Technology invades the modern world

Chapter 70: Master Level Figures (Seeking Follow-up Reading)
This is one of the important reasons why mathematics requires seminars, an academic atmosphere, and guidance from masters.

Some cutting-edge papers, even if the authors do not write about how to prove or obtain something easily, and write the complete proof process clearly and exquisitely, most mathematicians will still find it confusing when reading them.

"Fuck, how could he think of this?"

You don’t need cutting-edge papers, just a slightly difficult high school math problem. Just looking at the detailed answer will make you wonder what the thinking process behind it is.

Not to mention the most cutting-edge theories.

Therefore, the content of Lin Ran's communication was still very informative, and everyone's attention was immediately shifted from the previous gossip to what Lin Ran was going to say now.

As he said, the mathematicians present had all made preparations in advance and had carefully read and reread the paper he had just published not long ago. They were very clear that the theory of linear logarithms could be applied to a large number of number theory problems.

So everyone is eager to know how Lin Ran came up with this theory, which may help them apply this theory to solve other number theory problems.

“Everyone knows that in addition to mathematics, I am also studying for a doctorate in philosophy with Professor Horkheimer, researching his critical theory, including his instrumental critical theory.

The task he assigned me was quite significant. Critical theory seeks to transcend existing social structures. So, while I was pondering Diophantus's question, I was also thinking: since transcendental numbers exist, could we transcend existing mathematical structures? Could we find a way to break free from the constraints of existing algebraic equations?
With this question in mind, I thought of the Gel'fond-Schneider Theorem, proved by Alexander Gel'fond and Theodor Schneider in 1934. As a solution to Hilbert's seventh problem, it is a theorem that almost every mathematician in Göttingen must know.

Only Professor Siegel returned to Göttingen.

If he were sitting in the audience, he would probably question his life. You know so much about the Göttingen School, have you really been to Göttingen? I'm just old and have forgotten it, right?

Lin Ran erased the linear form logarithm theory and started writing the Gel'fond-Schneider theorem:

"As you can see, these two mathematicians used the auxiliary function method when proving this theorem.

They proved that Λ\LambdaΛ is non-zero by constructing a function with high-order zeros at specific points and deriving a contradiction by analyzing its growth properties.

However, these results are limited to two logarithmic linear forms.

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

At the time I had only a vague idea that the core approach 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 can have high-order zeros at multiple points related to logαi and maintain controllable growth.

Generalizing from a single variable to multiple variables will definitely involve more complex tools.

So I thought of multivariate interpolation techniques. In Gel'fond-Schneider's work, the auxiliary functions were univariate, but for my work, I wanted to find more complex tools.

At this time, multivariable complex analysis and interpolation theory in algebraic geometry seem extremely suitable. If Siegel's lemma is added, it will be perfect! "

The entire seminar was originally scheduled for two topics. The first part was given to Lin Ran, and the second part was for Harvey Cohen to talk about his latest discoveries.

As a result, all the time was used up by Lin Ran. Everyone discussed the linear logarithmic theory for a whole day and a half, leaving no time for Harvey Kane at all.

Of course, there was no time left for Chen Jingrun. He never found a chance to be alone with Lin Ran from beginning to end.

We just chatted for a while while having dinner together in the evening.

"Dehui, long time no see." Lin Ran said.

Chen Jingrun was a little reserved: "Professor, Happy New Year." Lin Ran didn't say much, but turned to Harvey Cohen and said: "Professor Cohen, Chen is my student in Hong Kong. I originally planned to teach him myself, but you know, I may have to work in the White House.

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

Chen has a good talent. I think his talent in number theory is no less than that of Shiing-Shen Chern."

This evaluation is already very, very high.

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

As for Chen Jingrun, not to mention in America, even in China, Chen Jingrun was just a nobody.

Harvey Cohen had no doubts. "Chen is very talented. During the interview, his understanding and insights into the Goldbach Conjecture were even deeper than mine."

Generally, a doctoral interview requires you to talk about your academic direction, what issues you are interested in, and your ideas about the direction you want to pursue.

As a member of the former Chinese Academy of Sciences Number Theory Seminar (Goldbach Conjecture), Chen Jingrun's choice must have been the Goldbach Conjecture.

"Maybe he can really solve the Goldbach conjecture." Lin Ran said half-jokingly and half-seriously.

After the dinner, Harvey Cohen even asked Chen Jingrun to stay and chat for a while:
"Chen, how about it? What do you think after listening to Professor Lin's lecture today?"

After thinking for a moment, Chen Jingrun replied: "It is very exciting and has given me a lot of inspiration.

Professor Lin demonstrated to us the thinking process from intuition to systematic theory, which is most valuable for mathematicians.

Starting from the special achievements of Gel'fond-Schneider et al., through a deep understanding of the auxiliary function method they used, they creatively thought of combining multivariable interpolation, complex analysis and algebraic tools, and gradually extending it to general cases.

This includes drawing inspiration from key issues of transcendental numbers and Diophantine approximations, and innovatively constructing auxiliary functions suitable for polylogarithmic linear combinations.

The lower bound of Λ\LambdaΛ is derived through growth estimation and contradiction method.

I feel that Professor Lin has masterful attainments in analysis, algebra, and geometry, and he is able to skillfully combine these methods, which is very difficult."

Harvey Cohen added: “He even embodies Professor Horkheimer’s philosophy.

Lin is a master, and not just a master in the field of number theory.

So what I want to tell you is that Lin said that you have the potential to solve the Goldbach conjecture and that your talent is comparable to that of Shiing-Shen Chern. I don’t deny that you do have outstanding talent, but what you knew in the past was too narrow. Do you understand?

Your theories and methods are too limited. If we limit ourselves to the field of number theory, or even classical number theory, it will be difficult for us to produce valuable results.

Take Lin as an example. If he only knew number theory, would he realize the need to use knowledge of multivariable complex analysis and algebraic geometry?

So my arrangement for you is that you have to make up for the shortcomings in other areas first. Number theory is definitely not just number theory.

Talent is talent, whether you can realize it is the key."

(End of this chapter)