Principal Investigator: Xiaobo Liu
These projects emphasize Gromov-Witten invariants of compact symplectic manifolds, relations in the tautological rings of stable curves, and interactions of these topics with integrable systems. Roughly speaking, primary Gromov-Witten invariants count numbers of pseudo-holomorphic curves in compact symplectic manifolds. A larger class of invariants, called descendent Gromov-Witten invariants, can be defined by taking into consideration powers of first Chern classes of some tautological line bundles over the moduli space of pointed pseudo-holomorphic maps. In the case when the manifold is a point it was conjectured by Witten and proved by Kontsevich that descendant Gromov-Witten invariants are governed by the KdV hierarchy from integrable systems. The principal investigator is pursuing generalizations of the Witten conjecture and the Virasoro conjecture on by studying universal equations which come from relations in the tautological rings of the moduli spaces of stable curves.
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.
