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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2002-2003 MATHEMATICS COLLOQUIUM SERIES 09/13/02 Colloquium DR. JOHN BRYANT Florida State University Topological Homogeneity Abstract:   A topological space X is homogeneous if, for any two points x and y in X, there is a homeomorphism of X onto itself that takes x to y. Examples of homogeneous spaces include topological groups and connected topological manifolds (without boundary). Bing and Borsuk have conjectured that a certain class of "nice" homogeneous spaces are topological manifolds. We discuss this conjecture and relate it to another outstanding conjecture on homogeneity.