Mathematical Sciences: Rees Algebras, Associated Graded Rings, and Multiple Points of Finite Morphisms

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Ulrich 9623259 This proposal supports the work of Professor B. Ulrich to work on problems in commutative algebra and algebraic geometry. he intends to study the Rees algebra and its associated graded ring. In particular the proposed research investigates how graded rings associated with an ideal intertwine with the Cohen-Macaulay property. He also intends to study problems that circle around the classical problems of determining the classical join and secant varieties for certain factor rings. Finally, he intends to study the scheme associated to the source and target of finite morphisms between perfect R modules and certain Cohen-Macaulay rings. This research is on the boundary between commutative algebra and algebraic geometry. Both fields arose from the classical problem of studying mathematical objects defined in the plane by the simplest of equations, namely polynomials. Today, the fields use methods not only from algebra, but also from analysis and topology, and conversely are extensively used in those fields. Moreover it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.