We use adiabatic limits to study foliated manifolds. The Bott connection naturally shows up as the adiabatic limit of Levi-Civita connections. As an application, we then construct certain natural elliptic operators associated to the foliation and present a direct geometric proof of a vanshing theorem of Connes[Co], which extends the Lichnerowicz vanishing theorem [L] to foliated manifolds with spin leaves, for what we call almost Riemannian foliations. Several new vanishing theorems are also proved by using our method.

We study the asymptotic of the Bergman kernel of the spinc Dirac operator on high tensor powers of a line bundle.

We derive some Hodge integral identities by taking various limits of the Marino-Vafa formula using the cut-and-join equation. These identities include the formula of general λg-integrals, the formula of λg−1-integrals on -Mg,1, the formula of cubic λ integrals on -Mg, and the ELSV formula relating Hurwitz numbers and Hodge integrals. In particular, our proof of the MV formula by the cut-and-join equation leads to a new and simple proof of the g conjecture. We also present a proof of the ELSV formula completely parallel to our proof of the Marino-Vafa formula.

We outline a proof of a remarkable formula for Hodge integrals conjectured by Mar

We have developed a mathematical theory of the topological vertex-a theory that was originally proposed by M. Aganagic, A. Klemm, M. Marino, and C. Vafa 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.