The MRS conference is one of the regular academic activities of the American Materials Research Association, and it is also the most influential top conference in the field of materials science.

Its coverage covers almost all the research directions in the field of materials science, and its status is probably equivalent to the "International Congress of Mathematicians" in the field of materials science.

However, unlike the International Congress of Mathematicians, which is held every four years, MRS is twice a year, divided into spring and autumn. Spring is generally in Phoenix, Arizona, and autumn is more stable, usually in Boston, Massachusetts.

The purpose of the conference is mainly to exchange technology, present the results to the industry, connect rich companies with cash-hungry labs, and also provide a place for peers to tear up.

Yes, it's tearing.

It would not be surprising if someone threw shoes at the speaker at the meeting. On the contrary, no one quarreled at that meeting, and everyone calmly finished the meeting, exchanged views, reached a consensus, and praised the technology of peers...... That will definitely make people in the industry wonder if the sun is coming out in the west.

The bigger the bull, the more it is torn.

This is a difficult situation to see at the International Congress of Mathematicians.

In a sense, the style of mathematics is really different from that of other subjects.

As a mathematics professor, Lu Zhou has no interest in tearing up.

But for him, the meeting was an opportunity.

Moreover, MRS suddenly sent him an invitation letter, and many people must have been very interested in his research results.

Of course, even so, Lu Zhou has not forgotten his identity.

He is a professor of mathematics.

Even in order to see the "future era" after grade 10, he has to take a long-term view, and he can't leave behind the mathematics that determines the upper limit of grades in other subjects.

On the last day of August, Lu Zhou was in his office at the Institute for Advanced Study and tested two other students.

It is also ten questions, with a time limit of two hours.

After handing over the A4 paper with the title to the two of them, Lu Zhou picked out the book from his desk and flipped through it in his hand.

Minutes and seconds passed.

As the mobile phone rang, Lu Zhou snapped the book in his hand and looked at the two people who were struggling to write on the paper.

"When the time comes, let me see how you learn this month."

Hardy put down the ballpoint pen in his hand with a headache, and Qin Yue, who also stopped writing, was also nervous.

"Professor, the time you gave is too short," Hardy said with a bitter expression on his face as he got up and handed the test sheet to Lu Zhou's desk, "If I give me another ten minutes, I will definitely be able to write the next question." ”

"The length of time doesn't matter, and I'm not asking you to do every question so I can see what you're going to do."

Taking the test paper of the two of them, Lu Zhou said as he glanced at the questions on it.

For him, these are very simple things, and he probably has a number in his mind at a glance.

Qin Yue made six questions, and the seventh question was not finished, but there was no big problem with his thinking.

Overall, his condition is not bad, and this is also expected by Lu Zhou.

Hardy made five of them, barely meeting the passing standards, which was somewhat unexpected by Lu Zhou.

Lu Zhou originally thought that there was at least one person who would not be able to pass his test, and that person was most likely Hardy. Because of the three students, he has the most impetuous personality.

However, the situation now seems to be more optimistic, and all three have been qualified to participate in this project.

Putting the A4 paper aside, Lu Zhou cleared his throat and spoke.

"First of all, congratulations on joining my project."

Hearing this, Hardy, who was originally a little frustrated because he had only made five questions, widened his eyes in surprise. Qin Yue, who was next to him, also showed a surprised expression.

As if seeing through what they were surprised about, Lu Zhou explained in a relaxed tone: "The passing line I set is five questions, and I can make at least five questions, which shows that you have listened to the tasks I assigned, and at least the month and a half of time has not been wasted." ”

"I'll just briefly talk about the specifics of our topic."

After taking a sip of coffee, Lu Zhou stood up, walked to the whiteboard in the office, and picked up a marker.

Vera, who was sitting in the corner of the office silently collecting literature on the computer, also stopped what she was doing, and like the other two students, moved her chair to sit in front of the whiteboard, waiting for the boss to speak.

"A month and a half ago, I told you that our problem is related to hail."

"If you know anything about additive number theory, I'm sure you've probably guessed what this topic is."

Hardy and Qin Yue nodded one after another.

As Lu Zhou said, they have already guessed what the subject is.

As for Vera, there was no superfluous reaction. After all, as early as half a month ago, she had already passed the assessment, and she had even participated in the project a long time ago.

After a pause, Lu Zhou continued.

"The so-called hail conjecture, also known as the Kakutani conjecture, or the 3n+1 problem. The proposition described is that for arbitrarily determined positive integers N, after a finite number of consecutive effects of fokn(n)=1, they all fall into the numerical trap of {4,2,1} without exception. ”

"In layman's terms, choose an N, if N is an odd number, the next step is 3N+1, and if N is an even number, the next step becomes N/2. After a finite number of cycles, no matter how much its value expands during this period, it will eventually fall to the bottom of 1 like a hailstone. ”

Speaking of this, Lu Zhou paused for a moment, smiled and continued.

"It's like a black hole."

Compared with Goldbach's conjecture, the hail conjecture is undoubtedly more well-known in the United States.

In the seventies of the last century, almost all American college campuses could be seen people delving into this magical "numbers game". And this phenomenon has even appeared in the Washington Post, an old big newspaper in North America, and has formed a trend for a period of time.

Of course, for the average person, it's a numbers game, but for mathematicians, there's something deeper about it.

"This is a number theory problem, and it is a classic problem in additive number theory. But at the end of the day, it's a matter of complex analysis! ”

"Kakutani guess is your task for the next three years. I don't require you to prove this proposition completely, but you must at least complete a paper in this direction that is worthy of inclusion in the Annals of Mathematics......"

Lu Zhou thought for a while, picked up his pen, and wrote a line of equations on the whiteboard.

[h(z^3)=h(z^6)+{h(z^2)+λh(λz^2)+λ^2h(λ^2z^2)}/3z] (where λ=e^{2πi/3})

Seeing this line of calculations, Qin Yue immediately took out the notes he was carrying from his pocket, and Hardy quickly cheered up.

As for Vera, she listened as attentively as ever.

"Although the outside world is generally pessimistic about solving this problem, in fact, the number theory community is not without progress on this problem."

"In the nineties of the last century, in '94 to be exact, Professors L. Berg and Professors Dus proved that the 3n+1 conjecture is equivalent to the functional equation h(z^3), which is the equation I wrote on the upper panel."

"The emergence of this equation paved the way for the subsequent proof to the top of the mountain."

There are some things that cannot be said.

In response to the three pairs of expectant gazes, Lu Zhou turned around and continued to write on the whiteboard.

[g(z)=z/2+(1−cosπz)(z+1/2)/2+1/π(1/2−cosπz)sinπz+h(z)sin2πz satisfies: N⊂Φ(g). 】

【……】

Seeing these lines, Vera's eyes gradually brightened.

Qin Yue and Hadi showed thoughtful and non-understanding expressions respectively.

After stopping his pen, Lu Zhou gently placed the marker on the table next to him and smiled at his three students.

"This step is critical."

"If it can be proved that there is an integer function h(z), for g(z) above, there is z0∈D for every branch D of Φ(g) that contains a positive integer, so that [gok(z0)] converges to 1......"

After a pause, looking at the three pairs of expectant eyes, Lu Zhou smiled and said in an affirmative tone.

"From this, we can prove it."

"3N+1 is established!"

-

References: BergL, Meinardus G. Functional-equations-connected-ith-the-Collatz--in-Math, 1994, 25:1-12

Last time, I saw a children's shoe in the book review area proposing to post the literature, but to be honest, I don't think it makes much sense......