Gopakumar-Vafa large N duality is a correspondence between Chern-Simons invariants of a link in a 3-manifold and relative Gromov-Witten invariants of a 6-dimensional symplectic manifold relative to a Lagrangian submanifold. We address the correspondence between the Chern-Simons free energy of S3 with no link and the Gromov-Witten invariant of the resolved conifold in great detail. This case avo...

We apply ideas from crystal theory to affine Schubert calculus and flag Gromov–Witten invariants. By defining operators on certain decompositions of elements in the type-A affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams. We show how this crystal framework can be applied to study the product of a Schur function with a...

This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a 2-dimensional Artin group the Deligne complex is Gromov hyperbolic preci...

In this paper we define, for each aspherical orientable 3-manifold M endowed with a torus splitting T , a 2-dimensional fundamental l1-class [M ] whose l1norm has similar properties as the Gromov simplicial volume of M (additivity under torus splittings and isometry under finite covering maps). Next, we use the Gromov simplicial volume of M and the l1-norm of [M ] to give a complete characteriz...

where the Z action is generated by (x, s, θ) 7→ (f(x), s + 1, θ). In this paper we compute the Gromov invariants of the manifolds Xf and of fiber sums of the Xf with other symplectic manifolds. This is done by expressing the Gromov invariants in terms of the Lefschetz zeta function of f and, in special cases, in terms of the Alexander polynomials of knots. The result is a large set of interesti...

Let W → C be degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first constructed the stack of expanded degenerations of W . We then constructed the moduli space of stable morphisms to this stack, which provides a degeneration of the moduli spaces of stable morphisms associated to W/C. Using similar technique, for a pair of smooth varie...

The goal of this article is to motivate and describe how Gromov-Witten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from Gromov-Witten theory have led to both conjectures and theorems showing that the tautological part of the cohomology ring has a remarkable and profound structure. As an illustration, we describe a new approach to Fa...