Chapter 18: The First Cornerstone Of The Unification Of Mathematics
On the morning of January 31, 1960, the Columbia University Mathematics Department report hall was shrouded in the thin mist of New York’s winter.
Lin Ran stood at the podium waiting for the arrival of mathematicians from around the world.
Columbia University’s president Rothes personally supported him.
Such a grand event in the mathematics community, once proven, would crown Columbia University with solving a conjecture that had puzzled the mathematics community for centuries; with talent like Randolph Lin in the Mathematics Department, surpassing Princeton and Harvard in the field of mathematics was entirely possible.
Just thinking about mathematics overpowering the old rival stirred Rothes’s heart.
He had even planned that if this academic report gained unanimous recognition from the mathematicians, the old president must be invited to the subsequent celebration banquet.
The old president had a resounding name in Washington: Eisenhower.
After retiring from the army at the end of his military career, many companies hoped to invite Eisenhower to serve as CEO or chairman, but he ultimately chose to accept Columbia University’s offer, served for four years, and then returned to Washington.
Once the mathematicians in the audience were seated one after another, Grothendieck was right in the center of the first row.
He had just rushed over from Paris, and all the mathematicians voluntarily gave him the best seat.
Andrew Wiles was using a red and blue pencil to mark notes along the edges of the manuscript; Grothendieck was quietly discussing something with the accompanying Serre, and the black leather notebook was already open to page seventeen.
When the projection screen displayed Fermat’s Equation, the subtle murmurs throughout the hall abruptly stopped. Lin Ran pointed the chalk at the modular space parameter of the elliptic curve: “Assuming an integer solution (a, b, c) exists, the corresponding Frey Curve will lead to 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 evade 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. “Through 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 rule out the possibility of a solution.”
Wiles’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.”
At the forty-seventh minute, when Lin Ran introduced the Hecke algebra of automorphic forms acting on the Galois group, new mathematicians kept quietly entering from the side door in the back rows.
Andrew Wiles recalled a letter with a friend three months ago, which happened to include a conjecture about the correspondence between automorphic representations and the Galois group.
“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 the modular curve. “And I believe this bridge has a broader range of applications.
That is, the profound and precise correspondence between different fields of mathematics that many mathematicians have always hoped to find.
Such mappings should exist widely.”
The number theorists present had stiff necks but didn’t dare turn, fearing to miss even a bit of content.
Big shots spanning multiple fields rapidly wrote in their notebooks: “When Fermat’s Conjecture is transformed into a symmetry proposition about L-functions, it paves a path for the future development of mathematics.”
Grothendieck stood up, expressing hope for deeper thinking on the content on the blackboard: “I need to verify compatibility at the étale cohomology level.”
He quickly sketched the commutative diagram of the étale cohomology group on the blackboard. “If such a functorial correspondence exists, algebraic geometry will gain a coordinate card to enter the realm of automorphic forms.”
At noon, all mathematicians, even in the gaps in the canteen, hoped to gather around Lin Ran and discuss further theory on the proof of Fermat’s Conjecture with him.
However, most mathematicians didn’t have this opportunity; the other three people at the same table as Lin Ran were ones none of them could squeeze out.
The algebraic geometry pope Grothendieck, Columbia Mathematics Department Director Ralph Fox, and University of Göttingen Mathematics Department Director Hans Hermann Schwarz.
Schwarz had only become the University of Göttingen Mathematics Department Director in 1958, and it was only by attending this academic report that he learned his own student had proven Fermat’s Conjecture.
Regret—he truly regretted it.
The University of Göttingen after the war was far from its former glory as the mathematical holy land, now with just a few small shrimps.
Unlike the past with Gauss, Riemann, and Hilbert, where each generation had at least one top-tier mathematician of the time.
And Lin Ran had the potential to rival those three, yet such a hidden gem was not retained by their University of Göttingen and was picked up by Columbia University.
At three in the afternoon, sunlight slanted into the report hall, dust suspended before the blackboard like discrete mathematical symbols.
When Lin Ran began addressing the restrictive conditions of the inversion theorem on non-congruent subgroups, Wiles raised the densely annotated paper preprint: “Does the derivation in Section 4.2 involve a trick with choosing primes? I need to confirm if the traversal of the Schwartz space is sufficiently thorough.”
“This is precisely the essence of utilizing the Witt elimination theorem.” Lin Ran projected the numerical calculation results. “When the modular degree of the elliptic curve exceeds a certain threshold, its corresponding modular form must be cuspidal.”
Milnor from Princeton sketched a five-dimensional manifold diagram in his notebook and suddenly whispered to the neighboring Atiyah: “Can this idea be generalized to the classification of differential structures on four-dimensional manifolds?”
Discussion voices gradually rose like a diffusing topological vortex, until Lin Ran lightly tapped the chalk 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 Riemann Hypothesis?”
The entire academic conference lasted a full half month.
The final doubt came from Grothendieck, who felt the scope of applicability of the correspondence between elliptic curves and modular forms still needed discussion.
Lin Ran displayed the ultimate weapon specially prepared for this occasion: the mathematical framework of the globalized local correspondence relations from the Langlands Program.
The pulled-out vertical blackboard displayed the new mathematical map spawned by the proof of Fermat’s Conjecture, something specially prepared before the entire conference began, pointing out future content that the mathematicians present could work on.
Among them, the intersection zone of modular forms and algebraic geometry was marked as “the highway of correspondences between different fields.”
At the end of the session, Grothendieck still leaned against the wall revising his notes; Wiles’s proactively left question note was folded by Lin Ran into the Fermat work replicate edition specially given to him by Hans Hermann Schwarz.
Fox at the end of the corridor gazed at the Hudson River outside the window, the ripples on the river surface as if revealing the vibration spectrum of infinite-dimensional automorphic representations.
Everyone suddenly realized that the history of mathematics split into two segments at this moment: one ending with the period of Fermat’s Theorem, the other beginning with the infinite possibilities of reorganizing mathematics with new concepts.
“Randolph, you have found the first cornerstone for the unification of mathematics.”
