Mathematical Sciences: The Structure of Algebraic Varieties

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9622394 The principal investigator aims to study log varieties. These can be thought of as a pair consisting of a variety and a divisor on it. These objects arise in the study of several questions, for instance minimal model theory, moduli problems and the birational classification of rationally connected varieties. To a log variety one can associate various numerical invariants (for instance discrepancies and selfintersection numbers), and these invariants exhibit very interesting behaviour. For ordinary varieties these invariants are integers, but for log varieties they are rational numbers. In many cases not all rational numbers arise this way, and it is crucial to understand the set of rational numbers that arise. Even in dimension two this is a hard question,and the answer has immediate and deep applications to the moduli of surfaces (Alexeev, Koll\'ar,Mori), birational trasformations of Fano fiber spaces (Corti) and to diophantine geometry (Hassett). The principal investigator also aims to further explore recent applications of these and related ideas to the theoretical aspects of high-performance computing (joint work with Babai et al). The principal investigator aims to conduct research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, like lines, circles, ellipses. Nowadays the field makes use of methods not only from algebra and geometry, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.