Given two Lagrangians $L_1$ and $L_2$ that intersect either transversally or cleanly, one may construct a new Lagrangian through a surgery operation. When the intersection is transversal, Fukaya, Oh, Ohta and Ono described a concrete relation between J-holomorphic curves with boundary on these Lagrangians before and after the surgery.

We give a new perspective to this problem through Joel Fish's target-local compactness theorem, which allows us to generalize the result of FOOO to Lagrangian clean surgeries for monotone Lagrangian pairs. In particular, this implies the Huybrechts-Thomas conjecture on projective twists. This is a joint work with C.-Y. Mak and G. Xu.