Nonpositive Curvature and Geometric Rigidity

摘要DMS -0202536.该项目涉及两个主要主题。 第一个是紧流形上刚性定理（即度量唯一性）的研究。 例如，在这里我们考虑等谱问题：在何种程度上必须具有相同谱的空间（例如，特征值的拉普拉斯Beltrami算子，或Lebron的封闭测地线）是等距的。 这也包括度量刚性问题引起的共轭测地线流，以及逆散射问题。 另一类则认为无限群G在非正曲空间H（在Alexandrov意义下）上作用余紧。 本文研究了H的几何与G在H的理想边界上的诱导作用之间的关系。 这可以被认为是geometricgroup理论的一个方面，部分原因是格罗莫夫的一些问题。 除了这两个主要主题外，该建议还涉及作者对各种等周不等式的持续工作. 这些群表现为阿达玛空间H（包括非正曲率空间）的对称性。 该项目的第一个主题涉及一个空间是否可以由某组数据确定的问题。 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，其中一个确定的质量密度（或更准确地说，吸收系数）的对象从测量的总质量沿直线。 另一组测量值是声音在边界上任意两点之间传播所需的时间（这是建议中处理的边界刚性问题的特殊情况）。 所提出的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）足以完全确定所讨论的空间的几何形状。群自然地表现为各种空间的对称性。 这个项目的第二个主题是研究一类无限群.

ABSTRACT DMS - 0202536.This project concerns two major themes. The first is the study ofrigidity theorems (i.e. metric uniqueness) on compact manifolds. Herefor example we consider isospectral problems: to what extent must spaceswith the same spectra (e.g. eigenvalues of the Laplace Beltramioperator, or Lengths of closed geodesics) be isometric. This alsoincludes questions about metric rigidity induced by conjugacy ofgeodesic flows, as well as inverse scattering problems. The other themeconsiders infinite groups G acting cocompactly on nonpositively curvedspaces H (in the sense of Alexandrov). The project is to study therelationship between the geometry of H and the induced action of G onthe ideal boundary of H. This can be considered an aspect of geometricgroup theory and is partially motivated by some questions of Gromov. As well as these two major themes the proposal concerns the authorscontinuing work on various isoperimetric inequalities. These groups show up as symmetries of Hadamard spaces H (which include spaces of nonpositive curvature.) The first theme of the project concerns the question of whether a spacecan be determined by a certain set of data. One part of this relates toquestions of remote sensing. For example: can you determine the densityof an object (say a persons body or the moon) from measurements taken"from the outside"? The CAT scan is a practical example where onedetermines the mass density (or more accurately the absorptioncoefficient) of an object from measurements of the total mass alongstraight lines. An alternative set of measurements is the set of timesit takes for sound to travel between any two points on the boundary(this is a special case of the boundary rigidity question dealt with inthe proposal). The thrust of the proposed study is to determine underwhich circumstances certain sets of data (e.g. eigenvalues, lengths ofclosed geodesics, distances between boundary points) are sufficient tocompletely determine the geometry of the spaces in question. Groups show up naturally as symmetries of various spaces. The second theme of this project concerns the study a certain class of infinite groups.