Mathematical Sciences: Hodge Theory

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Eduardo Cattani and Aroldo Kaplan will continue their work in Hodge theory. This involves the study of topological and geometric invariants connected with an algebraic variety. Hodge theory yields a method of describing the space of differential forms on a manifold. This theory has seen very rapid developments in recent years. Most notably, on the construction of functorial Hodge structures on appropriate cohomology and homotopy spaces. The work of Cattani and Kaplan in cooperation with Wilfried Schmid has been at the forefront of research in variations of Hodge structures. Cattani and Kaplan will build on their recent results which bring the several variable case within reach of applications. They will apply these results to the study of the behaviour of integral cycles in an algebraic family of varieties. They will also study the structure of modular varieties. This work will focus on the characterization of maximal integral submanifolds for the horizontal distribution in a classifying space for Hodge structures.