How to classify reflexive Gorenstein cones

Two of my collaborations with Max Kreuzer involved classification problems related to string vacua. In 1992 we found all 10,839 classes of polynomials that lead to Landau–Ginzburg models with c=9 (Klemm and Schimmrigk also did this); 7,555 of them are related to Calabi–Yau hypersurfaces. Later we found all 473,800,776 reflexive polytopes in four dimensions; these give rise to Calabi–Yau hypersurfaces in toric varieties. The missing piece — toric constructions that need not be hypersurfaces — are the reflexive Gorenstein cones introduced by Batyrev and Borisov. I explain what they are, how they define the data for Witten's gauged linear sigma model, and how one can modify our classification ideas to apply to them. I also present results on the first and possibly most interesting step, the classification of certain basic weights systems, and discuss limitations to a complete classification.