Mathematical Sciences: Fixed-Point Problems

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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.