Eisenbud 9726460 This project is concerned with research in algebraic geometry, commutative algebra, symbolic computation, and computational statistics. In algebraic geometry, the main thrust is towards discovering connections between the geometry of sets defined by the vanishing of polynomials and the algebraic invariants derived by solving the linear equation over the polynomial ring whose coefficients are the defining equations. In commutative algebra, the principal investigator will work on the structure of syzygies in general, on the dimensions of the loci described by various sorts of equations, and on a problem coming from the part of number theory responsible for the recent proof of "Fermat's Last Theorem". In symbolic computation, he will work with the developers of Macaulay2 incorporating into scripts methods which will enable this software to address problems of importance to algebraic geometers. Finally, the principal investigator will study random walks on lattices, using techniques from algebraic geometry and commutative algebra to generate the walks and to study how fast they converge to their asymptotic distributions. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, 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 physics, theoretical computer science, and robotics.
