Modular curves are moduli spaces of elliptic curves equipped with certain level structures. This talk will be concerned with how the attendant theory has been used to answer questions about the modularity of elliptic curves over $\mathbb{Q}$ and over quadratic fields. In particular, we will outline two instances of the modularity switching technique over totally real fields: the 3-5 trick of Wiles and the 3-7 trick of Freitas, Le Hung and Siksek. The recent work of Caraiani and Newton over imaginary quadratic fields naturally leads one to consider the descent theory of 'twisted' modular curves, and this will be the focus of the final part of the talk.

Given a pair of metric tensors g_j ≥ g₀ on a Riemannian manifold, M, it is well known that Vol_j(M)≥Vol₀(M). Furthermore, the volumes are equal if and only if the metric tensors are the same, g_j=g₀. Here we prove that if for a sequence g_j, we have g_j≥g₀, Vol_j(M)→Vol₀(M) and diam(M_j) ≤ D then (M,g_j) converges to (M,g₀) in the volume preserving intrinsic flat sense. The previous result will then be applied to prove stability of a class of tori.
 

This talk is based on joint works of myself with: Allen and Sormani (https://arxiv.org/abs/2003.01172), and Cabrera Pacheco and Ketterer (https://arxiv.org/abs/1902.03458).