Mathematical Sciences: Some Problems in Geometric Topology

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The principal investigator will work on several problems in manifold theory. He plans to continue his work on topological rigidity with S. Weinberger (University of Chicago) and his work with his colleague Erik Peterson on controlled surgery. He is also interested in characterizing open subsets of Euclidean space topologically. In a thesis under his direction, Ms. T. Engel will study families of deformations in a situation where topological rigidity does not hold. These problems come down to probing the fundamental structure of the space in which we live and that of other abstractions of the physical world. How do these abstractions differ, and what is common to them all? To what extent do their basic properties follow from a few underlying assumptions without the possibility of any variation?