Ricci Curvature and Geometric Analysis

里奇曲率和几何分析

• 批准号: 1406259
• 负责人: Aaron Naber
• 金额: $ 19.73万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406259&HistoricalAwards=false
• 关键词: Ricci; Curvature; Geometric; Analysis

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.

该提案的目标是研究各种几何驱动方程及其应用。 该提案将主要围绕涉及里奇曲率的想法，但它的各个方面也将涉及调和图、谱分析、度量测量空间和随机分析。 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进展，但仍然存在大量未知问题，这些问题的解决方案将在数学和物理学的许多分支中得到应用。 该提案总共分为三个部分，涉及九个项目和七位合著者。每个项目将讨论明年预计取得的第一个进展，然后是超越该进展的目标。提案的第一部分集中于研究里奇曲率和路径空间无限维分析之间的联系。 最近的突破让研究人员认识到两者是密切相关的，希望对这一领域的进一步理解能够开辟新的研究领域，并解决涉及有界里奇曲率空间的许多问题。 该提案的这一部分有两个项目。 本文的第二部分重点研究具有下界里奇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 epsilon 正则定理，并表明具有下里奇曲率界的度量测量空间是可校正的。 该提案的第二部分共有三个项目。 该提案的最后部分包括来自几何分析各个领域的四个项目。 这包括欧几里得空间椭圆方程这一非常经典的主题的项目。 令人惊讶的是，但这一领域仍然存在许多悬而未决的问题。 特别是，该项目侧重于在非常粗糙的系数下研究此类方程的临界集。