Mathematical Sciences: Geometry and Analysis on Manifolds

小行星9625714 拟议的研究涉及辛几何，谱理论，伪微分算子和指数理论的各个方面。研究人员计划通过辛约化研究群表示的权重多重性，并研究具有边界的紧致流形中伪微分算子代数的K-理论。之间的联系等变corbodism理论辛流形和orbifolds是要追求的精确结构的核心的解决方案运营商科恩的del-bar-Neumann问题的严格pseudoconvex域是要审查以及。 经典力学中的许多问题都可以在辛几何和偏微分方程组的数学背景下表述。在这样一个制定过程中，研究人员面临的一个反复出现的问题是与偏微分方程族相关的许多不同的空间。在指数理论中，这些空间通过称为指数的离散数集来区分，给出了它们的粗略但实用的分类。

9625714 Melrose The proposed research has to do with various aspects of symplectic geometry, spectral theory, pseudo-differential operators and index theory. The investigator plans to study the weight multiplicities of group representations via symplectic reduction and to examine the K-theory of algebras of pseudo-differential operators in compact manifolds with boundaries. The link between equivariant corbodism theory for symplectic manifolds and orbifolds is to be pursued; the precise structure of the kernel of the solution operator to Kohn's del-bar-Neumann problem on strictly pseudoconvex domains is to be examined as well. Many problems in classical mechanics can be formulated in the mathematical context of symplectic geometry and systems of partial differential equations. In the course of such a formulation one recurring problem researchers face is a multitude of different spaces associated with families of partial differential equations. In index theory, these spaces are distinguished by means of a discrete set of numbers called indices, giving a coarse but practical classification of them.