Geometric Rigidity and Isoperimetric Inequalities

几何刚度和等周不等式

• 批准号: 1003679
• 负责人: Christopher Croke
• 金额: $ 36.4万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1003679&HistoricalAwards=false
• 关键词: Geometric; Rigidity; Isoperimetric; Inequalities

This project involves a number of topics. The first topic is the study rigidity theorems (i.e. metric uniqueness) of compact manifolds. Here for example is considered isospectral problems: to what extent must spaces with the same spectra (e.g. eigenvalues of the Laplace Beltrami operator, or Lengths of closed geodesics) be isometric. This also includes questions about metric rigidity induced by conjugacy of geodesic flows, as well as inverse scattering problems. The second topic involves isoperimetric inequalities. The ideal is to find sharp isoperimetric inequalities and study the equality case. This at times ties in with the first topic. It also involves systolic inequalities and other inequalities between geometric quantities. Another topic concerns infinite groups G acting cocompactly on nonpositively curved spaces X (in the sense of Alexandrov). The project is to study the relationship between the geometry of X and the induced action of G on the ideal boundary of X. This can be considered an aspect of geometric group theory. The final topic is geometric optics. This topic involves using differential geometric techniques to design mirrors and lenses to accomplish prescribe optics functions.The rigidity theme of the project is a continuing project with many different aspects. In general these problems concern the question of whether a space can be determined by a prescribed set of data. One aspect of this concerns questions of remote sensing. For example: can you determine the density of an object (say a brain or the earth) from measurements taken "from the outside"? The CAT scan is a practical example where one determines the mass density (or more accurately the absorption coefficient) of an object from measurements of the total mass along straight lines. An alternative set of measurements is the set of times it takes for sound to travel between any two points on the boundary (this is a special case of the boundary rigidity question dealt with in the proposal). A related set of measurements is to record the exit times and directions of geodesics given their entry directions (this is the "geodesic lens" or "scattering" data). The thrust of the proposed study is to determine under which circumstances certain sets of data (e.g. eigenvalues, lengths of closed geodesics, distances between boundary points, lens data) are sufficient to completely determine the geometry of the spaces in question. In some cases it is non-uniqueness that is interesting. For example, in cloaking the goal is to make it the space in question (the object to be cloaked) appear from the outside like a different space (empty space). One aspect of the optics theme of the proposal is the design of multiple mirror (or lens) systems. An example to consider with multiple mirrors is a periscope. By designing appropriately curved mirrors in the periscope one can make for example a non-distorting wide angle periscope or a non-distorting magnifying periscope.

这个项目涉及多个主题。第一个主题是研究紧致流形的刚性定理(即度量唯一性)。例如，这里考虑等谱问题：具有相同谱的空间(例如，Laplace Beltrami算子的特征值或闭测地线的长度)在多大程度上必须是等距的。这也包括由测地线流的共轭引起的度量刚性问题，以及逆散射问题。第二个话题涉及等周不平等。理想的做法是找到尖锐的等周不等式，并研究等式情形。这有时与第一个主题有关。它还涉及收缩不等式和几何量之间的其他不等式。另一个主题是关于无限群G在非正弯曲空间X(在Alexandrov意义下)上的余紧作用。本课题是研究X的几何与G在X的理想边界上的诱导作用之间的关系。这可以被认为是几何群论的一个方面。最后一个话题是几何光学。本课题涉及使用微分几何技术设计反射镜和透镜以完成规定的光学功能。该项目的刚性主题是一个具有许多不同方面的持续项目。一般而言，这些问题涉及空间是否可以由规定的数据集确定的问题。这其中的一个方面涉及遥感问题。例如：你能从“外部”的测量中确定一个物体(比如大脑或地球)的密度吗？CAT扫描是一个实际的例子，人们通过沿直线测量总质量来确定物体的质量密度(或者更准确地说，吸收系数)。另一组测量是声音在边界上的任何两点之间传播所需的时间集(这是提案中处理的边界刚性问题的一个特例)。一组相关的测量是记录给定入射方向的测地线的离开时间和方向(这是“测地线透镜”或“散射”数据)。拟议研究的主旨是确定在哪些情况下某些数据集(如特征值、闭合测地线的长度、边界点之间的距离、透镜数据)足以完全确定有关空间的几何形状。在某些情况下，有趣的是非唯一性。例如，在遮盖中，目标是使它成为有问题的空间(要遮盖的对象)，从外部看起来像是一个不同的空间(空白空间)。该提案的光学主题的一个方面是多镜(或透镜)系统的设计。使用多面镜子需要考虑的一个例子是潜望镜。通过在潜望镜中设计适当的曲面反射镜，例如，可以制造不失真的广角潜望镜或不失真的放大潜望镜。