Mathematical Sciences: Topics in Commutative Algebra

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This award supports research on two topics in commutative algebra. The first is concerned with excellent rings, formal fibres and the relationship between certain prime ideals of the generic formal fibre of an excellent local domain R and some birational extensions of R. The second is related to a conjecture by Lech concerning the behavior of the Hilbert function under faithfully flat morphisms of local noetherian rings. 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 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.