Mathematical Sciences: Fixed-Point Problems

这个项目是在连续不动点理论，一个领域， 几何拓扑学 中心主题是以下经典 未解决的问题 是不是每一个没有把平面分开的平面连续体 具有定点性吗 主要研究者最近试图回答这个问题， 这个问题导致了几个不动点定理的建立 用于变形和弧分量保持贴图。 这些 结果提出了一个新的概念，涉及分解的一个 连续统为唯一弧连通集。 之一 首席研究员的目标是建立一个纯粹的 拓扑定理基于这一概念，将推广 庞加莱-本迪克森定理 令人惊讶的是，一个简单的问题， 这架飞机多年来一直无法解决。 技术 为它的解决方案开发的可以预期是基本的， 有启发性，在其他地方也可以使用。

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.