Title: Hypersurface skeleta

Abstract: Rene Thom's proof of the Lefschetz hyperplane theorem shows that a smooth affine variety X of complex dimension d can be made to deformation retract onto a cell complex of real dimension d. I will describe an explicit such "skeleton" when X is an affine hypersurface. The skeleton has another natural realization as a singular Legendrian subset of S^d x T^{d+1} with a natural contact structure. Because of this, the skeleton carries a sheaf of triangulated categories, the "Kashiwara-Schapira sheaf." I will explain these things and their relevance in the homological mirror symmetry program. This talk is based on joint work with Ruddat, Sibilla and Zaslow, as well as on earlier joint work with Fang, Liu and Zaslow.