The principal investigator on this award will consider the following local questions in commutative algebra. (1) Is the symbolic algebra of a monomial prime P(a,b,c) finitely generated? (2) Is there a solution to the Riemann-Roch problem for high powers of a nef divisor on a nonsingular projective variety of dimension greater than or equal to three? (3) Suppose that R and S are regular local rings with the same quotient field K, and V is a valuation ring of K such that V dominates R and S? Is there a regular local ring T such that T can be obtained from both R and S by sequences of monodial transformations, and such that V dominates T? 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.
