Time：14:30-15:30, Monday, June 16 2025
Venue：E4-201
Host：Jiyuan Han, ITS
Speaker：Hang Yuan, BIMSA
Title：A mathematical formulation of SYZ conjecture and applications
Abstrct：The SYZ conjecture predicts that a Calabi-Yau manifold with a Lagrangian fibration admits a dual torus fibration whose total space forms the mirror Calabi-Yau. We propose a rigorous formulation in which the dual is a non-Archimedean torus fibration. This proposal is inspired by the toy SYZ model comparing the complex logarithm map (interpreted as a torus fibration over Euclidean space with a trivial integral affine structure) and its non-Archimedean counterpart, the tropicalization map.
Using Fukaya's A_\infty algebras for Lagrangian fibers, we construct a canonical dual fibration, unique up to isomorphism, that locally recovers this model. The original and dual fibrations share the same base and induce identical integral affine structures, as governed by Arnold-Liouville's theorem and Kontsevich-Soibelman's results.
We will review basic aspects of non-Archimedean geometry and discuss applications of this non-Archimedean formulation to symplectic geometry, such as the existence of pseudo-holomorphic disks in certain cases.
