Separating sets in tree products, and applications

最近首席研究员在平面拓扑学方面取得的进展，在很大程度上依赖于对两棵树的(笛卡尔)乘积中分离集的观察。也就是说，如果一个人把一棵树作为“x轴”，另一棵树作为“y轴”，人们可以研究什么样的(x，y)对集合将把所有对的集合分成两个或更多个部分。现在人们发现，对这种分离集的更复杂和更全面的理解将导致在连续统理论和平面拓扑学中其他重要的开放问题上的进一步进展。我们集中在两个古老而众所周知的问题上：确定平面上所有的齐次(拓扑均匀或“对称”)空间是否都已被发现(目前只有三个紧致连通例子：圆和两个非常奇异的空间)，以及确定非分离平面连续体到其自身的任何映射是否有不动点；即，在没有洞的连通平面集合内的连续“运动”是否必须保持某个点不动。解决这些突出的悬而未决的问题将大大阐明平面拓扑的性质，它是许多数学领域的基础，包括(复)分析、动力学、几何函数论和(黎曼)几何。我们展示了一些关于树木产品中的分离物的可接近的研究方向和问题，并描述了它们与上述问题和其他问题的直接联系。借助于大量准确的计算机生成的图，我们将仔细研究现有文献中分离集的例子，并产生新的例子和结果，这将有助于分离和阐明平面拓扑学中重要公开问题的核心的困难组合学。

Recent advances by the principal investigator in topology of the plane have depended crucially on observations pertaining to a separating set in the (Cartesian) product of two trees. That is, if one lays down a tree as the "x-axis", and another as the "y-axis", one can study what sorts of sets of (x,y) pairs will separate the set of all pairs into two or more pieces. It is now being found that a more sophisticated and comprehensive understanding of such separating sets will lead to further advances on other significant open questions in continuum theory and plane topology. We focus on two old and well-known problems: determining whether all the homogeneous (topologically uniform, or "symmetrical") spaces in the plane have already been discovered (at present there are only three compact connected examples: the circle, and two very exotic spaces), and determining whether any map of a non-separating plane continuum to itself has a fixed point; that is, whether a continuous "motion" within a connected plane set with no holes must leave some point stationary. Settling these striking open questions would substantially illuminate properties of the topology of the plane, which underlies a number of mathematical fields, including (complex) analysis, dynamics, geometric function theory, and (Riemannian) geometry. We exhibit a number of approachable directions of study and questions on separators in tree products, and describe their direct connections to the above problems, and others. With the aid of large and accurate computer-generated graphs, we will carefully study examples of separating sets in existing literature, and generate new examples and results, which will help isolate and illuminate the difficult combinatorics at the heart of significant open problems in plane topology.