Mathematical Sciences: Embeddings of Continua

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9400945 Lewis Lewis will investigate a number of problems related to embeddings of continua in manifolds and other spaces with a uniform structure. Techniques involving inverse limits, accessibility, and prime end structure will be studied. He is seeking characterizations of when various inverse limits can be embedded in the plane and when homeomorphisms of continua can be extended to homeomorphisms of the plane. He is also investigating continua with the property that all of their nondegenerate proper subcontinua are equivalently embedded, or which admit rigid embeddings in the plane, and various homogeneous embeddings in real 3-space. Determining the existence of an indecomposable continuum which is almost homogeneously embedded in the plane will be important in the study of dynamics. Questions relating embeddings and extensions of maps to computation of Hausdorff dimension and to attractors and cofrontiers are among dynamic topics related to this project. Lewis hopes to narrow the currently large gap between known theorems and existing examples in the theory of continua by broadening the base of the theory and looking for generic classes of examples and construction techniques. Recent work in discrete dynamics indicates that considerations of embeddings and extensions of maps are crucial in identifying which sets can be attractors or closures of orbits, how sensitive dynamic properties are to perturbation, and what type dynamics is possible or even generic. These investigations should yield more information about the continua themselves and new ways of approaching problems which until now have proven slow to yield to attacks. Possibilities for new approaches lie in the areas of decompositions, homogeneity, characterizations of spaces, and investigations of subcontinua. ***