Mathematical Sciences: Applications of Analysis to Problems in Geometry

一名研究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等周不等式。第二名研究员将研究偏微分方程组的退化系统、黎曼结构的等谱变形以及球面空间形式的最小等距嵌入等领域的相关问题。本课题所研究的问题是由两个非等距的等谱流形的Milnor引起的。虽然最初的例子是16维的，但最近已经构建了三维的例子。类似地，有人可能会问，如果两个圆鼓有相同的粗糙形状和敲起来的声音相同，它们是否具有完全相同的形状。事实上，他们必须这么做。这两位研究者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.