Gromov-Witten Invariants and Isoparametric submanifolds

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Abstract Award: DMS-0071372 Principal Investigator: Xiaobo Liu This research program addresses problems in two areas. The Virasoro conjecture in symplectic geometry asserts that the generating function of the Gromov-Witten invariants for a compact symplectic manifold must be annihilated by a sequence of differential operators in the Virasoro algebra. The principal investigator will study whether the genus-one Virasoro conjecture and perhaps more general constraints hold for all projective varieties. It is even possible that the generating function of genus-one primary Gromov-Witten invariants can be computed explicitly from the genus-zero invariants, although this is more than the Virasoro conjecture implies. The second area to be addressed is submanifold geometry, especially the properties of isoparametric submanifolds, including infinite dimensional submanifolds. We know some good constructions of isoparametric submanifolds, but important aspects of the classification theory are in early stages of development. Geometric descriptions of classical or quantum mechanical systems are based in symplectic geometry, which takes as its basic measurement the area of two-dimensional surfaces. The Gromov-Witten invariants of a symplectic manifold can be described as a family of number that count the number of surfaces of a nice sort that satisfy certain constraints, such as intersecting specified subsets of the manifold. Enumerative invariants of this kind are an ancient concern in algebraic geometry, where we count the number of solutions to a system of equations, and the subject was recently found to be connected to high energy physical theory, where the partition functions of certain quantum systems on a manifold have been found to be related to generating functions that collect families of Gromov-Witten invariants into a manageable form. The Virasoro conjecture proposed by a group of physicists and mathematicians asserts that these generating functions satisfy a family of constraints that simplify computations and connect the Gromov-Witten invariants to the mathematics of completely different problems. A different kind of geometry arises in the study of submanifolds as curved objects, where we use several measurements of bending to capture the intuitive sense that a sphere in three-space is more curved if its radius is small than if its radius is large. An isoparametric submanifold is an imbedded object which has particularly simple curvature properties with respect to the larger space, and the character of this specialty is driven by the fact that this local condition frequently imposes strong symmetry properties the whole submanifold.