Research on Geometric Scattering

这个奖项将支持以构造散射算子为中心的数学研究，并研究具有完备黎曼度量的紧致光滑流形上的性质。这包括渐近双曲流形和具有尖点的流形上的动力学散射理论。这使得散射矩阵可以看作是与流形上的算子相关联的Dirichlet到Neumann映射的类似。另一个项目是证明渐近双曲流形上拉普拉斯函数谱的某些迹公式，并研究共振的分布。涉及几何有限双曲流形上的散射算子的某些其他性质也将被研究。散射理论有很长的历史，是数学物理中的一个中心话题。近年来，该理论的许多方面的数学分析都有了很大的进步，这个项目将调查各种目前尚未解决的数学问题。

Research on Geometric Scattering.Antonio Sa BarretoPurdue University.AbstractThis award will support mathematical research centering on the constructionof scattering operators and the investigation of their properties on compactsmooth manifolds with a complete Riemannian metric. This includes topics indynamical scattering theory on asymptotically hyperbolic manifolds andmanifolds with cusps. This allows the scattering matrices to be viewedas an analogue of Dirichlet to Neumann map associated with an operator on amanifold. Another project is to prove certain trace formulae for thespectrum of the Laplacian on asymptotically hyperbolic manifolds and toinvestigate the distribution of resonances. Certain other propertiesinvolving the scattering operator on geometrically finite hyperbolicmanifolds will also be studied.Scattering theory has a long history and was a central topic in mathematicalphysics. The mathematical analysis of many aspects of the theory hasundergone significant advances in recent years and this project willinvestigate a variety of currently open mathematical questions.