Du Bois singularities, rational singularities, and beyond

We survey some extensions of the classical notions of Du Bois and rational singularities, known as the k-Du Bois and k-rational singularities. By now, these notions are well-understood for local complete intersections (lci). We explain the difficulties beyond the lci case, and propose new definitions in general to make further progress in the theory. This is joint work with Matthew Satriano, Sridhar Venkatesh and Anh Duc Vo.