Mathematical Sciences: Research Project in Geometry

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This research is in the broad area of complex algebraic geometry and involves three investigators. The unifying theme of the research is the geometry of subvarieties of moduli spaces. Carlson will study subvarieties of the Griffiths period domains i.e. variations of Hodge structures. Clemens will consider subvarieties of a fixed variety and their Abel-Jacobi theory. Toledo will examine the metric and function-theoretic properties of subvarieties of symmetric spaces as well as of more general domains. These subjects are closely related and much work of a joint nature will result. The subject of this proposal is algebraic geometry, the study of the geometric objects obtained as the set of solutions of systems of polynomial equations. The emphasis here is the study of their geometric (as opposed to their algebraic) properties. It still represents a nice blend of algebraic, analytic and, of course, geometric ideas. The three individuals work together as a powerful team to produce much of great interest.