Mathematical Sciences: Studies in Commutative Algebra and Algebraic Geometry

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This award supports the research in commutative algebra and algebraic geometry of Professors Melvin Hochster, Igor Dolgachev, and Kevin Coombes of the University of Michigan at Ann Arbor. Dr. Hochster has introduced the fruitful notion of tight closure of rings, and will continue his study of this concept, as well as several problems in local algebra. Dr. Dolgachev will be working on the classification of Enriques surfaces with finitely many automorphisms over fields of arbitrary characteristic, and will investigate a number of other questions in the theory of surfaces. Dr. Coombes will be studying more number-theoretic questions, namely the geometry of arithmetic surfaces, especially the theory of branched covers of arithmetic surfaces and of curves defined over fields of arithmetic interest. This is research in the field of Algebraic Geometry, one of the oldest parts of modern 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, namely polynomials. Nowadays the field makes use of methods not only from Algebra, but from Analysis and Topology, and conversely is finding application in those fields as well as in theoretical Computer Science and in Robotics.