Skeleta of affine hypersurfaces and mirror symmetry

In this talk I will report on a joint project with H. Ruddat, D. Treumann and E. Zaslow, which constructs a 'skeleton' for  affine hypersurfaces in toric ambient space, and proves that
there exists a retraction of the hypersurface onto it. The skeleton, which is a half-dimensional CW complex, will be defined using toric data. Defining the retraction will involve techniques from log geometry. If time permits I will discuss applications to homological mirror symmetry for
degenerate Calabi-Yau hypersurfaces, with toric components.