Mathematical Sciences: Applications of Stratified Morse Theory and Intersection Homology

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1. Applications of Intersection Homology: A study will be made of the torsion-linking pairing which exists on the intersection homology peripheral group of a complex analytic variety, and, in particular, its relationship with the multiplicity of the characteristic variety will be studied. Piecewise linear differential forms with singularities will be constructed for which the L2 forms compute the intersection homology, on any simplicial complex which can be (P.L) stratified with even-codimension strata. Hecke correspondences which induce homomorphisms on intersection homology will be studied, and their Lefschetz numbers will be calculated using geometric techniques. 2. Applications of Stratified Morse Theory: N. Spaltenstein's conjecture relating the intersection homology of nilpotent varieties and the Poincare polynomial of the complement of certain arrangements of hyperplanes will be investigated using the geometric techniques of stratified Morse theory.