Abstract:

We will give a gentle introduction to some recent developments in the area of homological mirror symmetry. We will use simple examples to illustrate Kontsevich’s conjecture and its extension beyond the Calabi-Yau setting in which it was first formulated. We will mostly focus on a one-dimensional example, the pair of pants, to give a flavor of the geometric concepts involved in a general formulation of homological mirror symmetry. If time permits, we will then describe a program to prove homological mirror symmetry for essentially arbitrary complete intersections in toric varieties (joint work in progress with Mohammed Abouzaid).

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).