Isoperimetric Inequalities and Rigidity

这个项目有两个密切相关的主题。 第一个是研究有边界和无边界的紧致流形上的刚性定理（即度量唯一性）。 第二个是研究锐等周不等式。 这里的“等周不等式”应解释为整体几何量（如体积、边界体积、拉普拉斯贝尔特拉米算子的特征值、闭测地线的长度等）之间的不等式。而尖锐意味着不等式是最好的可能。 它们是相关的，在适当的尖锐等周不等式的平等的情况下，可以导致有趣的刚性的结果。我们认为，例如isospectral问题：在何种程度上必须空间具有相同的频谱（例如特征值的拉普拉斯Beltrami运营商，或Lounge的封闭测地线）是等距的。 这也包括度量刚性引起的共轭测地线流的问题，以及逆散射问题。 另一个主题的建议是研究无限群G的合作非积极弯曲空间H（在意义上的亚历山德罗夫）。 本文研究了H的几何形状与G在H的理想边界上的诱导作用之间的关系。 这是几何群论的一个方面。 该项目的刚性主题涉及一个空间是否可以由特定的数据集确定的问题。 其中一部分涉及遥感问题。 例如：你能从“外部”测量来确定一个物体（比如一个人的身体或月球）的密度吗？ CAT扫描是一个实际的例子，其中人们从沿沿着直线的总质量的测量确定物体的质量密度（或更准确地说是吸收系数）。 另一组测量值是声音在边界上任何两点之间传播所需的时间（这是提案中处理的边界刚性问题的特殊情况）。 一组相关的测量是记录给定测地线进入方向的测地线的退出时间和方向（这是“测地线透镜”或“散射”数据）。 拟议研究的主旨是确定在何种情况下某些数据集（例如特征值、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，在隐身中，目标是使所讨论的空间（要隐身的对象）从外部看起来像一个不同的空间（空的空间）。

There are two major closely related themes to this project. The first is the study of rigidity theorems (i.e. metric uniqueness) on compact manifolds with and without boundary. The second is the study of sharp isoperimetric inequalities. Here "isoperimetric inequality" should be interpreted as an inequality between global geometric quantities (such as volume, volume of the boundary, eigenvalues of the Laplace Beltrami operator, lengths of closed geodesics etc.) while sharp means that the inequalities are the best possible. They are related in that the case of equality in appropriate sharp isoperimetric inequalities can lead to interesting rigidity results.We consider for example 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. Another theme of the proposal is the study of infinite groups G acting cocompactly on nonpositively curved spaces H (in the sense of Alexandrov). The project is to study the relationship between the geometry of H and the induced action of G on the ideal boundary of H. This is an aspect of geometricgroup theory. The rigidity theme of the project concerns the question of whether a space can be determined by a certain set of data. One part of this relates to questions of remote sensing. For example: can you determine the density of an object (say a person's body or the moon) 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).