Separating sets in tree products, and applications

这位首席研究员在平面拓扑学方面的最新进展，在很大程度上依赖于对两棵树（笛卡尔）积中的分离集的观察。也就是说，如果将一棵树作为“x轴”，另一棵树作为“y轴”，那么就可以研究哪种（x,y）对的集合将把所有对的集合分成两个或更多部分。**现在发现，对这种分离集的更复杂和全面的理解将导致在连续统理论和平面拓扑中其他重要开放问题的进一步进展。我们关注两个古老而众所周知的问题：确定平面上是否已经发现了所有齐次（拓扑均匀或“对称”）空间（目前只有三个紧连通的例子：圆和两个非常奇特的空间），以及确定非分离平面连续体到自身的任何映射是否有一个不动点；也就是说，在一个没有孔的连通平面内的连续“运动”是否必须使某一点静止。解决这些引人注目的开放性问题将从本质上阐明平面拓扑的性质，这是许多数学领域的基础，包括（复杂）分析、动力学、几何函数理论和（黎曼）几何。**我们展示了一些可接近的研究方向和问题的分离产品，并描述了它们与上述问题的直接联系，以及其他。借助大型和精确的计算机生成图，我们将仔细研究现有文献中分离集的例子，并生成新的例子和结果，这将有助于隔离和阐明平面拓扑中重要开放问题核心的困难组合学

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.**