Conference: Spectral Theory and Partial Differential Equations; July 17-August 11, 2006; Cambridge, England

摘要奖：DMS-0542162首席研究员：理查德·B·梅尔罗斯，维克托·吉列明这项提议是为了支持美国参与者参加将于2006年夏天在英国剑桥牛顿研究所举行的偏微分方程式项目。程序的重点是谱理论：特别是最低本征值估计，Weyl渐近性，迹公式，散射现象和周期薛定谔算子的性质。这些都是非常活跃的研究领域。此外，在这些领域存在着几十年来被证明难以解决的重大问题(如劳赫的“热点”猜想或Bethe-Sommerfeld的光谱鸿沟猜想)，根据最近的事态发展，这些问题似乎即将得到解决，其根本现象最近也得到了更好的理解。因此，对于这些领域的研究人员来说，这个项目发生的时间非常令人兴奋，明年夏天我们应该有很多东西可以相互学习。一个物体的频谱是它的固有振动频率的集合。其中包括被分子发光(或吸收)的光谱线。在许多情况下，频率作为参数的一组值出现，对于该参数，偏微分方程有一个非平凡解。最重要的数学问题涉及这些频率对物体几何属性的依赖，以及从这些频率中可以恢复关于物体的信息的程度。几位世界领先的分析家将参加这个项目，对于年轻的研究人员来说，这是一个直接接触普通学科的重要机会。这一分析分支的未来研究前景对物理、化学和其他科学的应用至关重要，在很大程度上取决于二三十岁的人的涌入，而这种类型的暑期项目提供了机会，让人们有机会第一手了解仍然在设计板上、在书本和期刊上无法获得的发展。有关该计划的更多信息，请访问athttp://www.newton.cam.ac.uk/programmes/STP/index.html.

AbstractAward: DMS-0542162Principal Investigator: Richard B. Melrose, Victor GuilleminThis proposal is to support U.S. participants in a program inpartial differential equations to be held in summer 2006, inCambridge, England at the Newton Institute. The focus of theprogram is spectral theory: in particular, lowest eigenvalueestimates, Weyl asymptotics, trace formulae, scattering phenomenaand properties of periodic Schroedinger operators. These are allvery active areas of research. Moreover, there are celebratedproblems in these areas which have proved intractible for decades(such as the "hot spot" conjecture of Rauch or the spectral gapconjecture of Bethe-Sommerfeld) which seem, in light of recentdevelopments, be on the verge of being settled, and for which theunderlying phenomena have recently become much betterunderstood. Hence this program is occurring at a very excitingtime for researchers in these areas, and we should have a lot tolearn from each other next summer.The spectrum of an object is the set of its natural frequenciesof vibration. These include the spectral lines in the lightemitted (or absorbed) by molecules. In many situations thefrequencies arise as the set of values of a parameter for which apartial differential equation has a non-trivial solution. Themost important mathematical questions concern the dependence ofthese frequencies on the geometric attributes of the object andthe degree to which information about the object can be recoveredfrom the frequencies. Several of the world's leading analystswill participate in this program, and it is important opportunityfor younger researchers, to gain direct access to the generalsubject. The future prospects of research in this branch ofanalysis, which is vitally important for applications to physics,chemistry and the other sciences, depend very much on the influxof persons in their twenties and thirties, and summer programs ofthis type provide the opportunity to learn first hand aboutdevelopments that are still on the drawing boards, not accessiblein books and journals. More information on the program isavailable athttp://www.newton.cam.ac.uk/programmes/STP/index.html.