Chapter 19: The First Cornerstone of Unified Mathematics

On the morning of January 31, 1960, the mathematics lecture hall at Columbia University was shrouded in the thin winter fog of New York.

Lin Ran stood before the podium, waiting for mathematicians from around the world to arrive.

Columbia’s president, Roses, had personally come to support him.

If this breakthrough was proven, Columbia University would wear the crown of having solved a centuries-old mathematical conjecture; with a talent like Randolph Lin in its ranks, surpassing Princeton and Harvard in mathematics was entirely possible.

The thought of outshining its old rival sent a thrill through Roses’ heart.

He had even planned ahead: if this lecture received universal acclaim from mathematicians, he would invite the former president to the celebratory banquet.

The former president’s name in Washington was Eisenhower.

After retiring from military service, numerous companies sought to appoint him as CEO or chairman, but he ultimately accepted Columbia’s offer, serving four years before returning to Washington.

When the mathematicians in the audience gradually took their seats, the center of the front row belonged to Grothendieck.

He had just arrived from Paris, and all mathematicians had willingly given him the best seat.

Andrew Weil was annotating the margins of his manuscript with red and blue pencils; Grothendieck whispered to his companion Serre, his black leather notebook open to page seventeen.

As the projection screen displayed Fermat’s equation, the faint murmurs in the hall fell silent. Lin Ran pointed his pointer at the moduli space parameter of the elliptic curve: “Assume there exists an integer solution (a, b, c); then the corresponding Frey curve would induce a contradiction in the l-adic Galois representation.”

Grothendieck suddenly raised his notebook, on which German script read: “How does the structure of the Selmer group evade the constraints of the Hasse principle?”

After Serre translated it, Lin Ran said: “This is precisely the key to the symbiosis between modular forms and elliptic curves.”

Lin Ran gestured for his assistant to unfold the third blackboard: “By constructing Galois representations, Fermat’s equation has a solution if and only if the corresponding modular form does not exist—but the fact that the rank of the modular form space is zero completely locks out the possibility of any solution.”

Weil’s pencil froze mid-air; he interrupted: “Is the contradiction provided by the Frey curve sufficient to support a general proof?”

“Of course.”

At the forty-seventh minute, when Lin Ran introduced the action of the Hecke algebra of automorphic forms on the Galois group, a soft clink of coffee cup against tray echoed from the back. Mathematicians began slipping quietly into seats from the side doors.

Andrew Weil recalled a letter from three months ago, which contained precisely the conjecture linking automorphic representations with Galois groups.

“The essence of this proof is building a bridge between the world of modular forms and the Galois group,” Lin Ran switched blackboards to display the complex analytic structure of modular curves, “and I believe this bridge has far broader applications.”

It is the deep and precise correspondence between different mathematical fields that many mathematicians have long sought.

Such mappings ought to be widespread.”

The number theorists in the room stiffened their necks, refusing to turn away, afraid of missing even a single detail.

Leading figures spanning multiple fields scribbled furiously in their notebooks: “When Fermat’s conjecture is transformed into a symmetric proposition about L-functions, it opens a path for the future development of mathematics.”

When Grothendieck rose, the buttons of his overcoat brushed against the chair, producing a chime: “I need to verify compatibility at the cohomological level.”

He swiftly sketched a commutative diagram of étale cohomology groups on the board: “If such a functorial correspondence exists, algebraic geometry will gain a coordinate chart into the realm of automorphic forms.”

By noon, even during breaks in the cafeteria, every mathematician hoped to cluster around Lin Ran and discuss further theoretical implications of the proof of Fermat’s conjecture.

But most mathematicians had no such opportunity; the three others seated with Lin Ran were unmovable.

The Pope of Algebraic Geometry, Grothendieck; Columbia’s mathematics department chair, Ralph Fox; and Göttingen University’s mathematics department chair, Hans Hermann Schwarz.

Schwarz had only become Göttingen’s mathematics chair in 1958; he only learned that one of his own students had proven Fermat’s conjecture when he attended this lecture.

He truly regretted it.

After the war, Göttingen University was no longer the glorious mathematical holy land it once was; now it was home to only a few small fry.

Unlike the past, when Gauss, Riemann, and Hilbert each generation produced at least one world-leading mathematician.

Lin Ran was someone who could stand shoulder to shoulder with those three—and yet Göttingen University had failed to retain this gem, allowing Columbia to snatch him up.

By three in the afternoon, slanting sunlight streamed into the lecture hall, dust suspended before the blackboard like scattered mathematical symbols.

As Lin Ran began addressing the restrictions of the inversion theorem on non-congruence subgroups, Weil raised his densely annotated preprint: “Does the derivation in section 4.2 rely on a trick in selecting primes? I need to confirm whether the ergodicity over the Schwartz space is thorough enough.”

“This is precisely the essence of the Witt cancellation theorem,” Lin Ran projected the numerical computation results: “When the modular degree of an elliptic curve exceeds a certain threshold, its corresponding modular form must be a cusp form.”

Milnor from Princeton sketched a five-dimensional manifold diagram in his notebook and whispered to his neighbor Atiyah: “Could this approach be extended to classify differential structures on four-dimensional manifolds?”

Discussion rippled outward like a spreading topological vortex, until Lin Ran lightly tapped his pointer, refocusing everyone’s attention on the blackboard: “Does the finiteness of the Selmer group here play a role analogous to the Riemann Hypothesis?”

The entire academic conference lasted half a month.

The final objection came from Grothendieck, who questioned whether the correspondence between elliptic curves and modular forms was universally applicable.

Lin Ran unveiled his ultimate weapon, prepared specifically for this occasion: the globalized mathematical framework of the Langlands program’s local correspondences.

The upright blackboard displayed a newly mapped mathematical landscape born from the proof of Fermat’s conjecture—a map prepared in advance to show the audience what future research paths lay ahead.

The region where modular forms intersected with algebraic geometry was labeled: “The Highway of Correspondences Between Fields.”

As the audience dispersed, Grothendieck still leaned against the wall revising his notes; the note Weil had left behind was folded by Lin Ran into the facsimile edition of Fermat’s works, a gift specially presented by Hans Hermann Schwarz.

At the end of the corridor, Fox gazed out the window at the Hudson River, its ripples mirroring the vibration spectra of infinite-dimensional automorphic representations.

Everyone suddenly realized that the history of mathematics had split at this moment into two segments: one ending with the period of Fermat’s theorem, the other beginning with the infinite possibilities of reorganizing mathematics through new ideas.

“Randolph, congratulations—you have found the first cornerstone of unified mathematics.”