International Research Fellowship Program: Ricci Curvature and Metric Measure Spaces

0754379慕尼黑国际研究奖学金计划使美国科学家和工程师能够在国外进行9至24个月的研究。 该计划的奖项为联合研究以及使用国外独特或互补的设施、专业知识和实验条件提供了机会。该奖项将支持Michael R.博士为期24个月的研究奖学金。穆恩与彼得·M博士合作。英国沃里克大学数学研究所的Topping。该项目的目的是与Peter Topping博士和沃里克大学数学研究所的几何分析小组密切合作。首先，PI的目标是继续他的研究，研究体积增长对具有非负Ricci曲率的黎曼流形拓扑结构的影响。第二，PI的目的是研究更一般的度量测度空间中Ricci曲率下界的结果。这个概念是在几何分析研究的前沿，最近才得到发展。因此，有许多基本的未决问题。最后，PI旨在研究Ricci流在这种更一般的设置中的行为，并确定底层度量空间的任何拓扑或几何后果。最近，菲尔兹奖得主佩雷尔曼用汉密尔顿？的Ricci流来解决世纪的庞加莱猜想。这些领域正在积极追求的研究人员在数学研究所在沃里克，最值得注意的是由博士托普，从而使沃里克一个理想的地方，这项奖学金的研究。Topping博士是几何分析领域的领先专家，并成功地运用了几何流技术，如Ricci流和平均曲率流。Topping博士是研究度量测度空间上Ricci流性质的少数数学家之一。 度量空间中的Ricci曲率和Ricci流在几何分析和度量几何中具有重要意义。这些原始概念最近才被引入，并处于几何分析研究的前沿。在佩雷尔曼？的工作，他没有解决里奇流时，奇点的发展，这一直是一个重要的问题，汉密尔顿，洛特，和其他人。在最近的一篇论文中，Topping和McCann引入了度量空间的Ricci流的概念。这些进展是非常重要的，因为它们可以用来理解这些奇点的发展。私家侦探？目前的研究涉及佩雷尔曼？涉及Ricci曲率。PI还将这些结果推广到度量测度空间。数学研究所在沃里克是一个蓬勃发展的环境，并正在迅速扩大。研究所拥有大量资源，是一个理想的东道网站。 目前有很大的兴趣在发展这些新的想法里奇流在度量空间和这些技术的应用在整个物理科学。与里奇流有关的思想与平均曲率流以及晶体流的思想密切相关。在这种背景下理解奇点的发展是至关重要的。特别地，奇点描述了晶体中缺陷的发展。更深入地了解这项研究的意义将使PI成为一个更好的教授在一所大学的科学专业和工程师。

0754379MunnThe International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad.This award will support a twenty-four month research fellowship by Dr. Michael R. Munn to work with Dr. Peter M. Topping at Mathematics Institute at the University of Warwick in the UK.The aim of this project is to work closely with Dr. Peter Topping and the Geometric Analysis group of the Mathematics Institute at the University of Warwick. First, the PI aims to continue his research studying the influence of volume growth on the topology of Riemannian manifolds with nonnegative Ricci curvature. Second, the PI aims to study the consequences of a lower bound on Ricci curvature in the more general setting of metric measure spaces. This concept is on the forefront of research in geometric analysis and has only recently been developed. As such there are many fundamental open questions. Lastly, the PI aims to study the behavior of Ricci flow in this more general setting and determine any topological or geometric consequences for the underlying metric space. Recently, Fields Medalist Perelman has used Hamilton?s Ricci flow to solve the century old Poincare conjecture. These fields are being actively pursued by researchers within the Mathematics Institute at Warwick, most notably by Dr. Topping, thus making Warwick an ideal place for this fellowship research. Dr. Topping is a leading expert in geometric analysis and has employed the techniques of geometric flows, such as Ricci flow and mean curvature flow, with great success. Dr. Topping is one of only a few mathematicians studying e nature of Ricci flow on metric measure spaces. An understanding of Ricci curvature and Ricci flow in metric spaces is of great importance in geometric analysis and metric geometry. These original concepts have only recently been introduced and are on the forefront of research in geometric analysis. In Perelman?s work, he did not address the Ricci flow when singularities develop and this has been an important question of Hamilton, Lott, and others. In a very recent paper, Topping and McCann introduce a notion of Ricci flow for metric spaces. These advances are of great importance as they can be used to understand the development of these singularities. The PI?s current research involves prior work of Perelman?s involving Ricci curvature. The PI also extends these results to metric measure spaces. The Mathematics Institute at Warwick is a thriving environment and is undergoing rapid expansion. The large number of resources available within the Institute make this an ideal host site. Currently there is great interest in the development of these new ideas for Ricci flow in metric spaces and these techniques have applications across the physical sciences. The ideas related to Ricci flow are closely related to those of mean curvature flow as well as crystal flow. Understanding the development of singularities in this context is essential. In particular, singularities describe the development of flaws in crystals. A deeper understanding of the implications of this research will enable the PI to be a better professor at a university with science majors and engineers.