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Analysis on Graphs and its Applications

8th January 2007 to 29th June 2007

Organisers: Professor BM Brown (Cardiff), Professor P Exner (Czech Academy of Sciences), Professor P Kuchment (Texas) and Professor T Sunada (Meiji)

Scientific Advisory Committee:: Professor Y Colin de Verdière (Institut Fourier), Professor WD Evans (Cardiff), Professor JP Keating (Bristol), Professor B Pavlov (Auckland) and Professor A Teplyaev (Connecticut)

Programme Theme

Analysis on graphs and other discrete structures has been developing for quite some time, in particular due to applications to number theory, algebra, probability theory, spectral geometry, as well as to its usefulness in many practical problems. This area, however, has experienced recently a significant boost in terms of new important applications arising, new methods being developed, and new models introduced not studied before. This has happened due to numerous new applications arising in different areas of mathematics, sciences, and engineering. They swipe throughout a wide scientific landscape, which besides the fields already mentioned includes nanotechnology, microelectronics, quantum chemistry, superconductivity, optics, etc. New objects, so called quantum graphs have also emerged. These are graphs considered as one-dimensional CW-complexes and equipped with differential or pseudo-differential, rather than customary difference operators. Such graphs, besides being in many cases useful low-dimensional models of complex systems, are also used as toy models for studying difficult issues such as Anderson localisation, scattering, and quantum chaos. This has lead to interest in this research by scientists coming from different fields (physics, partial differential equations, algebra, combinatorics, number theory). The methods already used or expected to be useful in analysis on graphs come from a very wide range of topics: algebra, combinatorics, PDEs, spectral theory, micro-local and complex analysis, to name a few.

The program will assemble, essentially for the first time, a diverse group of prominent mathematicians and physicists, as well as young researchers working in or interested in entering this fast developing and fascinating area. The hope is to facilitate the process of communication and cooperation, thereby promoting accelerated progress in this burgeoning new area. Among the particular topics that will be addressed during the program are: Spectral analysis on combinatorial graphs and its applications to number theory, discrete groups, random walks, and other areas; Quantum graph models for wave propagation in thin structures and for quantum and optical switching and computing; Spectral analysis and quantum chaos on quantum graphs; Analysis on fractals.

Final Scientific Report: 

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons