Asymptotics, Homology and the Wave Equation

DMS-0104116的摘要本提案包括七项内容。第一项是两位主要研究者的联合项目，回到了以前关于带边界流形的波迹的共同工作，目的是通过定义波迹来锐化和简化早期的结果，而不是定义为波动方程初始数据空间上的迹，而是边界数据空间上的迹。第二个项目是吉列明和K.Okikiolu的联合项目，涉及对Szego估计的“两级”渐近性的最新结果的改进。第三个项目是Guillmin和C.Zara的合作项目，它涉及图论和GKM流形上的拓扑学之间的相互作用。第四项是Melrose和P.Loya的联合项目，将Atiyah-Patodi-Singer指数定理推广到带角的流形。第五项涉及Melrose和R.Mazzeo正在进行的工作，以及与D.Grieser用爆破和伪微分方法分析奇异代数簇上的拉普拉斯的项目的开始。第六个项目涉及MelRose，A.Hassell和A.Vasy在描述光滑到无穷大的势的散射时。最后一项是Melrose和J.Wunschin的联合项目，其中研究了具有圆锥奇点的流形上波的奇性传播。上述项目的一个共同主题是波动方程和相关技巧。例如，两位主要研究人员的联合项目的重要应用之一将是理解波在平面中域中的反射。对于一个凸平面域，这个项目的结果应该阐明一个著名的问题：“人们能听到鼓的形状吗”，也就是说，与鼓头对应的平面域的振动频率是否决定了它的形状？研究人员早期的工作表明，振动的频率决定了域的所谓“长度谱”，即最小周长的内切多边形的长度。上述调查的结果之一应该是确定这些多边形的形状。

Abstract for DMS - 0104116.This proposal consists of seven items. The first item is a jointproject of the two principal investigators and returns to previousjoint work on the wave trace for a manifold with boundary, in order tosharpen and simplify the earlier results by defining the wave trace,not as a trace on the space of initial data of the wave equation, butas a trace on the space of boundary data. The second item is a jointproject of Guillemin and K. Okikiolu and involves refinements of arecent result on ``two-tiered'' asymptotics for Szego estimates. Thethird item is a joint project of Guillemin with C. Zara, and concernsthe interplay between graph theory and topology on GKM manifolds. Thefourth item is a joint project of Melrose and P. Loya extending theAtiyah-Patodi-Singer index theorem to manifolds with corners. Thefifth item involves on-going work of Melrose and R. Mazzeo and thebeginning of a project with D. Grieser to analyze the Laplacian onsingular algebraic varieties by blow-up and pseudodifferentialmethods. The sixth project involves Melrose, A. Hassell and A. Vasy inthe description of scattering by potentials which are smooth up toinfinity. The final item is a joint project of Melrose and J. Wunschin which the propagation of singularities for waves on manifolds withconic singularities is investigated.A common theme of the items above is the wave equation and relatedtechniques. For instance, one of the important applications of the jointproject of the two principal investigators will be to the understanding ofthe reflection of waves in a domain in the plane. For a convex planardomain the results of this project should shed light on the celebratedproblem: "Can one hear the shape of a drum", that is, do the frequencies ofvibration of a planar domain, corresponding to the head of a drum,determine its shape? Earlier work of the investigators showed that thesefrequencies of vibration determine the so-called "length spectrum" of thedomain, namely the lengths of inscribed polygons of minimalcircumference. One result of the investigation above should be adetermination of the SHAPES of these polygons as well.