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.

 Is it possible to define gauge theories on singular spaces? The answer to this question is emphatically yes​, and the prime example of such spaces are two-dimensional orbifolds known as spindles​. First, I will introduce spindles from a symplectic geometry perspective. Then I will discuss the notion of orbi-bundles, which allows one to consistently describe regular gauge fields/spinors on orbifolds.