Chapter 215: Lin Ran’s Special Treatment
“When Professor Harvey first asked me to give an academic lecture, he said I could talk about whatever I wanted.
If there were no recent mathematical achievements, it would be fine to share insights and experiences with young mathematicians.
After all, the previous Birds and Frogs theory seemed quite popular with everyone.
But I said I wasn’t being modest; I myself am a young mathematician with limited experience. It would be more appropriate to discuss this when I turn seventy and retire from the mathematics community during my retirement speech.
So, rather than pick another day, upon hearing that Chen had made great achievements in the Goldbach Conjecture field, I told Harvey that I might as well share my thoughts on the Goldbach Conjecture.
After all, both are problems in the prime number subfield of the number theory field, and I have some accumulation. Even if I went on stage temporarily, I wouldn’t lack content to provoke everyone’s thinking.
But the news quickly changed; I originally just said to talk about thinking, but in the media and dissemination process, it turned into a proof.
So I said proof, then proof it is. I’ve been trying it these past few days.
The strong form of the Goldbach Conjecture is a bit difficult. During my thinking process, I kept getting stuck on the strong conjecture involving the sum of two primes, where the secondary arcs’ contribution is estimated as x over log x term, while the main arc’s contribution needs to be of higher order than this, and the error term is hard to control. So I had to settle for the next best thing and prove the weak form.”
(From Terence Tao’s 2012 blog article “Heuristic Limitations of the Circle Method,” which discusses in detail: the fundamental reason why the Goldbach Conjecture cannot evolve from the weak form conjecture to the strong form conjecture)
The audience erupted in an uproar. Is this even human language?
Because the weak form conjecture itself is a top number theory problem that has plagued the mathematics community for over two hundred years.
“As expected of Randolph; this is his style,” Seagull sighed. “He always casually explains how to solve a very important problem.”
Sitting in the front row with Seagull were, to the left, Fox, director of Columbia University Mathematics Department, and to the right, Grothendieck. Further out were Harvey, Andrew Wiles, and others.
Jean Pierre in Paris didn’t come, but he sent a student, instructing the student to fully record Lin Ran’s proof process and fax it back to Paris immediately.
He organized professors and PhDs doing number theory at Paris Normal School; everyone should not take vacation, and once the proof arrives, they would first hold a seminar to study and research it.
After listening, Grothendieck twitched the corner of his mouth: “Seagull, don’t you think that when Randolph talks about Birds and Frogs, what he does is all Frogs work?
I mean, he can do both Birds and Frogs at the same time, and he does both at the top level, but these years he seems to have stayed doing Frogs. Apart from the Randolph Program, he hasn’t done any Birds-like work.
Compared to solving specific problems themselves, I still want to see Randolph unify different branches of mathematics in some way.
Recently while writing “Algebraic Geometry,” I’ve increasingly felt the infinite mysteries within it. If I can, in my lifetime, achieve a good unification of algebra and geometry as envisioned, I’d be satisfied.
But Randolph is different from us; first, he’s still young, and second, his brain seems even better.
If Randolph put all his energy into mathematics, I believe he could achieve what I cannot.” Grothendieck’s voice was very soft, his speech rate very fast, and his voice floated into Seagull’s ears like a transmission from afar.
For a mathematician of Grothendieck’s level, a problem like the Goldbach Conjecture is certainly impressive enough, but he hopes more to see the development of the mathematics community.
In his view, a framework theory capable of integrating different subfields of mathematics is obviously more worthy work for a mathematician of Lin Ran’s level than a single problem.
Seagull defended Lin Ran: “Alexander, Randolph is not like you and me; he only has small chunks of time to think about mathematical problems.
If he could detach from his NASA work and teach peacefully at Columbia University, I think he would definitely do Birds work.”
Fox quickly said from the side: “So, Professor Seagull, can you help persuade Randolph to focus entirely on mathematical work?
Anyone can do NASA work, but unifying different fields of mathematics—only Randolph can do that.”
Seagull shook his head, inwardly ranting hard: If I could persuade him, why wouldn’t I have advised him to come to Göttingen instead of your Columbia University?
Germany’s small city is no less suitable for undisturbed research than New York.
Princeton is in Princeton City, also a small city of only thirty thousand people, with even fewer people than Göttingen City where Göttingen is located.
For a moment, the three didn’t speak again; everyone knew this was an irreconcilable contradiction between reality and ideal.
This is not a problem that harmonic analysis can solve.
The mathematics community doesn’t have the energy to convince the White House to let him go.
On stage, Lin Ran had briefly introduced the distinction between the strong and weak forms of the Goldbach 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 primes.”
The above differs from the modern statement because Goldbach at the time followed the convention that ‘1 is also a prime.’ Now the mathematics community no longer considers 1 a prime, so the modern statement of Goldbach’s original conjecture is:
“Any integer greater than 5 can be written as the sum of three primes.”
This is the weak form of the Goldbach Conjecture.
Euler, in his reply, believed this conjecture had an equivalent version:
“Any even number greater than 2 can be written as the sum of two primes.”
And regarded this conjecture as a theorem, though Euler himself could not prove it.
The version commonly known to the public later is actually Euler’s version, which is also the strong form of the Goldbach Conjecture.
The strong form should more appropriately be called the Goldbach-Euler Conjecture.
In fact, these two conjectures are not equivalent.
Or rather, perhaps they are equivalent, but it would require proving another theorem to find a path equating the two.
“Talking about this has taken a bit long; let’s be more specific, starting from Ivan Vinogradov’s work in 1937.
Ivan Vinogradov was a Soviet mathematician, but not Alexander Vinogradov nor Askold Vinogradov, though those two are also famous.
These names are indeed easy to mix up, even though they are not the same person.
Ivan mainly proposed a technique for estimating sums of primes, and later the prototype of the bilinear form large sieve method, which everyone has been using around the Goldbach Conjecture, all stem from this method. Mathematicians have continuously improved upon this method.
Obviously, Chen’s previous work has already pushed this method to the extreme.
If we now want to solve the weak form using this method, it’s almost impossible.
So we need to introduce some new tools, especially to optimize on the secondary arcs, requiring improvements to the large sieve method, removing its extra factors to make its estimates more precise.
More importantly, we cannot merely use content from analytic number theory; we need to incorporate algebraic geometry content, using geometric structures to construct sums of primes and embed the problem into algebraic varieties.”
The mathematicians standing at the back of the audience had all 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 is doing it.
As mentioned earlier, the weak form of the Goldbach Conjecture was proved by Helffgott, a mathematician from Peru who graduated from Princeton.
But why is his work not well-known outside, when the weak form Goldbach Conjecture is also remarkable.
On one hand, because the paper hasn’t been published yet; after three versions of iteration, everyone thinks it’s probably correct, but no big shot has come out to definitively confirm it. His proof requires computer-assisted proof.
On the other hand, because Ivan Vinogradov proved in 1937 that all sufficiently large odd numbers are the sum of three primes. Helffgott’s contribution only closes the gap between sufficiently large and all numbers.
Ivan Vinogradov’s proof introduced the new concept of bilinear forms; Helffgott did not. He contributed to a specific subfield of analytic number theory related to explicit estimates, but not to larger fields.
In summary, Helffgott’s work lacks innovation.
And Lin Ran is absolutely not simply copying.
Simple copying is useless; if you directly use Helffgott’s achievements, in this era, computers fundamentally can’t verify it for you.
The audience is all mathematicians, the top contemporary mathematicians are all in the audience; no one would accept Helffgott’s result.
This is a fundamental improvement Lin Ran made on the basis of Helffgott; even if taken to 2020 spacetime, if Lin Ran were from Princeton, this would be Fields Medal-worthy achievement.
Lin Ran needs to improve Helffgott’s result to the point where no computer is needed for verification.
Lin Ran’s approach is to introduce algebraic geometry content, using this method to build a bridge for geometric modeling of primes.
This is a brand-new method, and in the present, it echoes the Randolph Program.
During the lunch break, Lin Ran came to the front row and was surrounded by the mathematicians.
Grothendieck said straightforwardly: “Randolph, I know aerospace is great, a great cause.
But compared to mathematics, it seems so insignificant.
I’m not saying it’s unimportant, but it’s not important enough to warrant a master-level figure like you doing it.
Such second-tier work should be left to those mathematicians who study applied mathematics.”
Lin Ran felt a bit awkward inwardly, because he himself originally did artificial intelligence, and in this group of top math big shots, the contempt chain might be several levels below applied mathematics.
But fortunately, he is now a pure mathematics Brahmin, and the most badass one among these pure mathematics Brahmins.
Back in 2020 spacetime, he just needs to post this improved version paper on Helffgott on Arxiv, and being a pure mathematics Brahmin would be effortless.
Lin Ran smiled: “Mathematics is the revelry of the spiritual world, while aerospace is the brilliant fireworks of the material world. For me, I want both.
Professor Alexander, you know, the same thing done by a genius and an ordinary person has completely different effects.”
Grothendieck fell silent.
He sighed: “Sigh, Randolph, if what you did wasn’t aerospace, but some other work, like scheming in the White House, I’d do everything to persuade you not to do it.
Alright, to be honest, the combination of number theory and algebraic geometry, from Gauss earliest connecting integer solutions of homogeneous polynomial equations with ideals, to later the Kronecker-Weber theorem and ideal theory trying to operate number theory using quotient rings of polynomial rings over integers.
Then to Richard Dedekind and Heinrich Webber applying algebraic methods to Riemann surfaces, establishing the analogy between number fields and function fields, providing an algebraic proof of the Riemann-Roch theorem.
And then to Andrew Wiles, Jean Pierre, and me systematically combining number theory and algebraic geometry, extending it to rational points research, number fields and function fields.
And you have helped us expand this boundary again.
First, in the Fermat’s Conjecture proof process, using the modularity theorem to connect elliptic curves and modular forms, then the Randolph Program conjecturing the connection between Galois representations and automorphic forms, and now applying geometric modeling to the Prime Number Theorem.
I always feel that we’re just a tiny bit short, a little inspiration, and we can fully incorporate number theory into the algebraic geometry framework.
Randolph, if you get inspiration during launching rockets, you can write to me anytime, tell me your inspiration, and I’ll help you verify it.”
Grothendieck said this personally: you have the idea, I’ll think along your lines—this even has a bit of the meaning of actively being Lin Ran’s assistant.
Lin Ran nodded: “Okay, Alexander, I’ll write to you anytime I have inspiration.”
Seagull added: “Randolph, the earlier part is no problem; I even regret that I’m old, with many inspirations but unable to do deep thinking anymore.
These sparks of inspiration have to quietly stay there and slowly extinguish.
My notes are left in Göttingen, but if you want them, just tell Döblin anytime. If Döblin has also retired, you can contact the director of Göttingen Mathematics Department; whoever it is, they will give you the original manuscripts.
The reason I’m not giving them to you now is because you’re still at NASA. Once you leave NASA, my manuscripts are yours.”
Seagull had wanted to say this last time but forgot.
Lin Ran thought, you still need to be badass enough.
Not having taught personally doesn’t matter, being Chinese doesn’t matter, even not being in Göttingen doesn’t matter.
If you’re badass enough, big shots will naturally proactively leave their manuscripts to you, making you the heir.
