This research project is based on a program that originated in 1912 with the Brouwer fixed-point theorem. The central problem of the program is to determine whether every plane continuum that does not separate the plane has the fixed-point property . R. H. Bing called this the most interesting problem in plane topology. Aside from providing a beautiful generalization to Brouwer's theorem, a solution to this problem would represent the final step in a historical research effort that has involved many of our brightest mathematicians; P. Alexandroff, R. H. Bing, K. Borsuk, and K. Kuratowski are among the international giants who have worked extensively on this famous problem. Through the years a list of related problems has been developed, each of which has taken on a significance of its own. Recently, the PI has solved two of these problems, establishing the fixed-point property for every simply-connected plane continuum and proving that every deformation of a tree-like continuum has a fixed point. The techniques developed should lead to further advances in this program. Topology is the study of properties that persist when geometric objects are bent, folded, shrunk, stretched, turned, twisted, or in any other way continuously transformed. For example, when points are marked on a rubber band and it is stretched, the order in which the points appear does not change. Other topological facts are more subtle. Among these are the fixed-point theorems, results about points that remain in their original position after an object has been transformed. Consider a cup of coffee that has been stirred (but not whipped) so that the surface of the liquid remains on top. The Brouwer fixed-point theorem tells us that when the liquid stops moving, some point on the surface will be in the same place that it was before we started stirring. There are a wide variety of interesting applications of this theorem in the literature. It has been used to prove a well-known theorem on the existence of roots of complex polynomials, the Fundamental Theorem of Algebra, as well as the existence of equilibrium points in an economy. Fixed points of deformations also appear in a variety of scientific applications. For example, in electromagnetic-wave theory, they are used to show that there are no isotropic antennas and explain why most magnetic plasma containers are tori instead of spheres. This project will deal with generalizations of this theorem.
