Mathematical Sciences: Fixed-Point Problems

这个项目是在连续不动点理论，一个区域的几何拓扑。中心主题是以下经典的未解决问题。是不是每一个不分离平面的平面连续体都有不动点的性质？最近，首席研究员试图回答这个问题，并建立了形变和弧分量保持映射的几个不动点定理。这些结果提出了将连续统分解为唯一弧连通集的新概念。主要研究者的目标之一是建立一个基于这个概念的纯拓扑定理，它将推广庞加莱-本迪克森定理。令人惊讶的是，一个很容易提出的关于飞机的问题多年来一直没有答案。为解决这一问题而开发的技术有望成为基础的、具有启发性的、适用于其他地方的技术。

This project is in continuum fixed-point theory, an area of geometric topology. The central topic is the following classical unsolved problem. Does every plane continuum that does not separate the plane have the fixed-point property? Recent attempts by the principal investigator to answer this question led to the establishment of several fixed-point theorems for deformations and arc-component-preserving maps. These results suggest a new concept involving the decomposition of a continuum into uniquely arcwise connected sets. One of the principal investigator's goals is to establish a purely topological theorem based on this concept that will generalize the Poincare-Bendixson theorem. It is surprising that an easily formulated question about the plane has defied solution for many years. Techniques developed for its solution can be expected to be basic, illuminating, and employable elsewhere.