Explore advanced topics in commutative algebra and number theory through this 57-minute lecture by Srikanth Iyengar at the Hausdorff Center for Mathematics. Delve into the extension of Wiles' numerical criterion for isomorphism of complete intersections to higher relative dimensions. Examine the work of Diamond generalizing Wiles' and Lenstra's results to criteria for module freeness. Investigate joint research with Khare and Manning that further expands these concepts. Focus on the commutative algebra aspects while also considering applications, including unconditional integral R = T theorems at non-minimal level for Hecke algebras acting on weight one cohomology of Shimura curves. Gain insights into congruence modules, contangent modules, and their roles in modularity lifting and related mathematical structures.