Given a Fano variety X with smooth anticanonical divisor D, one may consider the enumerative geometry of X, of the pair (X,D) or of D. A-model Quantum Lefschetz, Residue and Serre relate counts of genus 0 curves in X,  (X,D) and D. While the A-model statements are fairly involved, they become standard integral transforms when formulated as B-model correspondences within the Intrinsic Mirror Construction of Gross-Siebert. I will explain how this works. Time permitting, I will explain how for K-polystable del Pezzo surfaces, genus 0 log BPS instanton expansions transform into modular forms.

In this talk I will give a hopefully not too technical introduction to one of the techniques that allowed Taylor and Wiles to prove the modularity theorem that was the final step for proving Fermat's Last Theorem.
After explaining how the patching works, I will present some generalisations of the method to different contexts. If time permits, I will also briefly explain how patching was used to produce a candidate for the p-adic local Langlands correspondence.