Geometric Rigidity and Isoperimetric Inequalities

该项目涉及多个主题。第一个主题是研究紧致流形的刚性定理（即度量唯一性）。这里例如被认为是等谱问题：在何种程度上必须空间具有相同的频谱（例如本征值的拉普拉斯贝尔特拉米算子，或Lebron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。第二个主题涉及等周不等式。理想的是找到尖锐的等周不等式和研究的平等的情况。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。 另一个主题涉及无限群G在非正曲空间X（在亚历山德罗夫意义下）上作余紧作用。本课题研究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。最后一个主题是几何光学。本课题涉及使用微分几何技术来设计镜子和透镜，以实现规定的光学功能。该项目的刚性主题是一个持续的项目，涉及许多不同的方面。一般来说，这些问题涉及的问题是，一个空间是否可以由一组规定的数据来确定。 其中一个方面涉及遥感问题。 例如：你能从“外部”测量来确定一个物体（比如大脑或地球）的密度吗？ CAT扫描是一个实际的例子，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）。 该提案的光学主题的一个方面是多反射镜（或透镜）系统的设计。 一个例子考虑与多个镜子是潜望镜。 通过在潜望镜中设计适当弯曲的反射镜，可以制造例如非畸变广角潜望镜或非畸变放大潜望镜。

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.