A genius? I just love studying.

Chapter 177 Cooperation
The first day's exchange did not last very long. Everyone was tired from traveling and didn't have much energy for in-depth exchanges. It was just a preliminary discussion. The better time to make friends would be at the cocktail party on the first night of the conference.

After spending more than an hour in the coffee shop, Chen Hui returned to his room to continue studying his thesis.

The next day was Christmas Day in the West. The December snow gently kissed the banks of the Danube, and Budapest was transformed into an amber inlaid with gold. The cobblestone streets of the old town were covered with a layer of silver frost, and strings of lights with rattan balls hung between the Gothic spires and Baroque arcades, like a galaxy cascading onto the 19th-century stone walls.

The Christmas market in front of Wojdács Castle is the heart of the place—the aroma of smoky sausages wafts from the wooden stalls, hand-painted ceramics and woolen felt hats pile up like colorful hills, and the clanging of old Hungarian craftsmen hammering tinware mingles with the accordion melody of "Silent Night," weaving a warm net in the white mist of their breath.

In St. Stephen's Cathedral Square, a twenty-meter-tall Christmas tree is adorned with crystal-like glass balls, children chase each other around an ice sculpture of the Virgin Mary holding sugar gingerbread men, and across the Chain Bridge, the Buda Castle glows with a warm golden silhouette in the twilight, like a candelabra palace suspended in the night.

When Terence Tao knocked on Hui Chen's door, he was studying a paper on string theory AdS/CFT duality.

"Oh God, what day is it today? You're still reading papers. Come on, let's go for a walk in the streets and feel the warmth of human life. That's the real meaning of our mathematical research!"

"Let's go, today we won't talk about academics. In Budapest today, there is no mathematics, only life!"

Although Terence Tao is of Chinese descent, his long-term life in the West has led him to speak in a somewhat exaggerated manner, while retaining the humility typical of Chinese people. He is not a Western gentleman in the traditional sense, but rather more like a gentleman in the traditional Chinese cultural concept.

After thinking it over, Chen Hui did not refuse Tao Zhexuan's invitation. Just then, the resounding brass music on the street exploded in his ears like thunder, which even he could not completely ignore.

This is also his first time experiencing a Western Christmas. I'm sure Rui Rui will love such a lively scene.

Stepping out of the New York Palace Hotel, the streets were already packed with people. Local dancers, draped in traditional embroidered cloaks, were treading on their boots, their skirts swirling like flames. The crowd cheered, stomping their feet and exhaling hot breath that condensed into frost on their wool scarves.

In the café on the first floor, under the crystal chandelier, a harpist strums the melody of "White Christmas," while waiters carry silver platters among the gilded pillars. The frosted cakes and the tailcoats of the old gentlemen transport you back to the golden age of the Austro-Hungarian Empire.

"That man swaying on the dance floor is Fefferman, the head of the Department of Mathematics at Princeton."

Tao Zhexuan, holding a wine glass, introduced Chen Hui, "It's a pity that Mr. Wei Teng didn't come. Otherwise, you could have discussed string theory with him. Given that old man's temperament, you would surely have been deeply impressed by him."

Chen Hui didn't drink alcohol. Instead, he made a video call to his aunt and then, with the camera on, gave Rui Rui and Qing Shan an immersive Christmas experience in Budapest.

"Wow, brother, did you go to the palace?"

Li Qingrui's eyes were sparkling as she joined Chen Hui in the joy of Christmas.

Li Qingshan remained expressionless beside Li Qingrui, his eyes fixed on his phone screen without blinking.

Chen Hui then remembered that over the years, his uncle and aunt had never taken Rui Rui and Qing Shan out to play during holidays because they were busy making money. This was because it was usually the time when wages were highest, and they couldn't bear to waste time.

Chen Hui was also immersed in his studies, so there was no way he would go out to play.

This must be the first time Li Qingrui and her companion have ever witnessed such a grand festival!
In just over a month, it will be the Chinese New Year...

"Is this your daughter?"

Tao Zhexuan, his cheeks flushed from drinking, came to Chen Hui's side and, slurring his words, excitedly said...

But as soon as he finished speaking, he realized something was wrong. Looking at Chen Hui's youthful face, he knew that this didn't seem like someone who had a daughter.

"My sister."

Chen Hui gave an awkward but polite smile.

"Ha ha!"

Terence Tao laughed, "So cute! My daughter is about the same age."

As he spoke, he waved at Chen Hui's phone camera and greeted Li Qingrui in the video, saying, "Hello."

"This is Terence Tao."

Chen Hui adjusted the position of his phone camera and introduced the two to Li Qingrui.

"Hello, Brother Zhexuan." Li Qingrui waved happily in response, unaware of what the name Tao Zhexuan meant.

Li Qingshan replied as if facing a formidable enemy, "Hello, brother."

Chen Hui was surprised to find that Tao Zhexuan already had a daughter; he still looked very young.

Terence Tao was suddenly surprised to find that Chen Hui was actually a little guy of his daughter's generation, and this little guy had already published a paper in the Annals of Mathematics.

After a brief exchange, Terence Tao didn't disturb Chen Hui's conversation with his family and joined the next group of people with his wine glass in hand.

Just then, Adrien walked over with a wine glass in hand. "Merry Christmas!"

“Merry Christmas!”

Chen Hui responded with a smile.

"I've seen Bryant and Oliver these past few days."

Edrian came to Chen Hui's side and whispered a reminder, "You need to be careful they don't try to trip you up."

Chen Hui recognized the two names, but he was puzzled, "What kind of articles could they possibly write?"

“I don’t know either, but you know, based on my assessment of their character, it’s quite possible.”

Edrian was deeply worried.

Chen Hui didn't care at all. He was now fully focused on proving the existence of the Yang-Mills equation and didn't want to waste his energy on other things. But if the other party kept bothering him, he wouldn't mind putting in some effort.

He doesn't provoke trouble, but he's not afraid of it either. He's open to talking, but if it's a fight, he'll fight to the bitter end!
There are more than one or two obvious points in the paper's proof.

Seeing this, Adelian said no more. After talking with Chen Hui, he was unconsciously infected by Chen Hui's confidence, and his worries immediately lessened.

The party lasted until the early hours of the morning. Chen Hui returned to his room around 11 p.m. There was still a lot of noise, but it was within an acceptable range. He sat down at his desk and continued to study his thesis.

……

In a luxurious suite at the Palace Hotel in New York, Tian Yang sat at his desk, on which lay a thick stack of draft paper covered with intricate symbols.

Instead of joining in the Western revelry, he devoted himself to studying and researching here.

After an unknown amount of time, Tian Yang rubbed his slightly swollen temples, took off his reading glasses, and looked haggard and tired.

How can we solve the obstacle class problem in group expansion?

Tian Yang gazed at the brilliant lights outside the window and murmured to himself.

At the time, they all keenly noticed the problems in Oliver and Chen Hui's two papers. Whoever could prove the problem first would undoubtedly have an absolute advantage in this debate.

Unfortunately, several months have passed and the problem remains unresolved. He really is getting old!

I wonder how far the other side has progressed in their research?
After hesitating for a moment, Tian Yang got up, opened the door, and walked towards the room of his former teacher, now his mortal enemy.

Qiu Chengwu also came; this naturally couldn't be kept from him.

He didn't know if the other party had come for Chen Hui or simply didn't want to miss this grand event.

But since the other party has proposed the competition, they will surely spend time on this issue and gain something from it.

The meeting will officially begin tomorrow, and there are only three days left until Chen Hui's report. Those people are likely to cause trouble during Chen Hui's report.

He wasn't confident he could resolve the issue within three days.

They must resolve this issue within three days. China has finally produced a promising talent, and he cannot stand by and watch this talent be affected by such a thing.

He had seen geniuses who, after suffering setbacks, never recovered.

For this reason, he did not hesitate to bow down to that person.

Tian Yang walked down the dimly lit corridor, his expression not very pleasant, but his attitude was very resolute.

However, as soon as he turned the corner, Tian Yang stopped and stood still.

Opposite him, at the end of the corridor, stood another old man—Qiu Chengwu.

The two elderly people appeared here at the same time, as if they were telepathically connected.

What is he doing here?

"Could he have come for that matter as well? Should I speak first?"

"Should we pretend we're just going downstairs for dinner?"

"But the way downstairs isn't this way, it seems too deliberate."

Tian Yang's mind raced with thoughts, but he quickly put those thoughts aside, raised his head, and walked firmly toward the other side.

Almost simultaneously, Qiu Chengwu also walked towards him.

At that moment, the two looked at each other, understood each other's intentions, and the barrier between them seemed to loosen a little.

The two were located at opposite ends of the corridor, and were moving at the same speed, so they naturally met in the middle of the corridor.

Tian Yang spoke first, “The modular series N is often related to the symmetry of the system. Intuitively, n should be related to the factors of N, but this conclusion is not obvious. We need to establish a strict correspondence between crystal symmetry and modular parameters, and prove the primeness constraint of the denominator.”

Without saying anything more, he believed that the two of them had come for this very reason.

"Let's talk inside."

Qiu Chengwu did not answer, but instead cautiously observed the surroundings of the corridor.

Tian Yang subconsciously turned to go back to his room, and Qiu Chengwu did almost the same thing. After taking a few steps, they went in opposite directions and then both turned back in surprise.

After staring at each other for a few seconds from a distance of several meters, Qiu Chengwu finally spoke, "Yunwei is with me too."

Without further hesitation, Tian Yang stepped forward and followed behind Qiu Chengwu.

"Tell me, where is it stuck?"

Entering the room, Qiu Chengwu walked to his desk without looking back. His desk was also covered with draft papers, and Yun Wei was sitting at the desk.

Tian Yang suddenly felt a little dazed, as if time had flowed back to thirty years ago when he was still a student.

He encountered a problem that had troubled him for a long time, and when he went to his teacher, the teacher asked him the same question.

After not hearing a sound for a long time, Qiu Chengwu turned around, his brows furrowed slightly. Just as he was about to question him, he suddenly remembered that this was not his office thirty years ago.

"I have completed the proof of the primeness of the denominator of fractional Chern numbers. However, there are still some parts that I haven't figured out when establishing a strict correspondence between crystal symmetry and modal parameters."

After a brief moment of stunned silence, Tian Yang regained his composure and spoke.

"You haven't read your students' papers, have you?"

As soon as he opened his mouth, Qiu Chengwu's tone was full of sarcasm. The inertia of so many years could not be eliminated in a short time. "The establishment of crystal symmetry can be achieved by introducing the Langlands dual group, embedding the crystal symmetry group Cn into the Langlands dual group L_G, using the spinor representation of the crystal group, constructing a nontrivial homomorphism ρ:CnL_G from Cn to L_G, and then elevating ρ to Galois representation through local Langlands correspondence."

As he spoke, he pulled a piece of draft paper with a string of formulas written on it from his desk and handed it to Tian Yang.

ρ~:Gal(Qˉ/Q)→LG
Tian Yang's tightly furrowed brows slowly relaxed, then furrowed again. "How do we construct non-divergent representations?"

"Have you forgotten the Taylor-Wiles theorem?"

Chiu Cheng-wu's voice became stern, as if he were reprimanding his students, "By using the new compatibility conditions in the p-adic Hodge theory and extending the Taylor-Wiles theorem to the crystal symmetry group, we can prove that when n is a prime number, ρ is non-bidding at p=n."

"That's right, ρ~ is unbidding at p=n. According to the Néron-Ogg-Shafarevich criterion, the modular series N must be divisible by n. If n is not a prime number, then ρ~ is bifurcation at p|n, which leads to a_p = 0, contradicting the nonzero property of fractional Chern numbers!"

Therefore, the modular series N must contain a prime factor p=n!

Tian Yang has been studying this problem for some time, and even though he is getting old, he is still quick-witted and understands things very easily.

Yun Wei stared at the scene before him in a daze. Who would have thought that the two teachers and students who had broken off relations twenty years ago would one day reunite in a foreign land and discuss academic issues together, just like when they were still teachers and students thirty years ago.

Tian Yang and his companion failed to notice this. At this moment, they felt enlightened, and the bottlenecks that had been hindering them for so long were being overcome one by one during their discussion.

Yunwei quietly recorded the content of their discussion, and finally summarized it, and the proof of that problem gradually took shape.

When the two elderly people stopped their discussion and the room fell silent again, Yunwei had already written the proof of completion on his draft paper. He wrote the last sentence on the draft paper.

In summary, for any two-dimensional fractional Chern insulator with C_n rotational symmetry, the denominator n of its fractional Chern number must be a prime factor of the corresponding modular series N, that is, there exists a prime number p such that p | N and p = n.

The two elderly people looked at Yunwei's draft paper and smiled knowingly. They had finally solved this problem!

But after the initial excitement, the atmosphere in the room suddenly became awkward.

After glancing at Qiu Chengwu, Tian Yang walked out of the room without saying a word.

Yunwei checked the time; it was already past 1 a.m. The two elderly people looked exhausted. After tidying up the draft paper, they also got up and left the room.

(End of this chapter)