Xu 9700491 This project is concerned with research in algebraic geometry. Based on the mirror symmetry principle, physicists proposed several years ago a formula to predict the numbers of rational curves of various degrees on a general quintic threefold. Recently, Givental showed that the numbers suggested by the physicists agree with the numbers of J-holomorphic curves of genus 0 of various degrees on a general quintic threefold for a generic almost complex structure J. The principal investigator will work on Clemens' conjecture about the finiteness of rational curves on a general quintic threefold in each degree. He is also working towards an understanding of the local positivity of ample line bundles on a smooth projective variety through the study of Seshadri constants, which is related to a conjecture of Nagata on a relation between the singularity of a plane algebraic curve and the degree of the curve. In addition, the principal investigator will study the algebraic hyperbolicity of the complement of a general hypersurface and the enumeration of algebraic curves on smooth projective surfaces. Finally, he will work on the quantum cohomology algebras of Fano manifolds, to see whether the quantum cohomology algebras of Fano hypersurfaces are semi-simple in the sense of Dubrovin. 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.
