Chapter 19: The First Cornerstone of Unified Mathematics

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

Lin Ran stood before the podium, waiting for the arrival of mathematicians from all over the world.

Columbia University President Rosset personally stood for him.

In such a grand event in the mathematics community, once proven, Columbia University would wear the crown of solving a centuries-old conjecture in mathematics. With talent like Randolph Lin in the mathematics department, it was entirely possible to surpass Princeton and Harvard in the field of mathematics.

At the thought that mathematics could suppress old rivals, Rosset felt a surge of excitement in his heart.

He had even thought it through: if this academic lecture received the unanimous approval of the mathematicians, then the subsequent celebratory banquet must include the old president.

The old president had a resounding name in Washington: Eisenhower.

After Eisenhower retired from his military career, many companies hoped to invite him to serve as CEO or chairman, but he finally chose to accept Columbia University's offer, serving for four years before returning to Washington.

After the mathematicians in the audience had taken their seats one after another, sitting in the very middle of the first row was Grothendieck.

The other party had just arrived from Paris, and all the mathematicians had voluntarily given up the best seats to him.

André Wei Yi was marking annotations on the edge of his manuscript with a red and blue pencil, while Grothendieck was whispering to Serre, who was accompanying him, and his black leather-bound notebook was already opened to the seventeenth page.

When the projection screen displayed the Fermat equation, the subtle discussion in the hall ceased abruptly. Lin Ran pointed his pointer at the moduli space parameters of the elliptic curve: "Assuming there exists an integer solution (a, b, c), the corresponding Frey curve will trigger a contradiction in the l-adic Galois representation."

Grothendieck suddenly 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 said: "This is precisely the key to the symbiosis of modular forms and elliptic curves."

Lin Ran signaled 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 will completely lock out the possibility of a solution existing."

Wei Yi's pencil suddenly stopped in mid-air, and he interrupted: "Is the contradiction provided by the Frey curve sufficient to support a general proof?"

"Of course."

In the forty-seventh minute, when Lin Ran introduced the action of the Hecke algebra of automorphic forms on the Galois group, a light sound of coffee cups colliding with saucers came from the back row. Mathematicians kept entering from the side door quietly.

André Wei Yi recalled his correspondence with a friend three months ago, which happened to contain a conjecture about 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 wider range of applications.

That is, what many mathematicians have always hoped to find: that there exists a profound and precise correspondence between different fields of mathematics.

Such mappings should exist widely."

The mathematicians present who were working in number theory had necks so stiff they didn't dare to turn, for fear of missing even a tiny bit of content.

A big shot spanning multiple fields wrote rapidly in his notebook: "When the Fermat Conjecture is transformed into a proposition about the symmetry of L-functions, it finds a path for the future development of mathematics."

When Grothendieck stood up, the buttons of his trench coat brushed against the seat, making a sound: "I need to verify the compatibility at the level of cohomology sheaves."

He quickly sketched out the commutative diagram of the étale cohomology group on the blackboard. "If such a functorial correspondence exists, then algebraic geometry will obtain a coordinate card to enter the field of automorphic forms."

At noon, all the mathematicians, even in the gaps in the cafeteria, hoped to surround Lin Ran and discuss further theories regarding the proof of the Fermat Conjecture.

However, most mathematicians did not have this opportunity; they couldn't squeeze out any of the other three people at the table with Lin Ran.

The Pope of Algebraic Geometry, Grothendieck; the chair of the Columbia University mathematics department, Ralph Fox; and the chair of the University of Göttingen mathematics department, Hans-Hermann Schwartz.

Schwartz only became the chair of the University of Göttingen mathematics department in 1958, and it was only by attending this academic lecture that he learned that a student from his own school had proven the Fermat Conjecture.

Regret, truly regret.

The University of Göttingen after the war was far from the grand occasion of being a mathematical holy land in the past; now there were only a few small fry.

It was completely different from the past, when there were Gauss, Li Man, and Hilbert, with at least one top mathematician of the time in every generation.

And Lin Ran had the hope of being on par with the three above, yet their University of Göttingen had failed to keep such a pearl, letting Columbia University pick up the leak.

By three o'clock in the afternoon, the sunlight slanted into the lecture hall, and dust floated in front of the blackboard like discrete mathematical symbols.

When Lin Ran began to deal with the constraints of the inversion theorem on non-congruence subgroups, Wei Yi raised the preprint of the paper, which was densely marked: "Is there a trick of choosing primes in the derivation of Section 4.2? I need to confirm whether the traversal of the Schwartz space is thorough enough."

"This is precisely the essence of using the Witt cancellation theorem." Lin Ran projected the numerical calculation 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 drew a five-dimensional manifold diagram in his notebook and suddenly whispered to Atiyah next to him: "Can this idea be generalized to the classification of differential structures of four-dimensional manifolds?"

The discussion gradually rose like a diffusing topological vortex, until Lin Ran tapped his pointer to refocus everyone's visual attention on the blackboard: "Does the finiteness of the Selmer group play a controlling role here similar to that in the Li Man Hypothesis?"

The entire academic conference lasted for a full half-month.

The final questioning came from Grothendieck, who felt that the scope of the correspondence between elliptic curves and modular forms remained to be discussed.

Lin Ran displayed the ultimate weapon he had prepared specifically for this time: the mathematical framework after the local correspondence of the Langlands Program was globalized.

The unfolded vertical blackboard displayed a new mathematical map born from the proof of the Fermat Conjecture; this was something prepared specifically before the start of the entire conference, pointing out to the mathematicians present what could be done in the future.

Among them, the intersection zone of modular forms and algebraic geometry was marked as "the highway corresponding to different fields."

Grothendieck was still leaning against the wall modifying his notes when the meeting adjourned, and the question note left by Wei Yi was folded by Lin Ran into the Fermat work reprint specifically given to him by Hans-Hermann Schwartz.

At the end of the corridor, Fox gazed at the Hudson River outside the window, the ripples on the river surface just like the vibration spectrum of infinite-dimensional automorphic representations.

Everyone suddenly realized that the history of mathematics had split into two segments at this moment: one ended at the period of Fermat's Theorem, and the other began with the infinite possibilities of reorganizing mathematics with new concepts.

"Randolph, congratulations, you have found the first cornerstone of unified mathematics." ===== CHAPTER 20 =====