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We introduce a joint project with Cheol-Hyun Cho on the construction ofquantum-corrected moduli of Lagrangian immersions. The construction hasimportant applications to mirror symmetry for pair-of-pants decompositions, SYZand wall-crossing. The key ingredient is Floer-theoretical gluing between localmoduli spaces of Lagrangians with different topologies.

Moduli of Lagrangian immersions with formal deformations