This proposal has two closely-related central themes. The first is the use of the Hodge theory for algebraic varieties to refine the study of questions of a topological nature. The Principal Investigator will attempt to prove the Hodge-theoretic version of the so-called Zucker Conjecture on the square-integrable cohomology of Shimura varieties (1980; settled affirmatively 1987), and to apply mixed Hodge theory to the analysis of the cohomology of their boundaries. The second theme, almost the reverse of the first, is to expand the scope of properties that are, effectively via Hodge theory, known for algebraic varieties. The best example of this is the attempt to obtain the powerful Decomposition Theorem for a class of spaces and mappings wider than morphisms of algebraic varieties.
When mathematicians look for examples of geometric spaces, they often turn to those that are the set of solutions of a polynomial equation in any number of variables, or more generally, the simultaneous solutions of a number of such equations. These spaces are the building blocks of the branch of mathematics known as algebraic geometry, whose spaces are called algebraic varieties. Although varieties are described in a fundamentally direct manner, it is often hard to see how this information gets converted to the significant geometric properties of the space. When W.V.D. Hodge came up with the theory that bears his name, it was primarily intended for applications in algebraic geometry. The research proposed by Professor Zucker will be on spaces having continuous symmetries, and on the algebraic varieties that are made by conceptually simple contructions on those spaces.
