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

摘要奖：DMS-0542162主要研究者：Richard B。梅尔罗斯，维克托吉列曼这一建议是为了支持美国的参与者在一个程序偏微分方程将于2006年夏季举行，在剑桥，英国牛顿研究所。该计划的重点是谱理论：特别是，最低特征值估计，外尔渐近，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，在这些领域中有一些著名的问题，几十年来一直被证明是难以解决的（如劳赫的“热点”猜想或贝特-索末菲的光谱间隙猜想），根据最近的发展，这些问题似乎即将得到解决，而这些问题的基本现象最近也得到了更好的理解。因此，这个项目是在这些领域的研究人员非常兴奋的时候进行的，明年夏天我们应该有很多东西可以互相学习。物体的光谱是它的自然振动频率的集合。这些包括分子发出（或吸收）的光的光谱线。在许多情况下，频率是作为一个参数的值的集合出现的，对于这个参数，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性，以及从频率中恢复物体信息的程度。几位世界领先的分析师将参与该计划，这是年轻研究人员直接接触一般主题的重要机会。这一分析的分支的研究对物理、化学和其他科学的应用至关重要，其未来的前景在很大程度上取决于二三十岁的人的流入，而这类暑期项目提供了了解第一手资料的机会，这些资料仍在绘图板上，无法在书籍和期刊上获得。有关该计划的更多信息，请访问http：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.