Asymptotics, Homology and the Wave Equation

DMS -0104116摘要。本建议书由七项内容组成。第一个项目是两位主要研究者的一个联合项目，它回到了以前关于带边界流形的波迹的联合工作，通过将波迹定义为边界数据空间上的迹，而不是波动方程初始数据空间上的迹，来锐化和简化以前的结果。 第二个项目是Guillemin和K. Okikiolu并涉及Szego估计的“两层”渐进性的最新结果的改进。第三个项目是Guillemin与C. Zara，并关注图论和拓扑之间的相互作用的GKM流形。第四项是Melrose和P. Loya的一个联合项目，将Atiyah-Patodi-Singer指标定理推广到了带角点的流形上。第五项涉及Melrose和R. Mazzeo和D. Grieser等人用blow-up和伪微分方法分析了奇异代数簇上的拉普拉斯算子。第六个项目涉及Melrose，A. Hassell和A. Vasy在描述散射的潜力是光滑的无穷大。最后一个项目是Melrose和J. Wunschin的一个联合项目，研究了具有圆锥奇点的流形上的波的奇点传播。例如，这两个主要研究者的联合项目的重要应用之一将是了解平面上一个区域中的波的反射。对于一个凸planardomain这个项目的结果应该阐明celebratedproblem：“可以听到鼓的形状”，也就是说，做的频率ofvibration的一个平面域，对应于头部的鼓，确定其形状？早期的研究表明，这些振动频率决定了所谓的“长度谱”的域，即长度的内接多边形的最小周长。上述研究的结果之一应该是这些多边形的形状的确定。

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.