Geometric Scattering Theory, and Propagation and Decay of Waves

首席研究员将研究粒子波对偶性学科中的数学问题。这种对偶性已经被成功地研究了很长一段时间，特别是在封闭系统的背景下，但它的许多重要表现却鲜为人知，特别是在开放和混合系统的背景下，这是主要研究者的重点。在开放系统中，首席研究者所研究的复标度方法在数值分析中以完全匹配层的形式出现，在纳米结构和微电子机械系统的工程中得到了应用。像首席研究员所研究的混合系统在电磁学、通信、光学、量子波导和半导体中都是重要的。主要研究者研究的主要动力学对象是囚禁集，这是始终保持在有界集中的粒子轨迹的集合。这一集合的几何形状在确定波演化的显著特征，如规律性、衰减性和渐近性方面具有核心重要性。首席研究人员将在重要但仍然相对神秘的情况下研究这些特征，这些情况包括重陷阱、粗糙系数和非紧致陷阱集。

The principal investigator will study mathematical problems in the subject of particle-wave duality. This duality has been successfully investigated for a long time, especially in the setting of closed systems, but many of its important manifestations are poorly understood, especially in the settings of open and mixed systems which are the principal investigator's focus. In open systems the method of complex scaling studied by the principal investigator appears in numerical analysis as the method of perfectly matched layers, where it has been useful in the engineering of nanostructures and microelectromechanical systems. Mixed systems like the ones studied by the principal investigator are important in electromagnetics, communications, optics, quantum waveguides, and semiconductors.The main dynamical object studied by the principal investigator is the trapped set; this is the set of particle trajectories which remain in a bounded set for all time. The geometry of this set is of central importance in determining salient features of wave evolution, such as regularity, decay, and asymptotics. The principal investigator will study these features in the important but still comparatively mysterious situations of heavy trapping, rough coefficients, and noncompact trapped sets.