Beyond Wiles´ proof of Fermat´s Last Theorem

The aim of our project is to generalize the method A. Wiles used to prove Fermat´s Last Theorem. Our Wallenberg Academy Fellowship pursues a general program of geometry-generated-by-groups.

By contrast, our specific goal here is to apply the general tools and ideas of that program to the particular, concrete problem of generalizing the results of Wiles et. al.Wiles´ strategy was to deduce Fermat´s Last Theorem from a much more abstract and sophisticated conjecture, that all elliptic curves defined over the rational numbers arise from modular forms.

In turn, the latter `modularity conjecture´ is logically a very special and basic case of a gigantic infrastructure of conjectures laid out by R.

Langlands, which explicitly connect (i) special analytic functions with extroardinary symmetry and (ii) algebra and geometry.

The Langlands program has single-handedly unified more branches of math than any other in history; it is also connected with physics and quantum computing.The efforts by R. Taylor and others to generalize the `Taylor-Wiles method´ are now hitting a wall.

Calegari & Geraghty proposed a program to push beyond the Taylor-Wiles method by improvements stemming from homological algebra.Our goal is to improve upon the Taylor-Wiles method in a different direction, by applying the group-theoretic ideas of our past, current and future work.

Generalization of the modularity conjecture will similarly lead to new applications in arithmetic, geometry and quantum computing.