We will demonstrate one direction of homological mirror symmetry

(HMS) for punctured Riemann surfaces -- the wrapped Fukaya category of a punctured Riemann surface is equivalent to the matrix factorization category MF(X,W) of the toric Landau-Ginzburg mirror (X, W). We will begin by introducing examples of HMS, that for cylinders and pairs of pants, which are spheres with three punctures.

The category MF(X,W) can be constructed from a Cech cover of (X,W) by local affine pieces that are mirrors of pairs of pants. We supply a suitable model for the wrapped Fukaya category for a punctured Rimemann surface so that it can also be explicitly computed in a sheaf-theoretic way, from the wrapped Fukaya categories of various pairs of pants in a decomposition.

The pieces are glued together in the sense that the restrictions of the wrapped Floer complexes from two adjacent pairs of pants to their adjoining cylindrical piece agree. We will then briefly discuss how this sheaf-theoretic method can be applied more generally.