Technology invades the modern world

Chapter 18 The First Cornerstone of Unified Mathematics

On the morning of January 1960, 1, the lecture hall of the Department of Mathematics at Columbia University was shrouded in a New York winter mist.

Lin Ran stood in front of the podium waiting for mathematicians from all over the world to arrive.

Columbia University President Ross personally supported it.

Once this grand event in the mathematical world is proven, Columbia University will be crowned with the title of having solved centuries-old conjectures in the mathematical world. With talents like Randolph Linn in the mathematics department, it is entirely possible for Columbia University to surpass Princeton and Harvard in the field of mathematics.

Roses felt excited at the thought that mathematics could surpass his old rival.

He even thought that if this academic report received unanimous approval from mathematicians, he would definitely invite the old principal to attend the subsequent celebration banquet.

The old principal has a famous name in Washington: Eisenhower.

After Eisenhower retired from the military, many companies wanted to invite him to serve as CEO or chairman, but he finally chose to accept the offer from Columbia University and returned to Washington after four years.

When the mathematicians in the audience arrived one after another, Grothendieck was sitting in the middle of the first row.

The other party had just arrived from Paris, and all the mathematicians took the initiative to give him the best positions.

Andrew Weil was making annotations on the margins of the manuscript with red and blue pencils. Grothendieck was whispering something with his companion, Seere, and his black leather notebook was open to page 17.

When the Fermat equation appeared on the projection screen, the quiet discussion in the room abruptly ceased. Lin Ran pointed with his pointer at the moduli space parameters of the elliptic curve: "Assuming there exists an integer solution (a, b, c), then the corresponding Frey curve will lead to a contradiction in the l-adic Galois representation."

Grothendieck suddenly held up his notebook. Written in German was the question: “How does the structure of the Selmer group circumvent the constraints of the Hasse principle?”

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

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

Weil's pencil suddenly stopped in mid-air, and he interrupted, "Does the contradiction provided by the Frey curve support a general proof?"

"of course."

In the 47th minute, when Lin Ran introduced the action of Hecke algebra of automorphic forms on Galois groups, new mathematicians kept quietly entering the back row through the side door.

Andrew Weil remembered a correspondence with a friend three months ago, which happened to include 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 the modular curve, "and I believe this bridge has a wider range of applications.

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

This kind of mapping should be widespread.”

The mathematicians working on number theory present had stiff necks and dared not turn their heads for fear of missing even a little bit of the content.

A multidisciplinary expert scribbled in his notebook: "When Fermat's conjecture was transformed into a symmetry proposition about L-functions, it found a path for the future development of mathematics."

Grothendieck stood up and expressed his desire to reflect more deeply on what was written on the blackboard: “I need to verify compatibility at the cohomology level.” He quickly sketched the commutative diagram of the étale cohomology group on the blackboard. “If such a functorization correspondence exists, then algebraic geometry will have a coordinate card into the realm of automorphic forms.”

At noon, all mathematicians, even during breaks in the cafeteria, wanted to gather around Lin Ran and discuss with him further theories about the proof of Fermat's conjecture.

However, most mathematicians do not have this opportunity, as they cannot squeeze out any of the other three people who can sit at the same table with Lin Ran.

Grothendieck, the Pope of algebraic geometry, Ralph Fuchs, the Chairman of the Department of Mathematics at Columbia University, and Hans Hermann Schwarz, the Chairman of the Department of Mathematics at the University of Göttingen.

Schwarz did not serve as the head of the Department of Mathematics at the University of Göttingen until 1958. It was only when he attended this academic lecture that he learned that students at the university had proved Fermat's conjecture.

I regret it. I really regret it.

After the war, the University of Göttingen was no longer as prosperous as it once was as a holy place of mathematics, and now it is only a few small ones.

This is completely different from the past when there was Gauss, Riemann and Hilbert, and each generation had at least one top mathematician of the time.

Lin Ran had the hope of being on par with the above three, but the University of Göttingen failed to keep this missed opportunity and it was picked up by Columbia University.

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

As Lin Ran began to address the restrictions on the inversion theorem on non-congruence subgroups, Weil held up the densely annotated preprint of his paper and asked: "Is there a trick in choosing prime numbers in the derivation of Section 4.2? I need to confirm whether the traversal of Schwarz space is thorough enough."

"This is the essence of using the Witter elimination theorem." Lin Ran pulled up the projection of the numerical calculation results. "When the modular degree of an elliptic curve exceeds a certain threshold, its corresponding modular form must be a sharp form."

Milnor from Princeton drew a diagram of a five-dimensional manifold in his notebook and suddenly whispered to Atiyah, who was sitting next to him, "Can this idea be extended to the classification of differential structures of four-dimensional manifolds?"

The discussion grew like a spreading topological vortex, until Lin Ran tapped his pointer to focus everyone's attention back on the blackboard. "Does the finiteness of the Selmer group play a controlling role similar to that in the Riemann hypothesis?"

The entire academic conference lasted for half a month.

The final question came from Grothendieck, who felt that the scope of application of the correspondence between elliptic curves and modular forms was still debatable.

Lin Ran demonstrated the ultimate weapon specially prepared for this time: the mathematical framework of the globalization of the local correspondence of the Langlands program.

The vertical blackboard that was pushed out displayed the new mathematical map spawned by the proof of Fermat's conjecture. This was something specially prepared before the entire meeting began, pointing out what the mathematicians present could do in the future.

The intersection of modular forms and algebraic geometry is marked as "highways corresponding to different fields."

Grothendieck was still leaning against the wall to revise his notes when the meeting ended. The note of question that Weil had left behind was folded by Lin Ran into the copy of Fermat's works specially given to him by Hans Hermann Schwarz.

At the end of the corridor, Fox gazed out the window at the Hudson River. The ripples on the river seemed to reveal the vibration spectrum of an infinite-dimensional automorphic representation.

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

"Randolph, you have found the first building block of unified mathematics."

(End of this chapter)