Mean curvature flow and Ricci flow

拟议的研究是在微分几何，偏微分方程，变分法，随机分析和广义相对论的交叉点。具体来说，主要的重点是两个几何版本的热方程：表面的演变，其平均曲率，和演变的弯曲空间的汉密尔顿的里奇流。平均曲率流模拟了许多物理过程，这些过程涉及到一个不断变化的表面或界面。这是最有效的方法来减少表面的面积，并向最佳的发展。相应地，Ricci流使弯曲空间朝向最佳形状变形。 虽然在这两种流动中已经获得了许多基本结果，但一个核心问题是奇点将在大多数相关情况下形成。例如，如果几何形状在初始时间看起来像哑铃，则颈部将夹断，从而防止以平滑的方式继续流动。所提出的研究的主要目标是提高我们的理解下的平均曲率流和里奇流的奇点的形成，并开发方法，继续流超过第一个奇异时间。这将促进数学内外的许多新应用。 一个长期的目标，我的研究平均曲率流是构造解决方案与手术一般的平均凸超曲面，广泛推广的估计和方法，我在我以前的工作与布鲁斯克莱纳。外科手术的想法是在奇点形成前不久小心地切割表面，并通过粘上合适的帽来治愈它。在与我的博士后和学生的联合工作中，我将研究手术平均曲率流的各种拓扑应用，特别是关于嵌入球体的模空间拓扑的高维Smale型拓扑。 我对Ricci流的研究（主要是与Aaron Naber联合）的一个长期目标是发展一种广义解理论，使我们能够继续通过奇点流动。在最近的一篇论文中，我们证明了一类新的估计的Ricci流是强大到足以刻画解决方案。根据我们的估计，我们可以为里奇流提供弱解的概念，它解决了一个长期存在的开放问题。 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，特别是在佩雷尔曼解决庞加莱猜想之后留下的几个几何解析解。 这项研究处于现代数学的前沿。我的主要目标之一是吸引最好的加拿大和国际学生在几何和分析来多伦多，并参与他们的研究项目。我将组织研讨会、讨论小组、专题班、暑期学校和会议。我将广泛传播这项研究，并将作许多简短的讲座。

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.