Singularity Models for Ricci Flow

摘要奖：DMS-0505920首席研究员：丹·F·克诺普夫该项目将通过几何热流来推进正则几何的研究。鉴于最近佩雷尔曼在哈密尔顿解决几何化和庞加莱猜想的计划中取得的里程碑式的进展，这一研究领域正在经历快速和富有成效的扩展。佩雷尔曼工作中的强大创新和深刻见解造就了利玛奇流作为研究黎曼和复流形的几何和拓扑的工具的非凡力量。在几乎所有已知的Ricci流应用中，对奇点形成机制的深入理解是至关重要的。因此，该项目将研究奇点形成的四个方面。选择这四个目标是为了建立在PI的先前结果和当前研究计划的基础上，并与RICCI FLOW的有前景的新应用高度相关。其目的是研究(1)Ricci流奇点形成的渐近性，(2)四维Ricci流奇点的分析，(3)Kaehler-Ricci流奇点模型的分析，以及(4)约化几何的结构。流形是一个物体，它像我们的宇宙一样，局部看起来像欧几里德空间，但它的全局拓扑和几何可能有很大的不同。这个项目的主要目标是找到对流形进行分类的最佳几何结构。所用的方法是某些称为几何热流的偏微分方程组。这个想法是让年龄测量对象在时间上进化，使其几何结构得到改进和简化，可能是在拓扑发生变化后。一种被称为Ricci流的几何热流刚刚在数学中两个最困难的公开问题上取得了重大突破。这些成功为将其应用于其他具有挑战性的开放问题提供了巨大的激励，并使其成为一个非常活跃和竞争激烈的研究领域。在Ricci流中研究的偏微分方程组的类型与用于模拟石油在页岩和薄膜中的运动、在多孔介质中的燃烧、热传播、雪崩、种群扩散、微观液滴的扩散以及等离子体物理中的某些效应的偏微分方程组有许多相似之处。因此，在这个项目中开发的方法可能会在应用数学的这些领域有重要的应用。该项目将重点放在理解这些方程变奇异时所需的精细分析上。这种分析应该有重要的广泛应用，其中包括以下几点。(I)所发展的方法，特别是渐近分析，应扩展到上述实际应用。(Ii)该项目将促进与物理学的跨学科互动，在物理学中有许多几何分类和流动技术的应用。例如，广义相对论的理论家们想要对四维时空的可能拓扑进行分类。弦理论和镜像对称性的研究人员对理解某些六维流形很感兴趣。Ricci流本身是量子场论中一个重要模型的热正化流的近似值。(Iii)该项目将有利于研究生教育，因为PI将投入时间帮助学生发展几何、分析和拓扑学相关领域的专业知识。

AbstractAward: DMS-0505920Principal Investigator: Dan F. KnopfThe project will advance the search for canonical geometries bymeans of geometric heat flows. In light of the landmark progressmade recently by Perelman in Hamilton's program to resolve theGeometrization and Poincare' Conjectures, this research area isundergoing a rapid and productive expansion. The powerfulinnovations and profound insights in Perelman's work contributeto the extraordinary power of Ricci flow as a tool forinvestigating the geometry and topology of Riemannian and complexmanifolds. In virtually all known applications of Ricci flow, itis critical to have a deep understanding of the mechanisms ofsingularity formation. Therefore, the project will investigatefour aspects of singularity formation. These four objectives arechosen to build upon the prior results and current researchprogram of the PI and to be highly relevant to promising newapplications of Ricci flow. The objectives are to study (1)asymptotics of Ricci flow singularity formation, (2) analysis ofRicci flow singularities in dimension four, (3) analysis ofsingularity models for Kaehler-Ricci flow, and (4) the structureof reduced geometry.A manifold is an object that - like our universe - looks likeEuclidean space locally, but whose global topology and geometrymay be much different. The broad goals of this project are tofind optimal geometric structures with which to categorizemanifolds. The methods used are certain partial differentialequations called geometric heat flows. The idea is to let ageometric object evolve in time in such a way that its geometryimproves and simplifies, possibly after a change in topology. Ageometric heat flow called the Ricci flow has just yielded majorbreakthroughs in two of the most difficult open problems inmathematics. These successes provide great incentives to apply itto other challenging open problems and make it a very active andcompetitive field of research. The types of partial differentialequations studied in Ricci flow have much in common with thoseused to model the movement of oil in shale and in thin films,combustion in porous media, heat propagation, avalanches,population dispersal, the spreading of microscopic droplets, andcertain effects in plasma physics. For this reason, methodsdeveloped in this project may have important applications tothose areas of applied mathematics. The project will focus on thedelicate analysis needed to understand such equations as theybecome singular. This analysis should have important broadapplications, among which are the following. (i) The methodsdeveloped, especially asymptotic analysis, should extend to thepractical applications mentioned above. (ii) The project willpromote interdisciplinary interactions with physics, where thereare many applications for geometric classification and flowtechniques. For example, theorists in general relativity want toclassify possible topologies of four-dimensionalspace-times. Researchers in string theory and mirror symmetry areinterested in understanding certain six-dimensionalmanifolds. The Ricci flow itself is an approximation to therenormalization flow for an important model in quantum fieldtheory. (iii) The project will benefit graduate education,because the PI will invest time helping students developexpertise in relevant areas of geometry, analysis, and topology.