Ricci Curvature and Geometric Analysis

该提案的目的是研究各种几何激励方程及其应用。该建议将主要围绕涉及Ricci曲率的思想，但它的一些方面也将涉及调和映射、谱分析、度量空间和随机分析。这些主题中的每一个都在某种程度上与高维几何的弯曲和内在结构有关。近年来，这些领域都取得了很大的进展，但仍有许多未知的问题，这些问题的解决方案将在数学和物理的许多分支中得到应用。该提案总共有三个部分，有九个项目和七个合著者。每个项目将首先讨论预期在明年取得的进展，然后讨论未来一年的目标。提案的第一部分集中在研究Ricci曲率和路径空间上的无限维分析之间的联系。最近的突破使研究人员意识到这两者是密切相关的，希望对这一领域的进一步了解应该开辟新的研究领域，并解决许多涉及有界Ricci曲率空间的问题。提案的这一部分有两个项目。本文的第二部分讨论了具有下有界Ricci曲率的流形的正则性。特别是，与几位合著者一起，这些项目包括证明爱因斯坦流形的新的正则性定理，以及证明具有较低Ricci曲率界的度量度量空间是可纠正的。提案的第二部分有三个项目。提案的最后部分包括几何分析各个领域的四个项目。其中包括欧几里德空间上椭圆型方程这一非常经典的主题的一个项目。这是令人惊讶的，但在这一领域仍有许多悬而未决的问题。特别是，该项目侧重于研究这类方程在非常粗糙的系数下的临界集。

The goal of the proposal is to study various geometrically motivated equations and their applications. The proposal will mostly center around ideas involving Ricci curvature, however aspects of it will also involve harmonic maps, spectral analysis, metric-measure spaces, and stochastic analysis. Each of these topics is somehow concerned with the bending and intrinsic structure of higher dimensional geometries. In recent years much progress has been made in each of these areas, but there are a great deal of unknown questions which remain, the solutions of which would have applications in many branches of mathematics and physics. In all there are three parts to the proposal with nine projects and seven coauthors. Each project will discuss first progress expected to be made over the next year, and then goals past that.The first part of the proposal centers on studying the connections between Ricci curvature and the infinite dimensional analysis on path space. Recent breakthroughs have allowed researchers to realize that the two are intimately connected, and the hope is that further understanding in this area should open up new areas of research as well as solve many questions involving spaces with bounded Ricci curvature. There are two projects in this part of the proposal. The second part of the paper focuses on the regularity of manifolds with lower and bounded Ricci curvature. In particular, together with several co-authors the projects include the proving of new epsilon-regularity theorems for Einstein manifolds and showing that metric-measure spaces with lower Ricci curvature bounds are rectifiable. There are three projects in the second part of the proposal. The final part of the proposal includes four projects from various areas of geometric analysis. This includes a project in the very classical topic of elliptic equations on Euclidean space. It is surprising, but there are still many open questions in this area. In particular, the project focuses on the study of critical sets of such equations under very rough coefficients.