Mathematical Sciences: Upstate New York Topology Seminar

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Douglas R. Anderson will work on several problems in the areas of boundedly controlled topology and its applications. Some of the problems are designed to expand the uses of the methods of boundedly controlled topology by applying them to problems in other areas, while others lie entirely within the domain of boundedly controlled topology itself. Attempts to expand the scope of the methods are intended to demonstrate their relevance and clarify their limitations and may expose additional questions within the subject which need to be addressed. The problems within the area itself represent extensions of research already done by Anderson and H. J. Munkholm of Odense, Denmark. The objectives are to understand more clearly the exact relationship between this part of topology and the versions of controlled topology introduced by such other researchers as T. A. Chapman, S. C. Ferry, and F. S. Quinn, and, if necessary, to enlarge the body of foundational results within boundedly controlled topology. Topology is the study of properties of geometric objects which are not destroyed by continuous transformations, such properties as connectedness, knottedness, and so forth. Topological properties are as basic and ubiquitous as the more familiar geometric properties of length and area, but they call for entirely different techniques to deal with them. By now the topology of high dimensions has been intensively developed, and the array of techniques available is quite extensive. From time to time in the development of a subject, it becomes necessary to organize its methods to make them more readily available to others and easier to pass on to future generations which will wish to use them without having been directly involved in their invention. Such a time has arisen in high dimensional topology, and the boundedly controlled approach is a natural codifying principle that is proving useful in making order in the toolbox, so to speak.