Analytic invariants of multiple points

. We develop an original approach in computing analytic invariants of zero-dimensional singularities, which is based essentially on the study of properties of differential forms and the cotangent complex of multiple points. Among other things, we consider a series of specific tasks and problems for zero-dimensional complete intersections, graded and gradient singularities, including the computation of cotangent homology and cohomology for certain types of such singularities. We also examine the unimodular families of gradient zero-dimensional singularities, compile an adjacency diagram and compute the primitive ideals of these families. Finally, we briefly discuss the problem of nonexistence of negative weighted derivations, some relationships between the Milnor and Tjurina numbers and estimates of these invariants in the case of zero-dimensional complete intersections.