Mathematical Sciences: Some Relations Between Partial Differential Equations and Geometry

首席研究员将分析各种问题 反映了偏微分方程之间的相互作用 以及流形的几何学和拓扑学。 他特别 将扩展他现有的关于L^2-上同调的工作，并导出 L^p-上同调空间的拓扑不变量 此外，本发明还提供了一种方法， 负曲流形小特征值问题 将被调查。 L^p-上同调与数论、代数 几何学、群表示和组合拓扑学。 的 首席研究员将运用他的L^2理论知识 解决两个困难但自然的问题， 考虑更一般的L^p情况。

The principal investigator will analyze various problems reflecting the interaction between partial differential equations and the geometry and topology of manifolds. Specifically, he will extend his existing work on L^2-cohomology and derive topological invariants for L^p-cohomology spaces. In addition, the problem of small eigenvalues of negatively curved manifolds will be investigated. L^p-cohomology has links with number theory, algebraic geometry, group representations, and combinatorial topology. The principal investigator will use his knowledge of the L^2 theory to solve two difficult but natural problems which arise from considering the more general L^p case.