关于几乎共线函数的Atiyah-Sutcliffe猜想和关于对称函数的一些新猜想
Atiyah-Sutcliffe Conjectures for Almost Collinear Configurations and Some New Conjectures for Symmetric Functions
摘要：
2001年，M。F. Atiyah提出了一个猜想（C1），后来与P. Sutcliffe提出了两个更强的猜想（C2）和（C3）。受物理学（量子力学的自旋统计定理）的启发，这些几何图形被几何定义为欧几里得三维空间中的任何点的配置。对n= 3，4证明了猜想（C1），对一般n仅对某些特殊构形（M. F. Atiyah，M. Eastwood和P. Norbury，D.德约科维奇）。本文将解释几种新的对称函数的结构，它们对几乎共线的组态包含（C_2）和（C_3）。在Maple和J. Stembridge的对称函数软件包SF的帮助下，可以进行n=6的计算。当n=4时，我们还对某些四面体的无限族证明了（C2）和（C3）。这是一个联合工作与我。乌尔比哈最后，我们提到，通过最小化几何定义的能量，在这些图表中计算，人们可以连接到一些复杂的物理理论，如Skyrmions和Fullerenes。
英文摘要：
In 2001 Sir M. F. Atiyah formulated a conjecture (C1) and later with P. Sutcliffe two stronger conjectures (C2) and (C3). These conjectures, inspired by physics (spin-statistics theorem of quantum mechanics), are geometrically defined for any configuration of points in the Euclidean three space. The conjecture (C1) is proved for n=3,4 and for general n only for some special configurations (M. F. Atiyah, M. Eastwood and P. Norbury, D. Djokovic). In this paper we shall explain some new conjectures for symmetric functions which imply (C2) and (C3) for almost collinear configurations. Computations up to n=6 are performed with a help of Maple and J. Stembridge's package SF for symmetric functions. For n=4 the conjectures (C2) and (C3) we have also verified for some infinite families of tetrahedra. This is a joint work with I. Urbiha. Finally we mention that by minimizing a geometrically defined energy, figuring in these conjectures, one gets a connection to some complicated physical theories, such as Skyrmions and Fullerenes.
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