积极曲率。具体来说，提议者计划进一步研究流形的正各向同性曲率（PIC）使用里奇流和极小曲面技术。他还希望证明领域定理下逐点捏捏条件捏略低于1/4。 在相对论，他计划研究几何静态物质的解决方案，希望表明，物质机构不能分开的凸集在这样的解决方案。他还计划研究一般的彭罗斯不等式，即黑洞初始集具有非零第二基本形式的约束不等式。他还将追求一系列的问题，关于满足自由边界条件和连接到特征值问题的极小子流形。最后，他计划继续他的研究最小拉格朗日和特殊拉格朗日子流形的卡勒-爱因斯坦流形。他将试图证明一个猜想关于不变性的子群的积分同源性的卡-丘流形这是由最小拉格朗日周期时，一个变形的环境卡-丘结构。了解空间的曲率性质是一个基本的想法在数学和科学。自然定律经常提供关于曲率性质的信息，我们必须从这些信息中推导出关于空间的信息。这方面的一个主要例子是广义相对论，其中的运动方程是由曲率描述的，我们必须从这些方程中确定宇宙的具体性质。该项目涉及一系列这类问题，例如黑洞如何从平滑的初始数据中形成。 此外，几何在诸如成像和计算机图形的应用领域中是重要的。该项目的研究将推进形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.