Chapter 70: Master-level Figure
Mathematics requires seminars, requires an academic atmosphere, and requires guidance from masters. One very important reason for this is exactly that.
Some cutting-edge papers, even if they don’t write “easy to prove” or “easy to obtain,” and give you the complete proof process written clearly and ingeniously, most mathematicians will still feel baffled when reading them.
“Holy crap, how did he think of this?”
No need for particularly cutting-edge papers; even for high school math problems that are slightly difficult, just looking at the detailed solution will make you marvel at what the thinking process behind it was like.
Not to mention the most cutting-edge theories.
Therefore, the exchange content that Lin Ran brought out still had a lot of substance, and everyone’s attention immediately shifted from the earlier gossip to the content that Lin Ran was about to discuss.
Just as he said, the mathematicians present had all prepared in advance and had carefully and repeatedly studied in depth the paper he had published not long ago, fully understanding that linear form in logarithms theory could be applied to a great many number theory problems.
So everyone was also eager to know how Lin Ran came up with this theory, as it might help them apply the theory to solve other number theory problems.
“Everyone knows that besides mathematics, I am also pursuing a PhD in philosophy with Professor Horkheimer, researching his critical theory, which includes his tools critical theory.
The tasks he assigned me were quite heavy. Critical theory pursues thinking that transcends the existing social structure, so while thinking about the Diophantine problem, I was also thinking: since the concept of transcendental number exists, could we transcend the existing mathematical structure? Find a way to break free from the shackles of existing algebraic equations?
With such doubts, I thought of the Gelfond-Schneider theorem, respectively proved by Alexander Gelfond and Theodore Schneider in 1934, as the solution to Hilbert’s seventh problem. This is a theorem that almost every Göttingen mathematician must know.”
It’s just that Professor Seagull returned to Göttingen.
If he were sitting in the audience, he would probably doubt his life: Kid, you know the Göttingen school so well—did you really stay in Göttingen, or have I just gotten old and forgotten?
Lin Ran erased the linear form in logarithms theory and began writing the Gelfond-Schneider theorem:
“Everyone can see that these two mathematicians used the auxiliary function method when proving this theorem.
They constructed a function with a high-order zero at a specific point, derived a contradiction through analysis of its growth properties, and proved that Λ is non-zero.
However, these achievements are limited to linear forms of two logarithms.
So, could I find a way to generalize this method, expanding it from a single form to a broader range to handle more general linear combinations of multiple logarithms?
At the time, I only had a vague idea that the core method of the Gelfond-Schneider theorem could definitely be extended to the case of multiple logarithms.
So at this point, I was looking for how to construct this auxiliary function so that it could have high-order zeros at multiple points related to log αi and maintain controllable growth.
Generalizing from a single variable to multiple variables would definitely involve more complex tools.
Therefore, I thought of multivariable interpolation techniques. In Gelfond-Schneider’s work, the auxiliary function was single-variable, but in my work, I needed more complex tools.
At this point, multivariable complex analysis and interpolation theory from algebraic geometry seemed perfectly suitable, and if Siegel’s lemma was added, it would be perfect!”
The entire seminar was originally arranged with two topics: the first session given to Lin Ran, and the second session for Harvey Cohen to talk about his latest discovery.
In the end, all the time was taken by Lin Ran. Everyone discussed the linear form in logarithms theory for a full half day, leaving no time at all for Harvey Cohen.
Of course, there was also no time left for Chen Jingrun; from beginning to end, he never found an opportunity to be alone with Lin Ran.
Only during casual chat at dinner in the evening did they exchange a couple of sentences.
“Dehui, long time no see,” Lin Ran said.
Chen Jingrun was somewhat reserved: “Professor, happy new year.”
Lin Ran didn’t say much more, but turned to Harvey Cohen and said: “Professor Cohen, Chen is my student from Hong Kong. Originally, I planned to mentor him personally, but as you know, I might go to the White House for a position.
I won’t have much time to teach him, so I’m handing him over to you.
Chen’s talent is good; I believe his talent in number theory is no less than Shiing-Shen Chern’s.”
This evaluation was already extremely high.
Shiing-Shen Chern had completed his most important work fifteen years earlier, proving the high-dimensional Gauss-Bonnet formula.
As for Chen Jingrun, let alone in America, even in China, Chen Jingrun was just an unknown nobody.
Harvey Cohen didn’t doubt it: “Chen has great talent. During the interview, his understanding and insights on the Goldbach conjecture were deeper than mine.”
A typical PhD interview requires you to talk about your academic direction, which aspects of problems you’re interested in, and your ideas on the direction you want to pursue.
As a member of the former Chinese Academy of Sciences number theory seminar (Goldbach conjecture), Chen Jingrun naturally chose the Goldbach conjecture.
“Maybe he really can solve the Goldbach conjecture,” Lin Ran said half-jokingly and half-seriously.
After the banquet ended, Harvey Cohen specifically kept Chen Jingrun behind to chat alone:
“Chen, so, after listening to Professor Lin’s lecture today, what are your thoughts?”
After thinking for a moment, Chen Jingrun replied: “Very brilliant, it gave me a lot of inspiration.
Professor Lin very well demonstrated to us the thinking process from intuition to systematic theory, which is the most valuable for mathematicians.
Starting from special achievements like those of Gelfond-Schneider and others, through profound understanding of the auxiliary function method they used, creatively thinking to combine multivariable interpolation, complex analysis, and algebraic tools, gradually generalizing to the general case.
Including drawing inspiration from key problems in transcendental numbers and Diophantine approximation, innovatively constructing an auxiliary function suitable for linear combinations of multiple logarithms.
Deriving a lower bound for Λ through growth estimates and proof by contradiction.
I feel that Professor Lin has master-level attainments in analysis, algebra, and geometry; his ability to cleverly combine these methods is too difficult.”
Harvey Cohen added: “He even embodied Professor Horkheimer’s philosophical thought.
Lin is a master-level figure, absolutely not just a master in the number theory field.
So what I want to tell you is: Lin said you might solve the Goldbach conjecture, that your talent can match Shiing-Shen Chern’s—I don’t deny that you indeed have outstanding talent, but what you’ve learned in the past is too narrow, do you understand?
The theories and methods you know are too limited. If we only stay within the number theory field, or even classical number theory, it’s hard for us to produce valuable achievements.
Take Lin as an example: if he only knew number theory, would he have realized to use knowledge from multivariable complex analysis and algebraic geometry?
So my arrangement for you is that you need to catch up first, fill in the gaps in other fields—number theory is absolutely not just number theory alone.
Talent is talent; whether it can be realized is the key.”
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