Mathematical Sciences: Upstate New York Topology Seminar

Douglas R. Anderson将研究有界控制拓扑及其应用领域的几个问题。其中一些问题旨在通过将有界控制拓扑方法应用于其他领域的问题来扩展有界控制拓扑方法的使用，而另一些问题则完全属于有界控制拓扑本身的领域。试图扩大这些方法的范围是为了证明它们的相关性和澄清它们的局限性，并可能暴露出需要处理的主题内的其他问题。该地区本身的问题是安德森和丹麦欧登塞的H. J. Munkholm已经做过的研究的延伸。目标是更清楚地理解这部分拓扑与其他研究者（如t.a. Chapman、s.c. Ferry和f.s. Quinn）引入的受控拓扑版本之间的确切关系，并在必要时扩大有界受控拓扑的基础结果主体。拓扑学研究的是不被连续变换破坏的几何对象的性质，如连通性、结性等。拓扑学性质和我们更熟悉的长度和面积的几何性质一样基本而普遍，但它们需要完全不同的技术来处理。到目前为止，高维拓扑已经得到了广泛的发展，可用的技术阵列也相当广泛。在一门学科的发展过程中，有必要不时地组织它的方法，使它们更容易为他人所用，更容易传递给希望使用它们而没有直接参与发明的后代。这样的时代已经出现在高维拓扑中，有界控制方法是一种自然的编码原则，可以这么说，它被证明对工具箱中的秩序是有用的。

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.