A genius? I just love studying.

Chapter 227 The Key to Unlocking the Navier-Stokes Equations

In the office, Chen Hui and his colleague had a heated discussion. Although new ideas kept popping up, they couldn't solve two key problems.

After a heated discussion, the office fell silent again as the two men stared at the whiteboard, deep in thought.

The two have been stuck on this problem for some time now, and it's unlikely they can resolve it in the short term.

Dennis pointed regretfully to the diagram of entangled vortices drawn next to the vorticity equation on the whiteboard: "Look, vortices are like rubber bands, stretching, twisting, and even breaking and reconnecting. Topological braid groups and homology classes can describe the state, but how do you characterize this 'deformation' process itself? Directly corresponding to these terms on the right side of the equation..."

Looking at the vortex diagram and then at the fiber bundle connection diagram he had drawn in his notebook, Chen Hui suddenly had a flash of inspiration. "Deformation? The connection of fiber bundles defines parallel movement, essentially describing how the fibers change with the underlying manifold. If we consider the vortex lines as the fibers of the fiber bundle..."

Dennis's eyes lit up, and he immediately drew a twisted cylinder (a deformation of the mundane cluster) on the whiteboard. "Like this! The bottom space is the fluid domain, the fiber is the vortex direction, but the deformation driven by the Navier-Stokes equations is too violent, ordinary connections..."

Chen Hui interrupted him, speaking faster: "Ordinary communication might not be enough! But if we consider communication with 'torsion,' in general relativity, torsion can describe the distortion of microstructures. Here, torsion might be able to characterize the local rotational effect caused by the stretching and twisting of vortex tubes, which directly corresponds to the (ω·)u term!"

Dennis slammed his hand on the table with a loud bang, making his coffee cup rattle. "Torsion! Yes! (ω·)u is vorticity being 'dragged along' by the velocity gradient, producing rotation and deformation. This is essentially a nonholonomic displacement, which is exactly what torsion describes! Defining the connection of vortex bundles as a form with specific torsion... My God, this might be a framework!"

The two remained silent for more than ten seconds, then quickly wrote down key formulas and sketches in their respective notebooks, with only the scratching sound of pens rubbing against paper filling the air.

After a series of deductions, the two looked up almost simultaneously, each seeing a brilliant light in the other's eyes.

They may have inadvertently opened a brand new door, an ultimate door to the mysteries of the Navier-Stokes equations.

Define a vortex fiber bundle E, a bottom space M fluid domain, fibers ≈ vortex line direction, and a connection including a torsion tensor T. Design T to match the deformation term (ω·)u in the vortex transport equation...

Dennis added, "The key is how to map the physical vortex field ω and velocity field u onto this geometry. ω should be associated with the fiber direction, cross section, u defines the movement of the bottom space, and the deformation (ω·)u is encoded into the torsion part T of the connection."

After each of them conducted their own deductions, the two engaged in another heated discussion. This is often the case; they might be stuck on a bottleneck for several years, but once they break through that barrier, everything else will fall into place naturally and they will achieve great success.

"We need a mapping that elevates the physical u to a horizontal elevation on the vortex, where the resulting torsion effect, defined by the parallel shift, must precisely correspond to the change in the vortex direction of (ω·)u."

Chen Hui added his own comments on the whiteboard.

Dennis nodded, erased part of the whiteboard, and began to draw a more complex diagram: the base space, the fiber, a vector field representing u, and the "flow" path it caused on the bundle. "This lifting mapping needs to satisfy compatibility conditions. Perhaps it can be defined from the conservation of vorticity or the Helmholtz theorem?"

Chen Hui's eyes lit up. "That's right! Helmholtz's theorem states that vortices move with the fluid, which is exactly the core idea of ​​the 'connection' definition of parallel movement in fiber bundles—keeping the fibers 'unchanged.' Corresponding to the materiality of the vortexes here, we can use this to construct that lifting mapping!"

Over the next few hours, the two engaged in a heated discussion in front of the whiteboard, constantly defining, negating, and revising mathematical objects, including the explicit expressions of the structure group, connection form ω, curvature Ω, and torsion T of a cluster.

The desktop was piled high with draft paper.

The core task is to transform the physical equation ω/t+(u·)ω-νΔω=(ω·)u into a geometric equation on a vortex bundle as defined by them.

At 10 p.m., outside Chen Hui's office,
Looking at the tightly closed door and hearing the occasional sounds of heated conversation coming from inside, Wang Qiming felt a pang of worry.

Dennis was a visiting scholar at Jiangcheng University, and also a highly respected senior figure in the mathematics community. He was over sixty years old, and the two of them discussed for seven or eight hours.

If something happens to Dennis, Jiangcheng University will have a hard time explaining themselves.

But he didn't dare knock on the door and go in at that moment. He knew what the two were discussing, and if he disturbed them and interrupted their inspiration, he would be in big trouble.

For a moment, Wang Qiming, standing outside the door, was caught in a dilemma.

Inside the office, the two men, who had been engaged in a heated discussion, stopped at three in the morning, looked at each other, smiled, and then burst into laughter.

This framework may still be a long way from completely solving the Navier-Stokes equations, but it is already the first important fruit of their intellectual collaboration.

This framework can potentially bypass certain singularities, provide a unified description of topological changes, connect quantum topological methods, or be applied to the analysis of specific turbulent structures.

"You are such a genius!"

Dennis exclaimed sincerely, "Partnering with you was the wisest decision I've ever made!"

"I couldn't have done this job without you!"

Chen Hui smiled happily and said, "Dennis, there is a lot I need to learn from your research and intuition in topology."

"Haha, let's not flatter each other."

Dennis was so excited he was practically jumping for joy. "We can work it out. This framework is enough to publish a top-tier paper, and it might be helpful to others as well."

"As for our next step, we will use this framework to solve the Navier-Stokes equations."

"Hui, you might become the only person in the world to solve two Millennium Prize Problems!"

Chen Hui shook his head. "It's too early to talk about solving the Navier-Stokes equations. Dennis, you should go and rest."

He doesn't have the habit of opening champagne halfway through a game, and 3 a.m. is just a regular occurrence for him, but Dennis is over sixty years old and can't handle that kind of ordeal.

"Haha, don't worry, this old man is in great health!"

Dennis waved his hand, saying that he usually paid great attention to exercise and that staying up late occasionally was no problem.

However, he really should rest.

Pushing open the office door,

"Dean Wang, what's going on?"

Dennis was somewhat puzzled as he looked at Wang Qiming standing outside the office.

It's 3 a.m.!
"Professor Dennis, come on, I'll take you to a place to rest."

Wang Qiming was thankful that Dennis was alright, and his anxiety finally subsided.

Dennis suddenly realized that she was worried about his old age and declining health.

The accommodation prepared for Dennis was at Luojia Mountain Villa. He didn't drive; instead, he took Dennis there on his electric scooter.

Dennis also adapted to local customs; he himself commuted to get off work by electric bicycle in Paris, so it wasn't unfamiliar to him.

After taking Dennis to the hotel to settle in, Wang Qiming rode his electric scooter back to Chen Hui's office.

What infuriated him was that Chen Hui didn't go back to rest, but instead sat down behind his desk and buried himself in working out the formulas.

You can't do that again!

Wang Qiming was so angry that he didn't care whether he would interrupt Chen Hui's inspiration and reprimanded him sternly.

"?"

Chen Hui turned around and looked at Wang Qiming blankly, not understanding what was going on.

"How old is Professor Dennis? What if something happens to him while you're all staying up all night discussing this?"

Chen Hui nodded, "Don't worry, I'll be more careful in the future."

Today, because of the new breakthrough, the two of them were so excited that they forgot about the time, and they won't be able to do it again later.

"Not just Dennis, you are too!"

Wang Qiming looked at the thick stack of draft papers on Chen Hui's desk, snatched the pen from Chen Hui's hand, and said, "You still have decades of life ahead of you, why bother arguing about this short-term gains and losses?"

"No more staying up late." "Okay."

Chen Hui agreed without hesitation, packed his things, and went back to his dormitory to rest.

Indeed, it's already 3 a.m., time to rest.

Watching Chen Hui leave, Wang Qiming felt somewhat helpless. He had a premonition that Chen Hui would not heed his advice at all.

He also knew that a mathematician's golden age is only a dozen or so years, and if he couldn't seize the time to make groundbreaking achievements, he might never have another chance in his life.

He has never seen a mathematician in his fifties or sixties make any groundbreaking achievements.

But he didn't want Chen Hui to ruin his health because of this.

He didn't understand why Chen Hui, who had already achieved such remarkable success, still needed to work so hard.

Jiangcheng University, 9:00 AM
"Good luck, may good luck bring you joy and love..."

A burst of loud music woke Wang Qiming up.

Still half asleep, Wang Qiming wanted nothing more than to throw his phone out the window. He had finished dealing with Chen Hui's matter at around 4 a.m. yesterday, and after washing up at home, he finally went to sleep. Now, being woken up, he felt a splitting headache.

But the last shred of reason prevented him from doing anything crazy; he glanced at the caller ID.

Zhang Pingwen? !

Wang Qiming instantly regained his senses, and even his eyes became much clearer.

"What about people?"

"I need to speak with you urgently. Come to my office right away."

Zhang Pingwen's voice, tinged with anger, came through the microphone. He had just come out of Wang Qiming's office and, not seeing anyone, naturally wouldn't give Wang Qiming a friendly look.

"what's up?"

Although Wang Qiming was asking questions, he had already started getting dressed.

"We'll talk when you get here," Zhang Pingwen said before hanging up the phone.

Wang Qiming was completely bewildered, but he still made the most of his time to wash up and rode his electric scooter to Zhang Pingwen's office.

"What time is it? Isn't today a weekday?"

"Doesn't the dean have to come to work?"

Half an hour later, Wang Qiming walked into the principal's office and was greeted with a barrage of insults.

Wang Qiming had a bad feeling; it seemed that Zhang Pingwen's visit to him was definitely not for anything good.

Having prepared himself mentally, Wang Qiming didn't argue. He lowered his head and waited patiently for Zhang Ping to get down to business, so he could respond accordingly.

After hurling insults for a while, Zhang Pingwen was also tired and knew that this approach was useless against an old hand like Wang Qiming. So he finally spoke up and said, "People from Chengdu Aircraft Industry Group and Shenyang Aircraft Corporation are here. They want to poach people. You can handle this."

"Chen Hui?"

"if not?"

"How to deal with it?"

"Do I have to teach you how to handle this?"

"understand!"

After Wang Qiming finished speaking, he walked straight out of Zhang Pingwen's office, but he breathed a sigh of relief. It turned out to be a poaching operation. He thought it was something big. Even if these two research institutes had recently developed a sixth-generation fighter jet, a national treasure, they couldn't possibly snatch people from him, could they?
Based on the information provided by Zhang Pingwen, we went to the conference room on the third floor of the administration building.

When Wang Qiming saw who had come to the conference room, his expression hardened, and he immediately understood why Zhang Pingwen had gotten angry.

This time, it wasn't just people from the Chengdu Aircraft Industry Group and Shenyang Aircraft Corporation research institutes who came; even the head of the Zhantang Hall came in person.

"Dean Wang Qiming?"

When the leader of the Battle Hall saw Wang Qiming, he took the initiative to come forward and greet him with a smile.

"We came here today to borrow someone from Dean Wang."

Zhan Tang's aura was overwhelming; he got straight to the point, and his words were like a sharp blade piercing Wang Qiming, leaving him no way to avoid or dodge.

"Recently, the research on national strategic equipment has hit a bottleneck, and we need Professor Chen Hui's help."

"As you know, we are racing against time with our competitors. Completing the research and development a day earlier or a day later could make a world of difference in the final outcome."

"Dean Wang will surely prioritize national affairs!"

The words came in rapid succession, like bullets raining down on them.

Although the other party said these words with a smile, Wang Qiming felt a chilling killing intent behind him, as if if he dared to say no, the other party would suddenly attack and snap his neck.

"The hall master is right!"

Wang Qiming composed himself and replied with a smile, "But Chen Hui can still contribute to the country while studying at Jiangcheng University."

Even under such pressure, he still bravely said no.

The leader of the Battle Hall's expression changed, his smile vanished, and he said seriously, "These things are classified. If we leave them at Jiangcheng University and they are leaked, who will take responsibility?"

"I come!"

Wang Qiming said with absolute certainty.

The leader of the Battle Hall shook his head, "You can't bear this responsibility!"

Wang Qiming looked at the Hall Master of the Battle Hall, saying nothing but not retreating an inch.

The leader of the Battle Hall had sharp eyes that seemed to pierce through Wang Qiming's body.

Silence fell over the meeting room.

"Then it's up to Chen Hui to decide. You can't force him to go, can you?"

Finally, Wang Qiming took a half step back.

If it were just people from the Chengdu Aircraft Industry Group and Shenyang Aircraft Industry Group Research Institute, Wang Qiming could have easily dealt with them, but the fact that the Hall Master of the Battle Hall came in person left him with no way out.

"Then take us to see Professor Chen!"

Without further ado, the leader of the Battle Hall gestured for Wang Qiming to lead the way.

(End of this chapter)