Gaffney, Le and Massey will continue their study of the singularities of complex analytic spaces and mappings. Gaffney will investigate the equisingularity of families of mappings using analytic invariants, extending his results to C- zero finitely determined germs, and he will continue to develop an infinitesimal theory of Whitney equisingularity using integral closure operations. With Massey, he will investigate the correct generalizations of the Le cycles to the complete intersection case. Using blowing up operations, he will study the topology of maps between low dimensional spaces. Le will investigate the relation between algebraic depth and rectified homotopical depth, continuing his study of Lefschetz type theorems for singular spaces. Le also plans to give a new proof of the Le-Ramanujam theorem using vector fields, and he plans to study the canonical decomposition of the local monodromy of complex hypersurfaces and its links with the resolution of singularities and the geometry of polars. Massey will investigate questions concerning a topologically invariant decomposition of a constructible function, the Euler characteristic of the Milnor fibre of f-f(p) at p, and the related question of how the sheaf of vanishing cycles of a map decomposes as a perverse sheaf. The machinery of the derived category should allow for a powerful attack on the structure of non-isolated hypersurface singularities. Understanding singular spaces is important for realistic models of physical phenomena. Perhaps the most widely known singularity is the cosmologists' "big bang."
