(Number theory seminar) Generalized Fermat Equations

The proof of Fermat’s Last Theorem pioneered a new approach to resolving families of ternary Diophantine equations using modularity of residual Galois representations attached to Frey curves. In the case of differing exponents, Darmon gave a framework for resolving generalized Fermat equations in one varying exponent using Frey varieties. In this talk, I will survey the methods and techniques of recent progress on Darmon’s program as well as the challenges that remain in the study of generalized Fermat equations.