  We have developed a mathematical theory of the topological vertex--a theory
that was original proposed by M. Aganagic, A. Klemm, M. Marino, and C. Vafa in
hep-th/0305132 on effectively computing Gromov-Witten invariants of smooth
toric Calabi-Yau threefolds derived from duality between open string theory of
smooth Calabi-Yau threefolds and Chern-Simons theory on three manifolds.
