Mathematical Sciences: "Uniform Bounds in Noetherian Rings, The Theory of Tight Closure, and Big Cohen-Macaulay Algebras"

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This award supports work on several projects in commutative algebra especially concentrating on two topics: the theory of tight closure and big Cohen-Macaulay algebras, and the existence of uniform bounds in noetherian rings. The first topic involves a continuing joint effort with Melvin Hochster. The principal investigator and Hochster have been developing the theory of tight closure over the last six years and plan on continuing this work. The latter topic is centered around issues arising in the proof of the uniform Artin-Rees property. The principal investigator would also like to continue work on several other topics, notably local cohomology, Rees algebras, computer algebra, and general hyperplane sections of algebraic varieties. This is reasearch 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 origins, 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 robotics.