Quantum theories and geometric topology: A chapter in physical mathematics

In recent years, the interaction between geometric topology and classical and quantum field theories has attracted a great deal of attention from both the mathematicians and physicists. We discuss some topics from low-dimensional topology where this has led to new viewpoints as well as new results. They include categorification of knot polynomials and a special case of the gauge theory to string theory correspondence in the Euclidean version of the theories, where exact results are available. We show how the Witten-Reshetikhin-Turaev invariant in SU(n) Chern-Simons theory on S-3 is related via conifold transition to the all-genus generating function of the topological string amplitudes on a Calabi-Yau manifold. This result can be thought of as an interpretation of TQFT as topological quantum gravity (TQG). After a brief discussion of Perelman's work on the geometrization conjecture and its relation to gravity, we comment on some recent work on black hole radiation and its relation to mock moonshine.