WEEK 2: RELATIVE AND LOGARITHMIC GROMOV-WITTEN THEORY
Gromov-Witten theory, developed in the early 1990s, provides a system of curve-counting invariants of symplectic manifolds or non-singular algebraic varieties. In the context of symplectic geometry, these invariants are defined via moduli spaces of J-holomorphic curves, where J is an almost complex structure on the target symplectic manifold. In algebraic geometry, these invariants are defined using the Kontsevich moduli space of stable maps.
The richer theory of relative Gromov-Witten invariants, developed in the late 1990s, allows the imposition of tangency conditions of the curves with a fixed smooth divisor. This has provided a very powerful tool for the computation and understanding of Gromov-Witten invariants, especially via the degeneration formula, which allows the Gromov-Witten invariants of a variety to be computed by degenerating the variety to a union of two varieties.
In the last fifteen years, a number of generalizations allowing imposition of tangency conditions with a fixed normal crossing divisor have been developed. This includes the exploded manifold approach of Brett Parker and the logarithmic approach of Abramovich, Chen, Gross and Siebert, leading to an even richer theory which already has found important applications, including to mirror symmetry constructions.
This week will include lecture series on these approaches, especially by Brett Parker on the exploded approach and by Bernd Siebert on the logarithmic approach, as well as a number of other lectures by experts.