Mathematical Sciences: Pseudodifferential Techniques for Degenerate Elliptic Equations in Geometry

首席研究员将研究退化椭圆方程的伪微分技术及其在几何中的应用。他将扩展技术，以确定作用于无限体积的完全渐近双曲流形和双曲空间的几何有限商的微分形式上的霍奇-拉普拉斯算子的谱分析。相关项目包括研究无穷远处拉普拉斯方程的唯一延拓定理与负曲率存在之间的关系。多年来，数学家们一直在研究扩展到无穷远的超曲面上某些微分算子的行为。这项研究起源于牛顿的微积分。在这个项目中，首席研究员将分析在“接近”无穷大的非常大的域上有很大变化的算子。过去，只在局部变化较小的算子上取得了进展。

The principal investigator will study pseudodifferential techniques in the study of degenerate elliptic equations with applications to geometry. He will extend techniques to determine a spectral analysis of the Hodge-Laplace operator acting on differential forms on complete asymptotic hyperbolic manifolds of infinite volume and on geometrically finite quotients of hyperbolic space. Related projects involve the study of the relationship between unique continuation theorems for the Laplacian at infinity and the presence of negative curvature. For many years mathematicians have studied the behavior of certain differential operators on hypersurfaces which extend off to infinity. This study has its origins in the calculus of Newton. In this project the principal investigator will analyze operators which have large variations over very large domains "near" infinity. In the past, only progress on operators which have small local variations was made.