CAREER: Investigating Ricci flow singularity formation

摘要奖：DMS-0545984首席研究员：Dan F.Knopf该研究员的研究计划的主要目标是使用曲率流找到最佳几何形状并对其进行分类。这一主题目前正受到人们的强烈兴趣和快速发展，灵感来自于佩雷尔曼在汉密尔顿的程序中的里程碑式的结果，该程序用利玛奇流解决了庞加莱和几何化猜想。在Ricci流的应用中，人们在流形上发展黎曼度量来简化和改进流形的几何。在许多情况下，这些改进涉及由奇点形成触发的拓扑变化。这样的改进是可能的，因为预期Ricci流的解(像许多其他非线性偏微分方程组)在发展奇点的时空邻域中表现出非常特殊的、高度对称的轮廓。因此，对奇点形成的透彻和深入的理解对于从Ricci流动行为中提取拓扑和几何信息至关重要。研究人员的近期目标集中在曲率流的奇点形成，特别是(1)奇点形成的分析方面(渐近性)，(2)四维奇点的结构，(3)Kaehler-Ricciflow的奇点，(4)Perelman熵和约化距离的结构，以及(5)几何流在物理问题中的应用。朝着这些目标的进展将导致Ricci流在流形的几何和拓扑中的新应用，对各种几何流之间的平行关系的富有成效的见解，以及曲率流在材料科学和物理所激发的问题中的新应用。这些目标非常适合与几何演化方程和相关领域的其他研究人员合作，如比较几何、低维拓扑和非线性分析。曲率流动中产生的偏微分方程组与用于模拟热传播、石油在页岩和薄膜中的运动、多孔介质中的燃烧和等离子体物理中的某些效应的方程非常相似。通过增加我们对这类方程奇点形成的理解，本项目的研究内容可能有助于在这些应用方面取得进展。该项目的教育要素与其研究重点紧密结合在一起。这一整合的基石是一系列年度讲习班：每个讲习班都将包括研究生教育部分、研究部分和本科生推广部分。教育部分包括一次有组织的研讨会，为毕业生参加研究部分做好准备。外展部分包括邀请当地本科生与访问研究人员互动，目的是吸引这些学生继续深造数学。该项目将在另外两个方面惠及本科教育：研究人员将开发一门新的最佳几何课程，并将其整合到德克萨斯大学的文科荣誉课程中；研究人员将组织和指导一个本科生通过发现学习的项目，在该项目中，学生将研究组合曲率流。

AbstractAward: DMS-0545984Principal Investigator: Dan F. KnopfThe broad goals of the investigator's research program are to find andclassify optimal geometries using curvature flows. This subject iscurrently enjoying a period of intense interest and rapid progress,inspired by Perelman's landmark results in Hamilton's program toresolve the Poincare and Geometrization Conjectures by Ricci flow. Inapplications of Ricci flow, one evolves a Riemannian metric on amanifold to simplify and improve its geometry. In many cases, theseimprovements involve changes in topology that are triggered bysingularity formation. Such improvements are possible becausesolutions of Ricci flow (like many other nonlinear PDE) are expectedto exhibit very special, highly symmetric profiles in a space-timeneighborhood of a developing singularity. A thorough and deepunderstanding of singularity formation is therefore of criticalimportance for extracting topological and geometric information fromRicci flow behavior. The investigator's immediate objectives focus onsingularity formation for curvature flows, particularly (1) analyticaspects (asymptotics) of singularity formation, (2) the structure ofsingularities in dimension four, (3) singularities of Kaehler-Ricciflow, (4) the structure of Perelman's entropy and reduced distance,and (5) applications of geometric flows to problems inphysics. Progress toward these objectives will lead to newapplications of Ricci flow in the geometry and topology of manifolds,productive insights into parallels between various geometric flows,and new applications of curvature flows to problems motivated bymaterials science and physics. These objectives are well suited tocollaborations with other researchers in geometric evolution equationsand in related fields like comparison geometry, low-dimensionaltopology, and nonlinear analysis.The partial differential equations that arise in curvature flows areremarkably similar to equations used to model heat propagation, themovement of oil in shale and thin films, combustion in porous media,and certain effects in plasma physics. By increasing our understandingof singularity formation for such equations, the research elements ofthis project may contribute to progress in these applications. Theeducation elements of the project are very tightly integrated with itsresearch focus. The cornerstone of this integration is a series ofannual workshops: each will contain a graduate education component, aresearch component, and an undergraduate outreach component. Theeducation component includes an organized seminar to prepare graduatestudents for participation in the research component. The outreachcomponent includes invitations to local undergraduate students tointeract with the visiting researchers with the goal of attractingthose students to further studies in mathematics. The project willbenefit undergraduate education in two other ways: the investigatorwill develop a new course in optimal geometry and integrate it intothe liberal-arts honors curriculum at the University of Texas, and theinvestigator will organize and direct an undergraduatelearning-by-discovery project in which students investigatecombinatorial curvature flows.