Mean curvature flow and Ricci flow

这项研究涉及到了微分几何、偏微分方程组、变分法、随机分析和广义相对论。具体地说，主要集中在热方程的两个几何版本上：通过曲面的平均曲率来演化曲面，以及通过汉密尔顿的Ricci流来演化曲面空间。平均曲率流模拟了许多涉及不断变化的表面或界面的物理过程。这是减少曲面面积并使其向最佳曲面演化的最有效方法。相应地，Ricci流将曲线空间变形为最优形状。*虽然这两种流都已获得许多基本结果，但一个中心问题是在最相关的情况下会形成奇点。例如，如果几何体在初始时间看起来像哑铃，则颈部将被掐断，从而阻止一个人以平滑的方式继续流动。这项研究的主要目的是加深我们对平均曲率流和Ricci流下奇点形成的理解，并发展出在第一奇点时间之后继续流动的方法。这将促进许多新的数学内外的应用。*我研究平均曲率流的一个长期目标是用外科手术来构造一般平均凸超曲面的解，广泛地推广了我在先前与Bruce Kleiner的工作中开发的估计和方法。手术的想法是在奇点形成前不久仔细切割表面，并通过在合适的帽子上粘合来治愈它。在与我的博士后和学生的合作中，我将研究平均曲率流在外科手术中的各种拓扑应用，特别是关于嵌入球体的模空间的拓扑的高维Smeal型猜想。*我研究Ricci流的一个长期目标(主要是与Aaron Naber联合)是发展一种广义解的理论，使我们能够通过奇点继续流动。在最近的一篇论文中，我们证明了关于Ricci流的一类新的估计，它们强到足以刻画解的特征。基于我们的估计，我们可以提供Ricci流的弱解的概念，它解决了一个长期悬而未决的问题。在接下来的5年里，我们计划发展这些弱解的理论。我还将研究几个应用，特别是几个几何分析猜想，这些猜想在佩雷尔曼解决庞加莱猜想后一直悬而未决。*拟议的研究处于现代数学的前沿。我的主要目标之一是吸引在几何和分析方面最好的加拿大和国际学生来多伦多，并让他们参与研究项目。我会组织研讨会、讨论小组、专题班、暑期班和会议。我将广泛传播这项研究，并将发表许多说明性演讲。**

The proposed research is at the intersection of differential geometry, partial differential equations, calculus of variations, stochastic analysis and general relativity. Specifically, the main focus is on two geometric versions of the heat equation: the evolution of surfaces by their mean curvature, and the evolution of curved spaces by Hamilton's Ricci flow. Mean curvature flow models many physical processes which involve an evolving surface, or interface. It is the most efficient way to decrease the area of surfaces and to evolve them towards optimal ones. Correspondingly, Ricci flow deforms curved spaces towards optimal shapes.******While many foundational results have been obtained on both flows, a central problem is that singularities will form in most relevant situations. For example if the geometry at the initial time looks like a dumbbell, then the neck will pinch off preventing one from continuing the flow in a smooth way. The main goal of the proposed research is to improve our understanding of the formation of singularities under mean curvature flow and Ricci flow, and to develop methods to continue the flow beyond the first singular time. This will facilitate many new applications both within and outside mathematics.******A long term goal of my research on mean curvature flow is to construct solutions with surgery for general mean convex hypersurfaces, widely generalizing the estimates and the methodology that I developed in my prior work with Bruce Kleiner. The idea of surgery is to carefully cut the surface shortly before a singularity forms and to heal it by gluing in suitable caps. In joint work with my postdocs and students I will investigate various topological applications of mean curvature flow with surgery, notably higher-dimensional Smale type conjectures about the topology of the moduli-space of embedded spheres.******A long term goal of my research on Ricci flow (mostly joint with Aaron Naber) is to develop a theory of generalized solutions that enable us to continue the flow through singularities. In a recent paper, we proved a new class of estimates for the Ricci flow that are strong enough to characterize solutions. Based on our estimates, we can provide a notion of weak solutions for the Ricci flow, which solves a longstanding open problem. Over the next 5 years we plan to develop the theory of these weak solutions. I'll also investigate several applications, in particular several geometric-analytic conjectures that have been left open after Perelman's solution of the Poincare conjecture.***The proposed research is at the forefront of modern mathematics. One of my main aims is to attract the best Canadian and international students in geometry and analysis to come to Toronto and to involve them in the research projects. I'll organize seminars, discussion groups, topics classes, summer schools and conferences. I'll disseminate the research broadly and will give many expository lectures.**