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Spectral theory and partial differential equations stand at a meeting point of several different parts of mathematics and physics. Within mathematics it links spectral properties of elliptic and parabolic operators to the geometry and topology of the underlying manifold. Within physics it links, for example, the stability of matter to the properties of the potentials in the Schrödinger operators.
Some of the fundamental problems of spectral theory have been quite well understood. These include, for instance, the relation between the asymptotic properties of various spectral quantities (spectral counting function, heat content and heat trace functions) and the geometry of the underlying manifold, the general properties of periodic and magnetic operators etc. On the other hand certain questions of spectral geometry (e.g. connection between the bottom of the spectrum, nodal lines, multiplicities of eigenvalues and the geometric properties of the region or manifold), and the theory of periodic operators (e.g. detailed properties of the band-gap spectrum, absolute continuity, the number of gaps) remain unanswered.
Some fundamental questions in the theory of multi-dimensional Schrödinger operators, such as absolute continuity, are open and at present substantial efforts are being made towards understanding these through the so-called trace formulae. Many of these questions have important applications in physics (solid state physics, statistical physics, large particle systems, quantum mechanics, photonic crystals).
The aim of the programme is to focus the expertise in Spectral Theory on the issues mentioned above and incite useful collaborations involving mathematicians from the UK and other countries.
The following is the list of people who have so far agreed to be participants on the Programme; M. Ashbaugh, R. Banuelos, M. Birman, E.B. Davies, A. Grigoryan, P.B.Gilkey, V. Guillemin, T. Hoffmann-Ostenhof, P. Kuchment, E. Lieb, M. Loss, V. Mazya, R. Melrose, N. Nadirashvili, P. Sarnak, B. Simon, J. Sjöstrand, U. Smilansky, M. Solomyak, S. Zelditch, M. Zworski.
Report: Download Report
Complex Osserman algebraic curvature tensors and Clifford families
Authors: Peter B Gilkey
The periodic magnetic Schrödinger operators: spectral gaps and tunneling effect
Authors: Bernard Helffer, Yuri Kordyukov
Strong diamagnetism for general domains and applications
Authors: Soeren Fournais, Bernard Helffer
Four lectures in semiclassical analysis for non self-adjoint problems with applications to hydrodynamic instability
Authors: Bernard Helffer
24 July 2006 to 28 July 2006
The Organisers would like to thank the following sponsors for their generous support of the event:
|Thursday 20th July 2006|
|14:00 to 15:00||
Peter B Gilkey University of Oregon
|15:30 to 16:30||
Leonid Polterovich Tel Aviv University
|Wednesday 2nd August 2006|
|14:00 to 15:00||Room 2|
|Thursday 3rd August 2006|
|14:00 to 15:00||
Dirk Hundertmark University of Illinois at Urbana-Champaign
|15:30 to 16:30||
Uzy Smilansky University of California, Irvine and Weizmann Institute of Science
|Thursday 10th August 2006|
|14:00 to 15:00||
Ira Herbst University of Virginia
|15:30 to 16:30||
Yuri Kordyukov Pereslavl-Zalessky of the Russian Academy of Sciences
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