This award provides funding to help defray the expenses of participants, especially women, graduate students, postdocs, and junior faculty, in the "The Weyl Law at 100" conference that will be held from September 19--22, 2012, at the Fields Institute for Research in Mathematical Sciences in Toronto, Canada. In 1912 Hermann Weyl published his paper establishing the Weyl law which gives the leading asymptotic description for the counting function of eigenvalues of the Dirichlet or Neumann Laplacian on a bounded domain in the Euclidean space. He later conjectured the form of the second term in asymptotics of the counting functions. After contributions by many mathematicians, among them, Courant, Hilbert, Agranovitch, Levitan, Hormander, Seeley, Duistermaat, Guillemin, Melrose and Sjostrand, the Weyl conjecture was solved by Ivrii in 1982. The workshop is intended as a forward-looking celebration of the 100th anniversary of Weyl's paper. Among the directions to be explored are the (conjectured) connection between random matrix theory and high energy distribution of differences between eigenvalues, probabilistic Weyl laws for non-self-adjoint operators, distribution of scattering resonances and fractal Weyl laws, and physical experiments related to quantum chaos and inverse problems.

All of the conference topics are central to analysis and are extremely active areas of research. The conference will bring together a broad spectrum of accomplished researchers thereby providing ample opportunities to develop collaborative interactions, and the format of the meeting is such that young people will have ample opportunities to speak and be otherwise engaged in the various conference activities.

Project Report

A string or drum oscillates at frequencies determined by its length or shape, respectively. More generally to geometric or physical objects we can associate frequencies/modes describing oscillating/excited states of the system. Counting these modes has a long tradition going back to Lord Rayleigh and the study of black body ratiation. The asymptotic formula for the number of modes was conjectured by Hendrik Lorentz and it was proved by Hermann Weyl in 1912. Weyl also conjectured a finer asymptotic formula which, after many advances in mathematical analysis was proved by Victor Ivrii in 1982. The three day workshop was a forward looking celebration of the centenary of the Weyl. In addition to Victor Ivrii, Richard Melrose whose work played crucial role in establishing the Weyl conjecture was there, as were many other researchers, ranging from senior establishment figures to undergraduates doing research in the area. We also had a participation of physicists and numerical analysts. The topics covered included quantum chaos, metropolis algorithm, scanning tunneling microscope experiments (see figure showing an agreement of the experiment with the Weyl law), microwave experiments, modified (fractal) Weyl laws when energy can escap to infinity. Stephane Nonnenmacher of Commissariat a l'Energie Atomique (French Nuclear Research Agency) gave an entertaining and well attended public lecture. A modest reception sponsored by the Department of Mathematics of the University of Toronto added to the celebratory spirit of the event.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Edward Taylor
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University of California Berkeley
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