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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2003-2004 MATHEMATICS COLLOQUIUM SERIES 03/05/04 Colloquium JAMES GRIFFIN DEPARTMENT OF MATHEMATICS IMPERIAL COLLEGE LONDON Generalizations of Chebyshev Polynomials and Polynomial Mappings   Abstract: The generalized Chebyshev Polynomials orthogonal on a set of  disjoint intervals, , with respect to the measure, are one example of orthogonal polynomials on several intervals for which we have an explicit representation in terms of theta functions.  We derive a new representation and then specialize to the case where the set  can be mapped onto by a polynomial mapping. The results allow us to construct more general sequences of orthogonal polynomials in an attempt to gain an analogous generalization of the Askey-Wilson polynomials.