Traces, Singularities and K-Theory

摘要奖：DMS-0408993主要研究人员：理查德·B·梅尔罗斯和维克托·吉列明主要研究与分析、微分几何和拓扑学有关的各种基本问题，特别是奇异现象的出现。在不同的合作中，他们计划研究具有奇性的流形上的椭圆微分和伪微分算子的可解性，并计算拓扑障碍，特别是通过K-理论来计算可逆性。同时，在其他合作中，他们还打算继续研究波在具有奇点的流形上的传播，通常是以奇异锋面的形式，以及这两个问题之间的关系。Kaehler和辛几何中的投影强调作用在这些空间上的群的动量映射和构造，如切割、粘合和按对称群约简。其中几个投影的特征之一是通过底层空间的微分拓扑来表示全球尺度上的分析对象。例如，许多几何空间带有描述波运动的方程的版本，空间形状的各个方面被记录在它们上的波的衰减和散射特性中。这项工作倾向于将数学和物理中的各种研究结合在一起，特别是量子化理论和辛几何，研究哈密顿力学可以在其上定义的空间。

AbstractAward: DMS-0408993Principal Investigator: Richard B. Melrose and Victor GuilleminThe principal investigators propose to study a variety of basicquestions relating analysis, differential geometry and topologyand especially concerned with the appearance of singularphenomena. In various collaborations, they plan to study thesolvability properties of elliptic differential andpseudodifferential operators on manifolds with singularities andto calculate the topological obstructions, particularly throughK-theory, to invertibility. Jointly, and in other collaborations,they also intend to continue the study of the propagation ofwaves, typically in the form of singular fronts, on manifoldswith singularities and the relationship between these twoproblems. Projects in Kaehler and symplectic geometry emphasizemoment maps for groups acting on these spaces and constructionssuch as cutting, gluing, and reduction by groups of symmetries.One of the features of several of these projects is theexpression of analytic objects on a global scale through thedifferential topology of the underlying space. For example, manygeometric spaces carry versions of the equation describing wavemotion, and aspects of the shapes of the spaces are recorded inthe decay and scattering properties of waves on them. This worktends to bring together various strands of research inmathematics and physics, in particular quantization theory andsymplectic geometry, the study of spaces on which Hamiltonianmechanics can be defined.