Mathematical Sciences: Applications of Analysis to Problems in Geometry

一名研究人员将研究黎曼流形的刚性定理、无共轭点流形的结构和尖锐的等周不等式等问题。第二个研究人员将研究退化偏微分方程组、黎曼结构的等谱变形和球面空间形式在球面上的最小等距嵌入等相关问题。在这个项目中要研究的问题来自于两个不等距的等谱流形的Milnor的例子。虽然最初的例子是16维的，但最近已经构造了三维例子。类似地，人们可能会问，两个形状相同、敲击时声音相同的圆鼓是否具有完全相同的形状。事实上，他们必须这样做。这两位研究人员将研究高维超曲面的相关深层问题。

One investigator will study problems in the areas of rigidity theorems for Riemannian manifolds, the structure of manifolds without conjugate points, and sharp isoperimetric inequalities. A second investigator will study related problems in the areas of degenerate systems of partial differential equations, isospectral deformations of Riemannian structures, and minimal isometric embeddings of spherical space forms in spheres. The problems to be investigated in this project arise from an example due to Milnor of two isospectral manifolds which are not isometric. While the original example was sixteen dimensional, more recently three dimensional examples have been constructed. Analogously, one might ask if two round drums which have the same rough shape and which sound the same when struck, have the same exact shape. In fact, they must. These two investigators will study related deep questions for higher dimensional hypersurfaces.