Principal Investigator: Richard B. Melrose, Victor Guillemin
This proposal is to support U.S. participants in a program in partial differential equations to be held in summer 2006, in Cambridge, England at the Newton Institute. The focus of the program is spectral theory: in particular, lowest eigenvalue estimates, Weyl asymptotics, trace formulae, scattering phenomena and properties of periodic Schroedinger operators. These are all very active areas of research. Moreover, there are celebrated problems in these areas which have proved intractible for decades (such as the "hot spot" conjecture of Rauch or the spectral gap conjecture of Bethe-Sommerfeld) which seem, in light of recent developments, be on the verge of being settled, and for which the underlying phenomena have recently become much better understood. Hence this program is occurring at a very exciting time for researchers in these areas, and we should have a lot to learn from each other next summer.
The spectrum of an object is the set of its natural frequencies of vibration. These include the spectral lines in the light emitted (or absorbed) by molecules. In many situations the frequencies arise as the set of values of a parameter for which a partial differential equation has a non-trivial solution. The most important mathematical questions concern the dependence of these frequencies on the geometric attributes of the object and the degree to which information about the object can be recovered from the frequencies. Several of the world's leading analysts will participate in this program, and it is important opportunity for younger researchers, to gain direct access to the general subject. The future prospects of research in this branch of analysis, which is vitally important for applications to physics, chemistry and the other sciences, depend very much on the influx of persons in their twenties and thirties, and summer programs of this type provide the opportunity to learn first hand about developments that are still on the drawing boards, not accessible in books and journals. More information on the program is available at www.newton.cam.ac.uk/programmes/STP/index.html.
