Chapter 71: Master-level Figure (Please keep reading)

Mathematics needs seminars, needs an academic atmosphere, and needs guidance from masters—this is a very important reason for this.

For some cutting-edge papers, even if the author doesn't write "easily proven" or "easily obtained" and writes the complete proof process clearly and exquisitely, the vast majority of mathematicians will still feel confused when reading them.

"Holy crap, how could he think of this?"

You don't even need overly cutting-edge papers; just high school math problems, if they are slightly difficult, you will sigh even if you only look at the detailed answers: what was the thought process behind this?

Not to mention the most cutting-edge theories.

Therefore, the exchange content Lin Ran brought out still had a lot of substance, and everyone's attention immediately shifted from the gossip just now to the content Lin Ran was about to tell.

Just as he said, the mathematicians present had all done preparations in advance and had carefully and repeatedly read the paper he had just published not long ago, knowing very well that the linear form logarithm theory could be applied to a very large number of number theory problems.

So everyone also urgently hoped to know how Lin Ran thought of this theory; this might be helpful for them to apply this theory to solve other number theory problems.

"Everyone knows that besides mathematics, I am also studying for a philosophy PhD with Professor Horkheimer, researching his critical theory, which includes his tool critical theory.

The task he assigned me is still very heavy. Critical theory pursues that thinking must transcend existing social structures, so when I was thinking about Diophantine problems, I was also thinking: since concepts like transcendental numbers exist, can we transcend existing mathematical structures? Find a way to get rid of the shackles of existing algebraic equations?

Carrying such a question, I thought of the Gel'fond-Schneider theorem proven by Alexander Gelfond and Theodor Schneider in 1934 respectively; as a solution method for Hilbert's seventh problem, this is almost a theorem that every Göttingen mathematician must know."

It's just that Professor Siegel returned to Göttingen.

If he were sitting in the audience, he would probably doubt life: you kid know the Göttingen school so well, did you really stay in Göttingen, and I'm just old and forgot?

Lin Ran erased the linear form logarithm theory and then began to write the Gel'fond-Schneider theorem:

"Everyone can see that these two mathematicians used the auxiliary function method when proving this theorem.

They deduced a contradiction by constructing a function with high-order zeros at specific points and analyzing its growth properties, proving that Λ is non-zero.

However, these achievements are limited to the linear form of two logarithms.

Then, can I find a way to generalize this method, expand it from a single form to a wider range, and handle more general multi-logarithm linear combinations?

At that time, I only had a vague idea; the core method of the Gel'fond-Schneider theorem could certainly be extended to the case of multiple logarithms.

So at this time, I was looking for how to construct this auxiliary function so that it could have high-order zeros at multiple points related to logαi and be able to maintain controllable growth.

Extending from a single variable to multiple variables certainly involves more complex tools.

Therefore, I thought of multi-variable interpolation technology. In Gel'fond-Schneider's work, the auxiliary function is single-variable, while in my work, I need to find more complex tools.

At this time, interpolation theory in multi-variable complex analysis and algebraic geometry seemed incomparably suitable, and if Siegel's Lemma were added, it would be perfect!"

The entire seminar originally arranged two topics; the first session was handed to Lin Ran, and the second session was for Harvey Cohen to talk about his latest discoveries.

As a result, all the time was used by Lin Ran, and everyone discussed around the linear form logarithm theory for a full half-day, leaving no time for Harvey Kane at all.

Of course, no time was left for Chen Jingrun either; he was never able to find a chance to be alone with Lin Ran from beginning to end.

It was only when everyone was eating together in the evening that they chatted a few words.

"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 his head and said to Harvey Cohen: "Professor Cohen, Chen is my student from Xiangjiang. I originally planned to bring him myself, but you know, I am very likely going to work in the White House.

I don't have much time to teach him either, so I'll leave him to you.

Chen's talent is not bad; I think his talent in number theory is no less than Shiing-Shen Chern's."

This evaluation was already very, very high.

Shiing-Shen Chern had already completed his most important work fifteen years ago, proving the high-dimensional Gauss-Bonnet formula.

And Chen Jingrun? Let alone in America, even in Cultural Huaguo, Chen Jingrun was just a nobody.

Harvey Cohen didn't doubt it: "Chen is very talented; during the interview, his understanding and insights into the Goldbach conjecture were deeper than mine."

A PhD interview generally requires you to talk about your academic direction, which problems you are interested in, and talk about your thoughts on the direction you want to do.

As a member of the former Cultural Huaguo Academy of Sciences number theory seminar (Goldbach conjecture), Chen Jingrun certainly 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 for a while:

"Chen, how is it? What are your thoughts after listening to Professor Lin's lecture today?"

Chen Jingrun thought for a moment and then answered: "Very wonderful, it gave me a lot of inspiration.

Professor Lin very well demonstrated to us the thought process from intuition to systematic theory, which is the most valuable thing for a mathematician.

Starting from the special achievements of Gel'fond-Schneider and others, through a profound understanding of the auxiliary function method they used, he creatively thought of combining multi-variable interpolation, complex analysis, and algebraic tools, gradually generalizing to the general case.

Including drawing inspiration from key problems in transcendental numbers and Diophantine approximation, he innovatively constructed an auxiliary function suitable for multi-logarithm linear combinations.

Deducing the lower bound of Λ through growth estimation and the contradiction method.

I feel that Professor Lin has master-level attainments in analysis, algebra, and geometry; he can skillfully combine these methods together, which is too difficult."

Harvey Cohen added: "He even embodied Professor Horkheimer's philosophical thoughts.

Lin is a master-level figure, not just a master in the field of number theory.

So what I want to tell you is, Lin said you might be able to solve the Goldbach conjecture, said your talent can be compared to Shiing-Shen Chern's—I don't deny that you indeed have extraordinary talent, but what you knew in the past was too narrow, do you understand?

The theories and methods you know are too limited. If we only limit ourselves to the field of number theory, or even classical number theory, then it is very difficult for us to make valuable achievements.

Taking Lin as an example, if he only knew number theory, would he be able to realize that he needed to use knowledge of multi-variable complex analysis and algebraic geometry?

So my arrangement for you is, you have to make up for lessons first, make up for the shortcomings in other fields; number theory is definitely not just number theory.

Talent is talent; whether it can be cashed in is the key." ===== CHAPTER 72 =====