The principal investigator will continue his studies of intersection cohomology while partially supported by this Presidential Young Investigator award. In particular, he will work towards a proof of the Borel conjecture which would extend his solution with M. Stern of the Zucker conjecture and an earlier solution to the MacPherson conjecture. In two other projects, he will study complete Kahler metrics on algebraic varieties with non-isolated singularities and intersection cohomologies which are direct summands. Cohomology is the dual of homology. In cohomology theory spaces of functions on topological spaces are studied, while in homology theory the spaces themselves are graded by their algebraic connectivity properties. The principal investigator will work toward solving a deep conjecture concerning square- integrable cohomology as related to intersection cohomology.