  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.
