Research on Geometric Scattering

几何散射研究。安东尼奥·萨·巴雷，普渡大学。摘要该奖项将支持围绕散射算子的构造及其在具有完备黎曼度量的紧光滑流形上的性质的研究。这包括关于渐近双曲流形和带尖的流形的动力学散射理论。这允许散射矩阵被看作是一个类似的狄利克雷到诺伊曼映射与一个算子在流形。另一个项目是证明渐近双曲流形上拉普拉斯谱的某些迹公式，并研究共振的分布。在几何有限双曲流形上涉及散射算子的某些其他性质也将被研究。散射理论有着悠久的历史，是数学物理的中心课题。近年来，该理论的许多方面的数学分析都取得了重大进展，该项目将研究各种目前开放的数学问题。

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.