Geometric Flows and Analysis on Metric Spaces

该项目旨在研究热方程的两个非线性类比，这两个类比几十年来一直是数学家们密集研究的主题。这项研究的主要目的是研究与这些方程解的奇点形成有关的问题，包括它们的结构和稳定性，以使数学家能够对空间变形的方式做出推断。几何和拓扑学中基本猜想的解决是这项工作的结果。研究计划的另一个组成部分是利用过去几年开发的分析工具，对具有自相似或分形特征的空间进行调查。这里的目标之一是将空间变形为最佳形式，如果可能的话，以揭示隐藏的对称，否则表明不存在隐藏的对称。这是过去15年来几个研究趋势汇合的一部分。类似想法的另一个应用是在理论计算机科学中嵌入问题。这项研究的主要目的是研究几何演化方程的奇点、嵌入问题、度量空间上的分析和几何群论。该方案中的演化方程为平均曲率流和Ricci流。对度量空间簇的分析研究主要集中在三个方面：(1)bilipschitz嵌入问题及相关问题；(2)满足Poincare不等式的空间结构；(3)Gromov双曲空间的边界结构。*这三个领域的共同主题都是满足Poincare不等式的空间，以及重新缩放导致奇异极限空间的论点。在一定的Banach空间中，bilipschitz嵌入是理论计算机科学家感兴趣的一个主题；在以前的工作中，PI和合作者改进了关于负型空间嵌入的最著名的结果，与Goemans-Linial猜想的量化版本有关。

The project aims to study two nonlinear analogs of the heat equation that have been the subject of intense investigation by mathematicians for decades. The primary objective of this research is to study issues related to the formation of singularities in the solutions of these equations, including their structure and stability, to enable mathematicians to make deductions about the ways in which space can be deformed.  The resolution of fundamental conjectures in geometry and topology is an outgrowth of this work.  An additional component of the research program is an investigation of spaces that have a self-similar or fractal character, using analytic tools that have been developed in the last few years.  Here one of the goals is to deform the space into an optimal form, if possible, to reveal hidden symmetries, and otherwise show that no hidden symmetries exist.  This is part of confluence of several research trends over the last fifteen years. Another application of similar ideas is to embedding problems in theoretical computer science. The primary objective of this research is to study singularities of geometric evolution equations, embedding problems, analysis on metric spaces, and geometric group theory. The evolution equations in the proposal are mean curvature flow and Ricci flow.  The research in analysis on metric spaces clusters in three areas: (1) bilipschitz embedding problems and related issues, (2) the structure of spaces satisfying Poincare inequalities, (3) the structure of boundaries of Gromov hyperbolic spaces.  Common themes in all three areas are spaces satisfying Poincare inequalities, and rescaling arguments leading to singular limit spaces. A bilipschitz embedding in a certain Banach space is a topic of interest to theoretical computer scientists; in previous work, the PI and collaborators improved the best-known results on the embedding of spaces of negative type, in connection with the quantitative version of the Goemans-Linial conjecture.