From smooth to almost complex

GAUGE-TOPOLOGY-SYMPLECTIC

Speaker:

Weiyi Zhang - Warwick

Abstract:

An almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. We will discuss differential topology of almost complex manifolds, explain how to use transversality statements for smooth manifolds to formulate and prove corresponding results for an arbitrary almost complex manifold. The examples include intersection of almost complex manifolds, structure of pseudoholomorphic maps and zero locus of certain harmonic forms.
One of the main technical tools is Taubes' notion of "positive cohomology assignment", which plays the role of local intersection number. I will begin with explaining its motivation through multiplicities of zeros of a smooth function.

Our results would lead to a notion of birational morphism between almost complex manifolds. Various birational invariants, including Kodaira dimension, for almost complex manifolds will be introduced and discussed (this part is joint with Haojie Chen).