Singularity Models for Ricci Flow

Ricci 流的奇点模型

• 批准号: 0505920
• 负责人: Dan Knopf
• 金额: $ 10.8万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505920&HistoricalAwards=false
• 关键词: Singularity; Models; Ricci; Flow

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.

Abstractaward：DMS-0505920原理研究者：Dan F. Knopf The项目将推动搜索几何热流的规范几何形状。鉴于佩雷尔曼（Perelman）最近在汉密尔顿（Hamilton）计划解决地球化和庞康卡雷（Poincare）的猜想的计划中进行了具有里程碑意义的进步，该研究领域正是快速而有效的扩展。 Perelman作品中强大的信息和深刻的见解促进了Ricci流的非凡力量，以此作为对Riemannian和Complexchmanifolds的几何形状和拓扑结构进行评估的工具。在Ricci流的几乎所有已知应用中，对形成的机制有深入的了解至关重要。因此，该项目将研究奇异性形成的四方面。这四个目标是基于PI的先前结果和当前的研究计划，并且与RICCI流动的有希望的新应用高度相关。目的是研究（1）RICCI流动奇异性形成的渐近学，（2）尺寸四个方面的流动流奇异性分析，（3）分析Kaehler -Ricci流动流量的分析，（4）降低的几何结构。降低的几何形式是我们的众所周知 - 像我们的宇宙一样 - 在各个区域中，都可以在各个区域中进行整体，但在各个区域中，但在各个区域中，但它的整体范围都不同，但是它的整体范围是众所周知的。该项目的广泛目标是通过分类的最佳几何结构。所使用的方法是某些称为几何热流的部分差分。这个想法是让年龄对象在及时的几何形状和简化拓扑变化之后的几何形状和简化的方式中演变。称为RICCI流量的年龄测量热流刚刚在两个最困难的开放问题中产生了重大破裂。这些成功提供了将IT应用于其他具有挑战性的开放问题的巨大激励措施，并使之成为一个非常活跃和竞争的研究领域。在RICCI流中研究的部分差异类型的类型与模拟页岩和薄膜中的油，多孔培养基中的燃烧，热传播，avalanches，种群分散，显微镜滴剂的扩散，以及质量物理学的效果的类型。因此，在该项目中开发的方法可能具有重要的应用程序，可以是应用数学的领域。该项目将重点放在理解诸如become单数之类的方程所需的分析上。该分析应具有重要的广泛应用，其中包括以下内容。 （i）被开发的方法，尤其是渐近分析，应扩展到上面提到的实践应用。 （ii）该项目将与物理学跨学科的互动，其中许多用于几何分类和流程技术的应用。例如，一般相对论中的理论家希望批量化四维空间的可能拓扑。弦理论和镜像对称性的研究人员对理解某些六维曼群的兴趣感兴趣。 RICCI流本身是量子现场理论中重要模型的晶体规范化流量的近似值。 （iii）该项目将使研究生教育受益，因为PI将花费时间帮助学生在几何，分析和拓扑的相关领域中脱颖而出。