Intersection of almost complex manifolds and pseudoholomorphic maps

Weiyi Zhang (Warwick)

Monday 19th February, 2018 16:00-17:00 Maths 311B


An almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex or symplectic manifold is an almost complex manifold, but not vice versa.

Transversality is the notion of general position in manifold topology. If two submanifolds intersect transversely in some ambient manifold, then their intersection is a manifold. 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, 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.

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