One possible strategy for understanding oriented smooth 4-manifolds is to break them up into more tractable classes of manifolds in a controlled manner. Situated in the intersection of complex, symplectic and Riemannian geometries, K\"ahler manifolds are the best known candidates to be pieces of such a decomposition. We have shown that this can be achieved for any closed oriented smooth 4-manifold X. To be precise, we can decompose X into two compact K\"ahler manifolds with strictly pseudoconvex boundaries, up to orientation, such that contact structures on the common boundary induced by the maximal complex distributions are isotopic. The decomposition gives rise to a folded K\"ahler structure on X, a globally defined 2-form, which is a particular generalization of a symplectic form. Moreover, folded Lefschetz fibrations, a certain analogue of Lefschetz fibrations, are shown to be the geometric counterpart of these structures. In this talk we would like to outline these existence results in the direction of generalizing the study of symplectic structures and Lefschetz fibrations on smooth 4-manifolds. Detailed proofs and examples can be found in the preprint at arXiv: “K\"ahler decomposition of 4-manifolds”, math.GT/0601396.

It is known that every smooth oriented closed 4-manifold with b+> 0 admits a near-symplectic structure, i.e a closed 2-form which vanishes in a particular way along a link and is non-degenerate on the complement. Motivated by Taubes’ programme of constructing smooth invariants via pseudo-holomorphiccurve counting in near-symplectic 4-manifolds, this subject recently became a big deal of interest among 4-manifold topologists. D. Gay and R. Kirby gave an explicit construction of these manifolds by using symplectic and contact surgery techniques, and D. Auroux, S. Donaldson and L. Katzarkov showed that these forms are supported by singular Lefschetz fibrations. This talk is a survey of these constructions, and certain follow-up ideas.

Generating functions of Gromov-Witten invariants of compact
symplectic manifolds behave very much like tau-functions of Integrable
systems. It was conjectured by Eguchi-Hori-Xiong and S. Katz that
Gromov-Witten invariants of smooth projective varieties should
satisfy the Virasoro constraints, which also exist for many integrable
systems (e.g, Gelfand-Dickey hierarchies). It was conjectured by Witten
that the generating functions on moduli spaces of spin curves are
tau-functions of Gelfand-Dickey hierarchy. In a joint work with Kimura, we
showed that it is possible to use Virasoro constraints of a point and
the sphere to derive universal equtions for Gromov-Witten invariants of
all compact symplectic manifolds. Such equations can also be used to
compute certain intersection numbers on moduli spaces of spin curves which
coincide with predictions of Witten's conjecture.

Phase transitions in classical spin systems are well understood,
however phase transitions in quantum spin systems are not ... at least
that is what most mathematicians would say. (If anything, they might
question how well we even understand classical spin systems.) Physicists,
on the other hand, say that if the classical model has a phase transition,
then the quantum model does as well. We will prove that, for some special
models. The main tools are reflection positivity, coherent states, and a
new result which generalizes the Berezin-Lieb inequality to the level of
matrix elements.

We shall show a general mean value theorem on Riemannian manifold and how it leads to new monotonicity formulae for evloving metrics. As an application we show a local regularity theorem for Ricci flow.