Mathematical Sciences: Hodge Theory and L2-Cohomology

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This research is in the area of locally symmetric spaces and the transcendental methods of algebraic geometry. More specifically it will concentrate on the Hodge theory for the intersection homology of varieties. These structures should be given simpler descriptions and they should be related to the Hodge structures provided by L2 harmonic forms with respect to good Kahler metrics. A related development along these themes is the appearance of two proofs of the principal investigator's 1980 conjecture equating the usual L2-cohomology of a locally symmetric variety with the intersection homology of its Baily- Borel-Satake compactification. This problem will receive more work for the nature of the result will be elucidated by generalizing it to its natural setting, which is probably to Satake compactifications with only equal-rank boundary components. The nature of this work is in the geometrical end of studying the sets of solutions of polynomial equations. Various invariants are studied which give deep understanding of these geometric objects. These in turn should give applications to many areas of mathematics, most especially in geometry and analysis.