Geometric Rigidity and Isoperimetric Inequalities

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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.