Minimal Energy Problems on Manifolds

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ABSTRACTThe minimum energy (or "ground state") configurations for N particles interacting via a pairwise repulsive interaction V and confined to a fixed manifold A is of great mathematical and physical interest and has been the subject of much effort over the years from a variety of workers. In particular, such configurations are useful for purposes of sampling data, computer graphics, best-packing and understanding the physics of self-assembling materials. The grant will enable the researchers to investigate:(i) how relationships between geometrical, topological, and combinatorial properties of a manifold are reflected in the asymptotics of minimal energy problems; (ii) whether there are properties of minimal energy configurations that are universal in the sense thatthey are insensitive to the choice of underlying potential function; and (iii) how the theory can be applied in the development of efficientalgorithms for generating large numbers of points on a surface that are uniformly distributed or have some non-uniform prescribed distribution. We expect the results of this research to help elucidate the ordering of matter on curved surfaces. Furthermore, the infrastructure for research and education will be enhanced by the participation of graduate students and post-doctoral researchers who will broaden their education through interactions with condensed matter physicists as well as participate in collaborations and international exchanges with members of the Australian Centre of Excellence in Mathematics and Statistics of Complex Systems' research team at the University of New South Wales (Sydney, Australia).