Mathematical Sciences: Homological Questions in Commutative Algebra

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This award is concerned with the study of certain invariants, known as local Chern characters, defined for modules over local rings. These invariants have been defined for bounded complexes of free modules. The principal investigator will study the contributions of individual matrices defined by these complexes. He will also consider questions concerning multiplicities and other ideal-theoretic properties. The results of these studies will be applied to homological conjectures and to related problems. 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.