Higher Dimensional Algebraic Varieties

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The purpose of the proposal is to study the connections between algebraic geometry, topology and number theory. Specifically, at issue is the relationship between thetopology of a variety and the Kodaira dimension. Furthermore,the project will investigate the degeneration of varietiesand their application to solving diophantine equations overlocal fields.The aim of the research here is to furnish a theoreticalunderstanding about which shapes can be describedefficiently with equations. Such shapes include curves and surfaces but alsohigher dimensional shapes in higher dimensional spaces. Of specialinterest is a description of parametrized shapes, since this provides fast algorithms for computational purposes.