Minimal Energy Problems on Manifolds

摘要：N个粒子通过配对排斥相互作用V相互作用并局限于固定流形a的最小能量（或“基态”）配置具有很大的数学和物理意义，并且多年来一直是各种工作者努力的主题。特别是，这种结构对于采样数据、计算机图形、最佳包装和理解自组装材料的物理特性非常有用。该基金将使研究人员能够研究：(i)流形的几何、拓扑和组合性质之间的关系如何反映在最小能量问题的渐近性中；（ii）是否存在对潜在势函数的选择不敏感的最小能量构型的普遍属性；（iii）如何将该理论应用于有效算法的开发中，以在均匀分布或具有某些非均匀规定分布的表面上生成大量点。我们期望这项研究的结果有助于阐明曲面上物质的顺序。此外，研究生和博士后研究人员的参与将加强研究和教育的基础设施，他们将通过与凝聚态物理学家的互动扩大他们的教育，并参与与新南威尔士大学（澳大利亚悉尼）澳大利亚复杂系统数学和统计卓越中心研究团队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).