Mathematical Sciences: Connections of Modern Analysis with Geometry and Topology

道格拉斯教授计划在两年内继续他的研究 领域：多变量谱理论和不变量的研究 对于流形上的微分算子， 叶状流形 他对多变量算子的研究 理论将涉及希尔伯特模的研究， 解析几何和代数几何的观点。 研究 微分算子的不变量将涉及陈 有界流形上算子的特征 本研究涉及希尔伯特空间理论 运营商 这些本质上是复数的无限矩阵 号码 这样的物体在各个领域都有应用， 应用科学以及纯数学。 这种类型的 研究是试图将某些重要类别的 希尔伯特空间算子

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.