Technology invades the modern world

Chapter 215 Lin Ran's Special Treatment (4k)

"When Professor Harvey Cohen first asked me to give academic lectures, he said I could talk about whatever I wanted."

If you haven't made any mathematical breakthroughs recently, you could share your insights with young mathematicians.

After all, the theory about birds and frogs seemed to be quite popular before.

But I'm not being modest. I'm a young mathematician with limited experience. It would be more appropriate to talk about this when I retire from the mathematics field at the age of seventy and give my retirement speech.

So, since there was no time like the present, after hearing that Chen had made great progress in the field of Goldbach's Conjecture, I said to Harvey Cohen, "Why don't I also talk about my views on Goldbach's Conjecture?"

After all, both are problems in the subfield of prime numbers in number theory. I have some background knowledge in this area, so even if I'm on stage unexpectedly, I'm sure I'll have something to offer that will spark your thinking.

But the news quickly changed. What I originally just said was to share my thoughts, but in the process of being disseminated by the media and the public, it became a testament.

Well then, I'll prove it. I've been trying to do it for the past few days.

The strong form of Goldbach's conjecture is quite difficult. I got stuck because it involves the sum of two prime numbers, and the contribution of the minor arc is expected to be x divided by logx, while the contribution of the major arc is of a higher order. The error term is hard to control, so I had to settle for proving the weak form.

(From Terence Tao's 2012 blog post "The Heuristic Limitations of the Circle Method," which discusses in detail the fundamental reason why Goldbach's Conjecture cannot evolve from a weak form to a strong form.)
The audience erupted in uproar. "Is this even human language?"

The weak form conjecture itself is a top number theory problem that has plagued the mathematical community for over two hundred years.

“That’s Randolph for you, that’s his style,” Siegel remarked. “He always manages to explain how to solve a very important problem so casually.”

Seated in the front row alongside Siegel were, to his left, the head of the Columbia University Mathematics Department, Fox, and to his right, Grothendieck. Further to the sides were figures like Harvey Cohen and Andrew Weil.

Jean-Pierre did not come from Paris, but he sent a student, instructing the student to record Lin Ran's testimony in its entirety and send it back to Paris by fax as soon as possible.

He organized a seminar for number theory professors and doctoral students at the École Normale Supérieure in Paris, telling them not to take a holiday and to wait for the proof to come up before holding a workshop to study and research it.

Grothendieck's lips twitched slightly after hearing this: "Siegel, don't you think Randolph is doing the work of a frog, just like he's talking about birds and frogs?"

What I mean is, he can do both bird and frog, and he's top-notch at both, but it seems like he's been stuck doing frog all these years, and apart from the Randolph Program, he hasn't done any bird-like work.

Rather than solving specific problems, I would prefer to see Randolph unify the different branches of mathematics in one way.

While writing "Algebraic Geometry" recently, I have become increasingly aware of its infinite mysteries. I am already satisfied if I can achieve a good unification of algebra and geometry in my lifetime, as I envisioned.

But Randolph was different from us. Firstly, he was still young, and secondly, his brain seemed to work better.

"If Randolph put all his energy into mathematics, I believe he could do what I couldn't." Grothendieck's voice was soft and fast, as if it were being transmitted through a distance to Siegel's ears.

For a mathematician of Grothendieck's caliber, problems like Goldbach's conjecture are certainly impressive, but he would prefer to see the development of the mathematical community.

In his view, a framework theory that can integrate mathematics from different subfields is obviously more worthwhile for mathematicians of Lin Ran's caliber to work on than a single problem.

Siegel defended Lin Ran, saying, "Alexander, Randolph is not like you or me; he only has small chunks of time to think about mathematical problems."

If he could step away from his work at NASA and focus on teaching at Columbia University, I think he would definitely do bird-related work.

Fox quickly interjected, "So, Professor Siegel, could you help persuade Randolph to focus entirely on his mathematical work?"

Anyone could do the job at NASA, but only Randolph could unify different fields of mathematics.

Siegel shook his head, inwardly cursing, "If I could persuade him, why wouldn't I have persuaded him to come to Göttingen and stay at Columbia University?"

Small towns in Germany are no better than New York for focused research.

Princeton, the city where Princeton is located, is also a small city with only 30,000 people, even fewer than the city of Göttingen, where Göttingen is located.

For a moment, the three of them fell silent. They all knew that this was an irreconcilable conflict between reality and ideals.

This is not a problem that harmonic analysis can solve.

The mathematics community doesn't have the power to persuade the White House to release him.

On stage, Lin Ran had already given a brief introduction to the differences between the strong and weak forms of Goldbach's Conjecture.

In 1742, Goldbach proposed the following conjecture in a letter to Euler:
"Any integer greater than 2 can be written as the sum of three prime numbers."

The above differs from the current statement because Goldbach followed the convention that "1 is also a prime number." However, the mathematical community no longer considers 1 to be a prime number. Therefore, the modern statement of Goldbach's original conjecture is:
"Any integer greater than 5 can be written as the sum of three prime numbers."

This is the weak form of Goldbach's conjecture.

In his reply, Euler suggested that this conjecture could have another equivalent version:

"Any even number greater than 2 can be written as the sum of two prime numbers."

He regarded this conjecture as a theorem, but Euler himself could not prove it.

The conjecture commonly known to the public in later generations is actually Euler's version, which is also a strong form of Goldbach's conjecture.

The strong form should be called the Goldbach-Euler conjecture.

In fact, these two conjectures are not equivalent.

Or perhaps they are equivalent, but a path to make them equivalent can only be found after another theorem is proven.

"It seems like this has been going on for a long time, so let's be more specific and start from Ivan Vinogradov's work in 1937."

Ivan Vinogradov was a Soviet mathematician, but not Alexander Vinogradov or Askold Vinogradov, although both of them are also very famous.

These names are indeed easy to confuse, even though they are not the same person.

Ivan mainly proposed a technique for estimating the sum of prime numbers. The prototype of the bilinear sieve method that has been used in the Goldbach Conjecture is this method, and mathematicians have continued to improve upon it.

Clearly, Chen had already used this method to its fullest extent in the previous work.

It's virtually impossible to solve weak forms using this method now. Therefore, we need to introduce new tools, especially for optimization on minor arcs. We need to improve the large sieve method by removing its extraneous factors to make its estimation more accurate.

More importantly, we cannot simply use the concepts from analytic number theory; we must incorporate elements of algebraic geometry, constructing prime sums through geometric structures and embedding the problem into algebraic varieties.

The mathematicians standing at the back of the stage had already stood up.

Because the combination of algebraic geometry and number theory is undoubtedly the most cutting-edge mathematical content at present, so cutting-edge that no one except Lin Ran has done it.

As mentioned earlier, the weak form of Goldbach's conjecture was proven by Helfgett, a mathematician from Peru who graduated from Princeton University.

But why is his work not well known to the outside world? The weak form of Goldbach's conjecture is also remarkable.

On the one hand, because the paper has not yet been published, after he iterated three versions, everyone thought it was probably correct, but no expert had come out to definitively say that it was definitely correct. His proof required computer-aided proof.

Secondly, Ivan Vinogradov proved in 1937 that all sufficiently large odd numbers are the sum of three prime numbers. Helfgett's contribution, however, only bridged the gap between "sufficiently large" and "all numbers".

Ivan Vinogradov's proof introduced a completely new concept of bilinear forms, which Helfgett did not. He contributed to a specific subfield of analytic number theory related to explicit estimation, but not to the larger field.

In summary, Helfgett's work lacked innovation.

Lin Ran's actions were by no means simply a matter of copying.

Simply copying is useless. If you directly use Helfgate's results, in this era, computers simply cannot verify them for you.

The audience was full of mathematicians, including some of the top mathematicians of our time. No one would accept Helfgott's results at all.

This is a fundamental improvement made by Lin Ran based on Helfgett's work. Even if we take it to the 2020 timeline, if Lin Ran were a Princeton graduate, this would be an achievement worthy of the Fields Medal.

Lin Ran needs to improve Helfgett's results so that they can be verified without a computer.

Lin Ran's approach is to introduce the content of algebraic geometry and use this method to build a bridge to construct a geometric model of prime numbers.

This is a completely new approach, and in the present day, it is an echo of the Randolph Programme.

During the lunch break, Lin Ran went to the front row and was surrounded by mathematicians.

Grothendieck stated frankly, "Randolph, I know that spaceflight is a great undertaking."

But compared to mathematics, it seems so insignificant.

I'm not saying it's unimportant, but rather that it's not important enough for a master like you to undertake such a task.

Such secondary work should be done by mathematicians specializing in applied mathematics.

Lin Ran felt a little awkward because he originally worked in artificial intelligence, and among these top mathematics experts, he was probably several levels below applied mathematics in terms of the hierarchy of contempt.

Fortunately, he is now a pure Brahmin, and the most powerful one among them.

If I were to return to the timeline of 2020, all I would need to do is post this paper on the improved form of Helfgate on Arxiv, and becoming a pure number Brahmin would be a piece of cake.

Lin Ran smiled and said, "Mathematics is a carnival of the spiritual world, while aerospace is a dazzling firework of the material world. For me, I want both."

Professor Alexander, you know, the same thing done by a genius and an ordinary person can produce drastically different results.

Grothendieck remained silent.

He sighed, "Sigh, Randolph, if you weren't doing spaceflight, but some other job, like the backstabbing in the White House, I would advise you to quit."

Okay, to be honest, the combination of number theory and algebraic geometry started with Gauss linking integer solutions and solutions to homogeneous polynomial equations, and later with the Kronecker-Weber theorem and divider theory attempting to manipulate number theory using polynomial ring quotients over integers.

Richard Dedekind and Heinrich Weber applied algebraic methods to Riemann surfaces, establishing an analogy between number fields and function fields, and providing an algebraic proof of the Riemann-Roch theorem.

Then, Andrew Weil, Jean Pierre, and I systematically combined number theory and algebraic geometry, extending it to the study of rational points, number fields, and function fields.

And you have helped us expand those boundaries once again.

First, in the process of proving Fermat's Last Theorem, the modular theorem was used to connect elliptic curves and modular forms. Then, the Randolph program conjectured the connection between Galois representations and automorphic forms. And now, geometric modeling has been applied to the prime number theorem.

I always felt that we were just one tiny spark of inspiration away from completely integrating number theory into the framework of algebraic geometry.

Randolph, if you have any inspiration during the rocket launch, feel free to write to me anytime and tell me your inspiration. I'll help you verify it.

Grothendieck said this himself, "You have an idea, and I'll think along those lines," which almost sounded like he was offering to be Lin Ran's assistant.

Lin Ran nodded and said, "Okay, Alexander, I'll write to you whenever I have an inspiration."

Siegel added, "Randolph, the beginning is fine. I even regret that I'm getting old. I have a lot of inspiration, but I can't continue to think deeply."

These sparks of inspiration had to be left to quietly extinguish themselves.

My notes are still in Göttingen, but if you want them, just ask Doiling anytime. If Doiling has also retired, you can contact the head of the mathematics department in Göttingen. Whoever it is, they will give you my original manuscript.

The reason I'm not giving it to you now is because you're still at NASA. Once you leave NASA, all my manuscripts will be yours.

Siegel had wanted to say that last time, but he forgot.

Lin Ran thought to himself, "I still need to be good enough."

It doesn't matter if you haven't personally taught me, it doesn't matter if you're Chinese, and it doesn't even matter if you're not in Göttingen.

If you're good enough, the big shots will naturally leave the manuscript to you and make you their successor.

　To elaborate further, aside from the earliest Fermat's Last Theorem, later works, whether it's the twin prime conjecture or the current Goldbach conjecture, are not direct copies of later researchers' work. The former didn't offer any work for you to copy, as it hasn't been fully proven yet. As for the latter, even if you copied it in the 60s, it wouldn't have been useful, since it was ultimately unusable.

　　
　
(End of this chapter)