Chern-Simons Theory, and Its Limiting Geometry and Topology

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DMS-0203129Stavros GaroufalidisStavros Garoufalidis plans to study applications of quantum field theoryto 3-dimensional topology and geometry. This work is motivated by the Chern-Simons path integral, its exact computation (in terms of quantum invariants of knots and 3-manifolds) and its perturbative expansion.Quantum field theory predicts that invariants of knots and 3-manifolds whichare defined via cut-and-paste topology carry geometric information of theunderlying structures; in particular of the fundamental group of a 3-manifoldand its representations to Lie groups. A key question is to recover thegeometric information (described in terms of surgery theory or Riemannianhyperbolic geometry) by taking an appropriate limit of quantum invariants.The above mentioned relation between physics and mathematics has deep rootsin the past century. Mathematics has strongly influenced Physics, as is evident from the development of Mechanics, Quantum Mechanics and GeneralRelativity, which describe our world in three quite different scales.Knotedness is a fundamental property that binds fundamental particlestogether, but also the DNA molecule.The most recent interaction between 3-dimensional quantum field theory and topology aims at describing knotedness in terms of geometric and topological language. Topology could be a unifying language to describethe building blocks of physics, but also of life itself.