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

项目编号：dms -0542162项目负责人：Richard B. Melrose， Victor guillemin本项目为2006年夏季在英国剑桥牛顿研究所举办的偏微分方程研究项目的美国参与者提供资助。该计划的重点是谱理论：特别是，最低特征值估计，Weyl渐近，迹公式，散射现象和周期性薛定谔算子的性质。这些都是非常活跃的研究领域。此外，这些领域中还有一些著名的问题，几十年来一直被证明是难以解决的（比如劳赫的“热点”猜想或贝斯-索默菲尔德的谱隙猜想），根据最近的发展，这些问题似乎即将得到解决，而且对这些问题的潜在现象最近也有了更好的理解。因此，这个项目对这些领域的研究人员来说是一个非常激动人心的时刻，明年夏天我们应该有很多东西可以相互学习。一个物体的频谱是其固有振动频率的集合。这些包括被分子发射（或吸收）的光谱线。在许多情况下，频率出现在偏微分方程具有非平凡解的参数值的集合中。最重要的数学问题涉及这些频率对物体几何属性的依赖，以及从频率中恢复物体信息的程度。几位世界领先的分析师将参加这个项目，这对年轻的研究人员来说是一个重要的机会，可以直接接触到一般的主题。这一分析分支的研究对物理、化学和其他科学的应用是至关重要的，它的未来前景在很大程度上取决于二三十岁年轻人的大量涌入，而这类暑期项目提供了第一手了解尚在规划阶段、无法在书本和期刊上看到的发展的机会。有关该计划的更多信息请访问：//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.