Based on the duality between open-string theory on noncompact Calabi-Yau
threefolds and Chern-Simons theory on three manifolds, M Marino and C Vafa
conjectured a formula of one-partition Hodge integrals in term of invariants of
the unknot (hep-th/0108064). Many Hodge integral identities, including the
lambda_g conjecture and the ELSV formula, can be obtained by taking limits of
the Marino-Vafa formula.
Motivated by the Marino-Vafa formula and formula of Gromov-Witten invariants
of local toric Calabi-Yau threefolds predicted by physicists, J Zhou
conjectured a formula of two-partition Hodge integrals in terms of invariants
of the Hopf link (math.AG/0310282) and used it to justify physicists'
predictions (math.AG/0310283).
In this expository article, we describe proofs and applications of these two
formulae of Hodge integrals based on joint works of K Liu, J Zhou and the
author (math.AG/0306257, math.AG/0306434, math.AG/0308015, math.AG/0310272).
This is an expansion of the author's talk of the same title at the BIRS
workshop: "The Interaction of Finite Type and Gromov-Witten Invariants",
November 15--20, 2003.