Minimal Energy Problems on Manifolds

N个粒子通过成对排斥相互作用V相互作用并被限制在固定的流形A中的最小能量(或“基态”)组态具有极大的数学和物理兴趣，多年来一直是各种工作者的研究课题。特别是，这种配置对于采样数据、计算机图形学、最佳包装和理解自组装材料的物理特性是有用的。这笔拨款将使研究人员能够研究：(I)流形的几何、拓扑和组合属性之间的关系如何反映在最小能量问题的渐近性中；(Ii)是否存在最小能量配置的普适性质，即它们对潜在势函数的选择不敏感；以及(Iii)如何将该理论应用于开发高效的算法，以在曲面上生成大量均匀分布或具有某些非均匀规定分布的点。我们期望这项研究的结果有助于阐明物质在曲面上的排列顺序。此外，研究和教育的基础设施将通过研究生和博士后研究人员的参与得到加强，他们将通过与凝聚态物理学家的互动来扩大他们的教育，并参与与澳大利亚新南威尔士大学(澳大利亚悉尼)复杂系统数学和统计英才中心研究小组成员的合作和国际交流。

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).