Quantum algebra and topology

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Proposal: DMS-0072342PI: Greg KuperbergThe purpose of the project is to study the properties and applications ofquantum groups, the Yang-Baxter equation, graph homology, and similarconstructions in algebra with geometric formalism. In particular the role ofthese constructions will be considered in two areas: In 3-dimensionaltopology, they arise in the guise of of perturbative and non-perturbativeinvariants of Chern-Simons field theory. There is a good chance that suchinvariants are related to the Thurston Geometrization Program, at least forhyperbolic manifolds. In quantum computation, the same non-perturbativeChern-Simons field theory produces an interesting method for fault tolerance,one which needs to be developed further.The purpose of the project is to develop and apply certain areas of algebra inwhich formulas have a geometric structure. Two famous examples of this are alinear algebra, in which matrices have some geometry, and Feynman diagrams,whose geometry is physically meaningful. Geometrization of algebraic formulashas been influential in enumerative combinatorics, in non-commutative algebra,and in geometric topology in the last 15 years. One of the first and mostimportant representatives of this recent trend is the Jones polynomial, aninvariant of knots. Although the Jones polynomial arose in algebra andmathematical, it and its generalization are at least as important in topology,and recently also in quantum computation. These applications of the new trendin algebra will be investigated.