Abstract: I will talk about recent work with Srikanth Iyengar and Jeff Manning on a higher codimension version of the Wiles--Lenstra--Diamond numerical criterion (the original version is in codimension $0$). The original version played a key role in Wiles’s work on the modularity of semistable elliptic curves over the rationals.
I will sketch some (conditional) applications of the commutative algebra we develop to proving integral $R=T$ theorems in positive defect (as arise when considering $2$-dimensional Galois representations over imaginary quadratic fields, a defect one situation), and other questions/perspectives the work leads to. There is an unconditional application to proving an analog of the Jacquet--Langlands correspondence for Hecke algebras acting on the cohomology of Shimura curves with coefficients in weight one sheaves. As these Hecke algebras typically have a lot of torsion, such results cannot be deduced from the classical Jacquet--Langlands correspondence for classical weight one forms.