Collapsing Calabi-Yau Manifolds: August 31 – September 4, 2015

Organizers: Radu Laza, Valentino Tosatti, Mark Gross, and Simon Donaldson

Dates: August 31 – September 4, 2015

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This workshop is organized in connection to the SCGP program on Moduli spaces and singularities in algebraic and Riemannian geometry. The main topic is the Riemannian collapsing and large complex structure limits of Calabi-Yau manifolds. This is a very active research direction in mathematics with connections to physics.

One of the main motivations for studying the collapsing of Ricci-flat Calabi-Yau manifolds comes from mirror symmetry. The Strominger-Yau-Zaslow proposal suggests that given a family of Calabi-Yau manifolds which are degenerating to a large complex structure limit, the manifolds close to the limit should admit a special Lagrangian torus fibration to a half-dimensional base space, and that the mirror family should be obtained by dualizing the special Lagrangian torus fibration. While carrying out this program remains a formidable task to this day, around 2000 Gross-Wilson, Kontsevich-Soibelman and Todorov proposed an approach towards SYZ using metric geometry. More precisely, they conjectured that for a polarized family of Calabi-Yau manifolds near a large complex structure limit, the canonical Ricci-flat metrics after renormalization to unit diameter, should collapse to a half-dimensional space in the Gromov-Hausdorff sense. This space in general is expected to have singularities, and away from these there will be a Riemannian metric with a compatible integral affine structure, and the mirror family should give rise to the dual affine structure. After foundational work by Gross-Wilson on K3 surfaces, recent progress was made recently on this approach, using analytical techniques. On the other hand, a new and rather successful algebro-geometric approach to SYZ mirror symmetry was proposed by Gross-Siebert, who used also techniques from tropical geometry and proved that one can reconstruct a degeneration of Calabi-Yau varieties from a sufficiently general affine manifold with singularities.

One of the goals if this workshop will be to try to bridge the gap between the algebro- geometric/tropical approach to mirror symmetry by Gross-Siebert, and the analytic/geometric study of collapsing of Ricci-flat metrics, by bringing together experts in these two areas and fostering communication between them.

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