Mathematical Sciences: Some Relations Between Partial Differential Equations and Geometry

首席研究员将分析反映偏微分方程与流形几何和拓扑之间相互作用的各种问题。具体而言，他将扩展他在L^2-上同调上的现有工作，并推导L^p-上同调空间的拓扑不变量。此外，还研究了负弯曲流形的小特征值问题。L^p-上同调与数论、代数几何、群表示和组合拓扑都有联系。首席研究员将利用他对L^2理论的知识来解决两个困难但自然的问题，这些问题是在考虑更一般的L^p情况时产生的。

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.