Side Door and Fermat's Last Theorem: Behind-the-Scenes Museums and a Breakthrough in Modularity

Two stories unfold in this episode: Side Door invites listeners behind the scenes of the Smithsonian, exploring ghost lore, late-night halls, and curiosities that never sit in a display case, while Quanta Magazine surveys a landmark advance in number theory that ties together elliptic curves, modular forms, and the Langlands program. The narrative follows a collaboration of mathematicians as they extend modularity to abelian surfaces, aided by ideas from Louis Pan and a Bonn conference in 2023, culminating in a 230-page preprint posted in 2025. The result opens new questions and directions, including analogs of Birch and Swinnerton-Dyer for abelian surfaces and a fresh horizon for arithmetic geometry.

Overview and context

Two strands of curiosity run through this episode, one playful and cultural, the other mathematical and foundational. The Side Door segment invites listeners to imagine what it is like behind the cordons of a museum or a research complex, exploring ghost lore, train-robbery forensics, and curiosities that never sit in a display case, while the Quanta story surveys a major advance in number theory that ties together elliptic curves, modular forms, and a Langlands-inspired program.

Behind the Scenes at the Smithsonian and Side Door's Mission

The Side Door portion frames curiosity as a driver of public understanding, promising a behind-the-scenes look at the world's largest museum and research complex, revealing stories of history, science, art, and culture that you won't find in a typical display. It emphasizes the thrill of peeking behind cordons and the value of science storytelling for broad audiences.

Modularity and the Bridge Between Worlds

The mathematical narrative explains how Fermat's Last Theorem was resolved through a modularity theorem linking elliptic curves to modular forms. It then explains that mathematicians extended this correspondence to abelian surfaces, a more complex object in higher dimensions. The idea of modularity is presented as a universal principle that could unify a broad class of mathematical objects. The Langlands program is introduced as a sweeping set of conjectures aiming to connect disparate areas of mathematics through mirror-like correspondences. The section highlights the difficulty of constructing modular forms that match abelian surfaces and why the problem resisted earlier attempts. The team chose an ordinary abelian surface as a starting point because it is more tractable.

"we mostly believe that all the conjectures are true, but it's so exciting to see it actually realized." - Anna Kaani, Imperial College London

The Bonn sprint and the 230-page proof

After long-running Zoom collaboration and a Bonn conference in 2023, the team converged on Pan's techniques and developed a bridge between two modular-form clocks. They produced a 230-page proof posted online in February 2025, showing that any ordinary abelian surface has an associated modular form. The journey included a visa denial and a basement session that felt like mining, a vivid metaphor Pilloni uses to describe their intense, round-the-clock work. The team continues to push toward non-ordinary cases and broader connections with Pan, with optimism about new conjectures that link to Birch and Swinnerton-Dyer for abelian surfaces.

"it's like working in a mine." - Vincent Pilloni

Implications and the future of the Langlands program

The breakthrough is framed as a step toward a grand unified theory of mathematics, with modularity offering a bridge to a wider class of equations and objects beyond elliptic curves. The ordinary abelian surface result provides a new analog of Birch and Swinnerton-Dyer and supports the Langlands program's aim to connect number theory and analysis. MIT's Andrew Sutherland describes how conjectures once thought out of reach are becoming accessible, and researchers anticipate extending modularity to more objects within this framework. This is presented as the beginning of a hunt that could reshape arithmetic geometry for years to come.

"Now they at least know that the analog makes sense for these ordinary surfaces." - Andrew Sutherland, MIT

Future directions and concluding reflections

Looking ahead, researchers plan to extend modularity to non-ordinary abelian surfaces, continue collaborations with Pan, and probe deeper questions that arise from the Birch and Swinnerton-Dyer analog. The narrative suggests the potential for modular forms to illuminate a broad spectrum of mathematical phenomena, expanding the Langlands program’s reach and offering new avenues for cross-disciplinary collaboration.

"It changes things." - Andrew Sutherland, MIT