We introduce a joint project with Cheol-Hyun Cho on the construction of
quantum-corrected moduli of Lagrangian immersions. The construction has
important applications to mirror symmetry for pair-of-pants decompositions, SYZ
and wall-crossing. The key ingredient is Floer-theoretical gluing between local
moduli spaces of Lagrangians with different topologies.Comment: 23 pages, 12 figures, partly presented in the Gokova
Geometry/Topology Conference 201