Chapter 18: The First Cornerstone of Unified Mathematics

On the morning of January 31, 1960, the lecture hall of the Mathematics Department at Columbia University was shrouded in the thin mist characteristic of a New York winter. Lin Ran stood at the podium, awaiting the arrival of mathematicians from around the world.

The president of Columbia University, Rosse, was personally there to endorse the event.

Such a monumental occasion in the mathematical community promised that, once proven, Columbia University would crown itself with the triumph of solving a centuries-old conjecture. With a talent like Randolph Lin in the mathematics department, surpassing Princeton and Harvard in the field of mathematics seemed entirely within reach.

The very thought of outshining their long-standing rivals stirred Rosse deeply.

He had even resolved that if this academic presentation gained unanimous acclaim among mathematicians, the subsequent celebratory banquet must include the former president, Dwight D. Eisenhower.

Eisenhower, after retiring from his military career, had been courted by numerous corporations to serve as CEO or chairman, but ultimately accepted Columbia University's offer before returning to Washington after four years.

As the mathematicians gradually took their seats, Alexander Grothendieck was seated in the very center of the first row.

Having just arrived from Paris, all the mathematicians had willingly ceded the best seats to him.

Andrew Wiles was marking notes in red and blue pencils on the margins of his manuscript, while Grothendieck quietly discussed something with his companion, Serre, the black leather notebook opened to page seventeen.

When the projection screen displayed the Fermat equation, the subtle murmurs across the room abruptly ceased. Lin Ran pointed with his pointer at the modular parameter space of the elliptic curve: "Assuming there exists an integer solution (a, b, c), the corresponding Frey curve would induce a contradiction within the l-adic Galois representation."

Suddenly, Grothendieck raised his notebook, on which was written in German: "How does the structure of the Selmer group circumvent the constraints of the Hasse principle?"

After Serre translated, Lin Ran responded, "This is precisely the key to the symbiosis between modular forms and elliptic curves."

Lin Ran gestured to his assistant to unfold the third blackboard. "By constructing the Galois representation, the Fermat equation has a solution if and only if the modular form corresponding to this representation does not exist — but the fact that the rank of the modular form space is zero completely rules out the possibility of a solution."

Wiles abruptly paused his pencil mid-air and interjected, "Is the contradiction provided by the Frey curve sufficient to support a general proof?"

"Absolutely."

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At the forty-seventh minute, as Lin Ran introduced the action of the Hecke algebra on the Galois group through automorphic forms, the light clinking of coffee cups and trays came from the back row. More mathematicians quietly entered through the side doors and took their seats.

Andrew Wiles recalled correspondence from three months earlier with a friend that happened to include conjectures on the correspondence between automorphic representations and Galois groups.

"The essence of this proof is to build a bridge between the world of modular forms and the Galois group," Lin Ran switched the blackboard to display the complex analytic structure of modular curves. "And I believe this bridge has a much broader scope of application.

"For years, many mathematicians have hoped to find a profound and precise correspondence between distinct areas of mathematics.

"This kind of mapping should exist widely."

The number theorists present sat rigidly, unwilling to look away for fear of missing even the smallest detail.

A leading figure working across multiple fields rapidly scribbled in his notebook: "When Fermat's conjecture is transformed into a symmetry proposition about L-functions, it paves a path for the future development of mathematics."

As Grothendieck rose, the buttons of his trench coat scraped the chair with a faint sound. "I need to verify the compatibility at the level of higher cohomology."

He quickly sketched a commutative diagram of étale cohomology groups on the blackboard. "If such a functorial correspondence exists, then algebraic geometry will gain a coordinate chart for entering the realm of automorphic forms."

By noon, even during breaks in the cafeteria, every mathematician hoped to gather around Lin Ran to discuss further theories related to the proof of Fermat's conjecture.

Yet most had no chance; the other three mathematicians sitting at the same table with Lin Ran were unyielding.

They were the pope of algebraic geometry, Grothendieck; the chair of the Columbia Mathematics Department, Ralph Fox; and Hans Hermann Schwartz, chair of the Mathematics Department at the University of Göttingen.

Schwartz had only become head of Göttingen's mathematics department in 1958, and it was at this very academic presentation that he learned a former student from his own university had proven Fermat's conjecture.

Regret—real regret—consumed him.

After the war, the University of Göttingen was a mere shadow of the once-mighty mathematical sanctuary it had been, now boasting only a handful of minor scholars.

Gone were the days of Gauss, Riemann, and Hilbert, when each generation produced at least one world-leading mathematician.

And Lin Ran had the promise to stand shoulder to shoulder with these giants, yet this hidden gem had slipped from Göttingen’s grasp and been claimed by Columbia.

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By three o’clock in the afternoon, sunlight slanted into the lecture hall, and dust motes floated before the blackboard like scattered mathematical symbols.

As Lin Ran began addressing the constraints of the inversion theorem on noncongruence subgroups, Wiles raised a densely annotated preprint: "Does the derivation in Section 4.2 involve a subtle choice of primes? I need to confirm whether the traversal of Schwartz space is sufficiently thorough."

"That is precisely the essence of applying the Witt elimination theorem," Lin Ran responded, projecting numerical results. "When the modular degree of an elliptic curve exceeds a certain threshold, its corresponding modular form must be a cusp form."

John Milnor from Princeton sketched a diagram of a five-dimensional manifold in his notebook and whispered to his neighbor, Michael Atiyah, "Could this idea be extended to the classification of differential structures on four-dimensional manifolds?"

The discussion gradually swelled like a diffusing topological vortex until Lin Ran gently tapped his pointer, refocusing everyone’s attention on the blackboard: "Does the finiteness of the Selmer group play a controlling role here, analogous to that in the Riemann hypothesis?"

The entire academic conference lasted a full fortnight.

The final challenge came from Grothendieck, who questioned the scope of applicability for the correspondence between elliptic curves and modular forms.

Lin Ran unveiled the ultimate weapon prepared for this occasion: the mathematical framework of the globalized local Langlands program.

Displayed on the standing blackboard was a new mathematical map born from the proof of Fermat’s conjecture, prepared before the conference began to guide the assembled mathematicians toward future research directions.

The intersection between modular forms and algebraic geometry was marked as the “highway linking distinct fields.”

As the conference adjourned, Grothendieck leaned against the wall, still revising notes, while Wiles’s question slip was folded by Lin Ran into a replica of Fermat’s works, specially presented to Hans Hermann Schwartz.

At the corridor’s end, Fox gazed out over the Hudson River, whose ripples resembled the vibrational spectra of infinite-dimensional automorphic representations.

Suddenly, everyone realized that history of mathematics had split into two: one ending with the period of Fermat’s Last Theorem, the other beginning with the infinite possibilities of reorganizing mathematics through new ideas.

“Randolph, congratulations. You have found the first cornerstone for unifying mathematics.”