WEEK 1: THE SYZ CONJECTURE AND COLLAPSING CALABI-YAU MANIFOLDS
The Strominger-Yau-Zaslow (SYZ) conjecture dates from 1996. It was proposed as a geometric mechanism underlying mirror symmetry for Calabi-Yau manifolds. The proposal is that, at least near the “large complex structure” limit in moduli space, a Calabi-Yau manifold has a fibration whose generic fibers are Special Lagrangian tori. The mirror manifold should then be obtained by taking the duals of the torus fibers.
The existence of such Special Lagrangian fibrations, and particularly the understanding of the singular fibers is a difficult problem, and many of the prominent developments around the SYZ conjecture over the past quarter century involve related algebro-geometric and symplectic-topological constructions. However there have also been great advances and activity in complex differential geometry and in the study of Special Lagrangian submanifolds, and calibrated submanifolds more generally. There are also new connections with non-Archimedean geometry.
One substantial recent advance towards the SYZ conjecture came in the work of Yang Li, who established (under reasonable hypotheses) the existence of a Special Lagrangian fibration outside a set of very small volume. In metric terms, working near the large complex structure limit means that fibers of the conjectured fibration are very small compared with the size of the base. This fits into a wider discussion of “collapsing” in Calabi-Yau geometry (and Riemannian geometry more generally). There have been many other recent developments in this area, for example the classification by Sun and Zhang of collapsed Gromov-Hausdorff limits of Calabi-Yau metrics on K3 surfaces and descriptions of metrics as the complex structure degenerates, involving multiscale collapsing. These are very relevant to questions of moduli space compactification.
The aim of this week of the SCGP summer programme will be to involve both experts in these differential geometry and geometric analysis developments and experts in relevant aspects of neighboring fields such as algebraic geometry and symplectic topology. This week will include lecture courses by Yang Li and Song Sun, and talks by Simon Donaldson and Ruobing Zhang.
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.
WEEK 3: INTEGRABLE SYSTEMS METHODS IN ENUMERATIVE GEOMETRY
In the early 90’s Witten, motivated by a matrix model description of topological gravity, conjectured that the intersection numbers of tautological classes on moduli spaces of curves are governed by the KdV integrable hierarchy. Witten’s conjecture was first proven by Kontsevich, and his proofs as well as later different proofs by Okounkov-Pandharipande, by Mirzakhani, have all led to a rich theory connecting moduli spaces of stable curves, and more generally Gromov-Witten theory, with 1+1 dimensional integrable systems.
More recently, symplectic field theory has provided another link with integrable systems, including quantum ones. As it was observed by Dubrovin the genus $0$ Gromov-Witten theories are related with the so-called dispersionless integrable hierarchies, while adding the dispersion requires full genus GW theories. On the other hand, while the genus $0$ SFT leads to the same dispersionless hierarchies, the full genus SFT yields quantization of dispersionless hierarchies, rather than introducing a dispersion parameter.
An area of active current research, also motivated by symplectic field theory, is the study of the
double ramification (DR) cycle. The DR viewpoint allows a new interpretation and construction of (quantum) integrable systems from cohomological field theories, and also yields a 2-parametric deformation of dispersionless hierarchies, where one parameter introduces a dispersion, while the second leads to a deformation quantization. A lot of current research revolves around relations of DR cycles with other geometric constructions on moduli of curves and differentials, and in field theories.
This week will include series of lectures by Danilo Lewanski and Dimitri Zvonkine on this circle of ideas, and some individual talks by experts.