From Almighty Scholar to Chief Scientist

【ζ(1/2+it)=O(t^e)】

【ζ(1/2+it)/(t^e)=O(√plnp)】

After writing these two formulas, Lin Xiao frowned and began to think.

He has an impression of the first line.

"This seems to be... Riemann's conjecture? It seems to be a weakened form of Riemann's conjecture?"

Thinking of this, Lin Xiao was shocked.

Among the weak forms of Riemann's conjecture, there is a Lindelof conjecture.

Lindelof's conjecture is a conjecture about the growth rate of the ζ function on the critical line, which shows that given any e greater than 0, when t tends to infinity, ζ(1/2+it) is equal to O(t^e) , which is a relatively weak form for the Riemann conjecture, and it can finally be deduced that "given any e greater than 0, for a sufficiently large n, Pn+1-Pn is less than Pn^e(1/2+e)".

However, Lin Xiao then turned his attention to the second row of formulas, and became suspicious again.

What does this mean?

√plnp?

Does it mean that the above formula can be deduced from the following equation after the form transformation?

But suddenly, a flash of inspiration flashed in his mind, and he remembered a weak form of Riemann's conjecture, that is, the large prime number gap conjecture, and this is a stronger conjecture than Lindelof's conjecture.

And the conjecture believes that if the Riemann conjecture is true, the gap between the prime number p and the next prime number should be O(√plnp).

That is to say, this second formula is equivalent to linking the weak forms of the two Riemann conjectures together?

Lin Xiao's eyes flickered.

Obviously, this is a rather miraculous discovery.

No matter from the matter of Riemann's conjecture, or he actually discovered such a thing related to Riemann's conjecture from the seemingly endless sea of ​​formulas in his mind.

However, for the latter, he also knows that he can't explore anything, and to study this thing is just to increase his curiosity. As for the former, he also has nothing to say, after all, he has nothing to do with the Riemann conjecture itself. Research, so even if he knows that these two weak-form conjectures can be linked together, it will probably take him a lot of time to study.

And he doesn't have time to study Riemann's conjecture now. After all, he still has more than a month to submit a manuscript to the International Congress of Mathematicians. Get it out?

In any case, the Riemann conjecture is too difficult. If he really wants to study it, he might as well study the Hodge conjecture, which is also one of the seven major problems of the millennium. At least, he has more research on the Hodge conjecture. Some.

However, speaking of Hodge's conjecture...

Lin Xiao frowned, and wrote out the other five formulas.

These five formulas are all related to Hodge's conjecture!

Of course, it is no coincidence that all five are related to the Hodge conjecture.

Because Lin Xiao was already prepared for the next two times when he spent truth points, so he specially selected these formulas related to Hodge's conjecture to memorize.

He is ready to try to do research on Hodge's conjecture during the spare time of the X-ray project.

Don’t ask for a solution. After all, the difficulty of Hodge’s conjecture is not necessarily lower than that of Riemann’s conjecture. Even compared to the Riemann’s conjecture that has been studied by countless people, Hodge’s conjecture is like a hard nut to crack. , although there are many talents in algebraic geometry, not many people dare to challenge this thing with confidence.

Just like until now, the millennium problem has only been solved.

Of course, even if he doesn't seek to solve it, Lin Xiao will try his best to try, what if it is solved?

So, looking at these five formulas, Lin Xiao fell into thinking.

For Hodge's conjecture, his research is not insignificant.

After all, in order to prove Lin's conjecture, he almost read all the famous papers related to Hodge's conjecture. As for the degree of research, of course it is also very deep.

As for these five formulas, except for three of which are relatively easy to understand, the other two formulas gave him a strange feeling.

【︶: H^(p, q)(X)＊H^p(X)→H^p＋p(X)】

[Hdg*(X)=⊕kHdg^k(x)]

Looking at these two strange formulas, Lin Xiao fell into thinking.

"The meaning of these two formulas is that in a closed-chain group of cohomology class, Hodge class and algebra are homeomorphic on the complex projective algebraic variety of singular generation?"

"Well……"

Lin Xiao patted his head, and then continued to write the formula.

Undoubtedly, these two formulas seemed a bit strange to him, which caused him some thoughts.

Hodge guessed that this kind of mathematical problem dancing between geometry and algebra is very difficult. Even with Lin Xiao's current ability, it is not easy to solve it.

Of course, he obtained these five formulas through a bug in the card system, which at least gave him some clues.

And so, as time passed, his research eventually got stuck.

"This... if the Hodge conjecture is integrated over C, will the full Chow group have smooth projective varieties of finite rank?"

After thinking for a while, he turned on the computer and searched for a paper. He remembered that this was mentioned in a paper he had read before.

With his powerful memory, he quickly found the paper.

"This is the article, "On the Trajectory of the Hodges"...Huh? Is it Professor Deligne's?"

When Lin Xiao saw the author of this paper, Lin Xiao couldn't help being taken aback. This paper was exactly the paper of Pierre Deligne.

As a bigwig in the field of algebraic geometry, Deligne certainly has the courage to challenge Hodge's conjecture.

Of course, he didn't prove it anyway.

Soon, Lin Xiao also finished reading the paper.

After reading it, he couldn't help feeling: "Professor Deligne is indeed a master in the field of algebraic geometry. This paper was written more than [-] years ago, but the content in it is still so powerful."

"However, according to the content here, choose a Kahler structure on X, and then decompose H(X, C)... you can use the formal role of C* to exchange with the Laplacian operator, thereby inducing C *The role in space, if this step is to define H2P (X, Z) through the Poincaré duality theorem..."

Thinking of this, Lin Xiao's eyes gradually lit up.

It seems feasible to go in this direction?

"However, there are still some problems."

"Well, it's better to ask Professor Deligne directly."

As soon as he thought of it, Lin Xiao took out his mobile phone and called Professor Deligne who was far away on the other side of the Pacific Ocean.

Soon, the phone was connected.

Deligne's hearty laughter came from the other end of the phone.

"Hey, Professor Lin, how come you have time to call me today?"

Lin Xiao smiled and said, "Professor Deligne, long time no see."

"Yeah, it's been a long time, but this year's International Conference of Mathematicians, we should see each other again, right?"

"Of course." Lin Xiao smiled.

"That's good." Deligne said, "I'm just waiting for you to win a Fields Medal this year."

Lin Xiao laughed: "It's not decided yet, I can't say."

If you talk about looking forward to it, then he must be looking forward to it. After all, if he can't get it this time, he will have to wait another four years.

"Hehe, if you still can't get it, the International Mathematical Union will be ashamed. We old guys are already waiting to drink your champagne." Deligne smiled, and then said: "Okay, I believe you must have something to do with me, right? Tell me."

"There is indeed one thing about a mathematical problem."

"Oh? Lonely artist, you also need to come and ask me questions?"

"Lonely artist? What is that?" Lin Xiao was taken aback for a moment. When did he have such a title again?

"It's you!" Deligne said with a smile: "Look at the many problems you have solved, almost which problem was not researched by yourself? Like our mathematics world, more problems are completed in cooperation Yes, a person like you who researches alone, and who can always research something, is not a lonely artist?"

"That's it..." Lin Xiao touched his nose, or could he be called...the lonely brave man?

Well, don't be a kid anymore.

Skipping over this topic, he talked about the purpose of coming to Deligne this time, and after telling Deligne the paper and his thoughts, Deligne fell into deep thought.

"Well...you wait for me for a while, and define H2P(X, Z) through the Poincaré duality theorem..."

Hearing the sound of writing from Deligne, Lin Xiao didn't wait, and started to think about it.

When two people study a mathematical problem together, it means that two people share inspiration, and this is very meaningful for mathematicians.

Especially when two top mathematicians collaborate.

283 chapters of the paper published in the shock brought about

As time passed slowly, these two world's top mathematicians forgot time in their thinking.

After more than ten minutes passed, Deligne's voice finally sounded again, "Do you know Bloch's conjecture?"

Lin Xiao thought for a while and said, "I know."

Bloch's conjecture, that is, Bloch's guess, and when Deligne mentioned this, Lin Xiao also understood what he meant.

"According to Bloch's conjecture, if X is a smooth projective surface on C and is Chow trivial, then H1(X)=0=Pg(X)."

"Yes." Deligne smiled, and at the same time felt in his heart that at such a young age, Lin Xiao can still be so knowledgeable and memorized, he can know a little knowledge point casually, and even he has decades of experience Accumulation is the only way to do it, so it's no wonder Lin Xiao can do it to this extent.

However, Lin Xiao did not stop there, but said again: "That's right! If we use the inverse prediction of Bloch's conjecture, if X is a smooth projective surface on C, and H1(X)=0=Pg(X) , then X is Chow ordinary, which is also established!"

Deligne was taken aback for a moment, and under Lin Xiao's explanation, he also quickly deduced this point.

There was no time to marvel at Lin Xiao's quick reaction. Sitting in the office at this time, he also quickly wrote down the inference Lin Xiao said on the paper, and then wrote down the inference he just said. After a while, he wrote an equivalent symbol in the middle.

equivalence!

For mathematicians, seeing this symbol moves them, and of course it may also make them afraid.

The moving situation usually occurs when they are proving a certain problem. Writing an equivalent symbol means that they have obtained a breakthrough on a key node, or even directly completed the proof.

As for the situation of not daring to move, it usually appears in the stem of a certain problem. For example, Hodge's conjecture is to draw an equivalent symbol between geometry and algebra, but obviously, its difficulty makes mathematicians dare not move.

But now, they belonged to the former, and their hearts were full of emotion at this time.

"Not bad! In this way, we are equivalent to connecting the Chow group with H1(X). Next, we can start from more directions."

Deligne said.

Lin Xiao also smiled and said, "Yes."

Calling Professor Deligne this time, he didn't expect to have such a harvest so soon, which made him quite satisfied.

"So where do you plan to start next?"

Lin Xiao thought for a while and said, "I think, we should start with non-homogeneous co-deployment."

"Is it non-homogeneous cohomology..." Deligne thought for a while, and finally said helplessly: "It seems that my thinking still can't keep up with you, but I remembered the Poincaré conjecture. Professor Perelman told Poincaré The proof of Lay's conjecture is somewhat related to Hodge's conjecture, a closed three-dimensional manifold must be homeomorphic to a three-dimensional sphere, and his solution to the geometric conjecture may inspire you."

As he said that, he sighed with emotion: "The more I understand this level, the more I can discover the extraordinary unity of mathematics. Maybe, our mathematics can also have a standard model, and then, like physics, everything can be unified. How about integrating it into it?"

Lin Xiao said with a smile: "Maybe, of course, I sometimes feel that the world must be highly unified."

"Well, it's a pity that I must not see it." Deligne sighed, and after a while, he suddenly asked with interest: "By the way, having said that, have you started studying Hodge's conjecture now? "

Lin Xiao smiled and said, "Yes."

"Should I really return to the mathematics world this time?"

Lin Xiao said helplessly: "I have always belonged to the mathematics world."

"Hahaha, in the eyes of some people in our mathematics world, you are a waste of talent." Deligne laughed: "Of course, I believe you, go and solve Hodge's conjecture, tell those people, even if you waste your talent , you can easily master mathematics."

"I'm embarrassed to say this, I can only say, just try."

"Then I will wait for your good news."

"Ok."

Lin Xiao smiled, then stopped talking and hung up the phone.

After hanging up the phone, he found that he had received a lot of text messages from the communication operator. After looking at them, they were all informing him that his mobile phone had been shut down for arrears.

Suddenly he was speechless.

Call charges of more than 100 yuan will be gone within 10 minutes.

I heard that international calls are calculated at 10 cents for six seconds, and 160 yuan for [-] minutes.

I knew I would use email.

Shaking his head, he immediately said that he was really black, and finally paid the phone bill honestly, and then continued to focus on the draft paper in front of him.