Mathematical Sciences: Topics in Commutative Algebra and Algebraic Geometry

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This award supports research in commutative algebra and algebraic geometry. Roger Wiegand and Sylvia Wiegand will study direct-sum decompostions of Cohen-Macaulay modules, particularly over Noetherian rings of dimension two. They will also study the prime ideal structure of two-dimensional rings. Brian Harbourne will investigate points in good position in the projective plane, with a goal of understanding the behavior of the dimension of the space of global sections of complete linear systems on blowings up of the plane at n points. Thomas Marley will study the depth properties of Rees algebras and associative graded rings of m-primary ideals of a local ring (R,m). 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.