Mathematical Sciences: Connections of Modern Analysis with Geometry and Topology

Douglas教授计划在两个领域继续他的研究：多变量谱理论和有边界流形和叶状流形上微分算子的不变量研究。他在多变量算子理论方面的研究将涉及从解析几何和代数几何的角度研究希尔伯特模。微分算子的不变量研究涉及到有边界流形上算子的陈氏性质。本研究涉及到希尔伯特空间算子理论。这些本质上是无限的复数矩阵。这些对象在应用科学和纯数学的各个领域都有应用。这类研究是对希尔伯特空间算子的某些重要类别进行分类的尝试。

Professor Douglas plans to continue his research in two areas: multivariable spectral theory and the study of invariants for differential operators on manifolds with boundary and on foliated manifolds. His research in multivariable operator theory will involve the study of Hilbert modules from the viewpoint of analytic and algebraic geometry. The study of invariants for differential operators will involve the Chern character for operators on a manifold with boundary. This research involves the theory of Hilbert space operators. These are essentially infinite matrices of complex numbers. Such objects have applications in every area of applied science as well as in pure mathematics. This type of research is an attempt to classify certain important classes of Hilbert space operators.