A genius? I just love studying.

Chapter 248 Topology is the bone, complex geometry is the soul
"surprise?"

"Navistokes equations".

"How is this possible?" Nakajima Kei exclaimed in shock. "Have you already completed the proof?"

Dennis shook his head. "We're not working together anymore. He's trying to use complex geometry, but I insist that topology is the right path."

He was very confident. These days, he had a burst of inspiration, and the topological method based on closed-chain homology and braid group theory was making great strides in explaining experimental phenomena. He had already seen the dawn of solving the Navier-Stokes equations.

"So he wants to solve the Navier-Stokes equations single-handedly in a short period of time?"

Nakajima Kei was dumbfounded, yet also found it somewhat amusing.

He acknowledged that Chen Hui was an unparalleled genius, but the Navier-Stokes equations were not a simple problem either. Chen Hui might be able to complete the proof, but it would take at least three to five months, or at least three to five years.

"He's very confident!"

Dennis recalled his impressions of Chen Hui, so he guessed that Chen Hui must be struggling to deduce the proof of the Navier-Stokes equations at the International Congress of Mathematicians.

He understands Chen Hui because he is the same kind of person!

"Mathematicians need confidence!"

Kei Nakajima shook his head. "Mathematicians do need confidence, but not blind confidence."

"Complex geometry is nothing but an elegant castle in the air; I don't think it has a good chance of solving the Navier-Stokes equations."

Kei Nakajima was also an excellent mathematician, and naturally he had his own insights.

What made him even more frustrated and amused was that he was watching Chen Hui waste time on the wrong path and was about to miss the deadline for submitting his paper to the conference, which was truly heartbreaking.

"maybe."

Dennis remained noncommittal. He firmly believed he was right, but he wasn't sure Chen Hui was wrong; that guy had already created many miracles.

"There are still three months until the conference starts, no need to rush."

Dennis was also eager to see what results Chen Hui could achieve. Although they had parted ways, it was only because of their different approaches. He still recognized Chen Hui's abilities, and his research had once again encountered a bottleneck.

"The deadline for submission is the day after tomorrow," Nakajima Kei said worriedly.

"The purpose of submitting the manuscript three months in advance is to expedite the review process. Chen Hui's work has already undergone nearly a year of peer review, so there's no need to submit it three months in advance."

"Dennis said."

Nakajima Kei remained frowning. "But that's how the rules are."

"Aren't rules meant to be broken?"

Dennis didn't linger after speaking, and walked out of the building. He still needed to go back to study his own research project. Although he looked forward to Chen Hui's results, he didn't want to be left behind by Chen Hui.

……

Outside Chen Hui's office at Jiangcheng University
Yuan Xinyi looked at the still tightly closed door and felt somewhat worried.

It's already late April, and there's only a month and a half left until the International Congress of Mathematicians, but Chen Hui still hasn't come out. If it weren't for the occasional noises coming from the room, he would worry that something might have happened to Chen Hui.

Chen Hui probably won't be able to participate in this year's Philippine Awards; let's just hope he can win something.

Yuan Xinyi sighed softly, left the place, and returned to his teaching and research office. Perhaps seeing Chen Hui's research on the Millennium Problem, he is now also studying the application of the Langlands Program to N-to-NP problems and has already achieved some results. He is also very busy now.

Chloe would come to the office every afternoon at three o'clock, hoping that the teacher would open the door and answer her questions as before, but each time she was greeted by a tightly closed door.

Wang Qiming had also inquired about the situation with Yuan Xinyi several times. This was the first time he had encountered a reclusive scholar like Chen Hui. Even when others were tackling some difficult problems, they would at least come into contact with outsiders.

This reminded him of Perelman, but it didn't seem like a good thing; he was somewhat worried about Chen Hui.

In the office, Chen Hui's eyes seemed to hold complex symbols and turbulent energy. The draft papers spread out on the table were piled up like a small mountain, each page densely covered with distorted complex structure symbols, difficult -Neumann operator estimations, and countless failed attempts.

He closed his eyes wearily, his fingers unconsciously stroking the warm, moist pen.

With a thought, the proficiency panel appeared before my eyes.

[Host: Chen Hui
Chinese Level 2 (63%)

Mathematics Level 5 (21%)

English Level 3 (90%)

Physics Level 4 (87%)

Chemistry Level 3 (26%)

Biology Level 2 (0%)

Physical Education Level 2 (0%)

Geography Level 2 (0%)

History Level 2 (0%)

Politics Level 4 (81%)

After two months of hard work, my proficiency in each subject has increased slightly, with math showing a particularly significant improvement. However, there is still a long way to go before reaching the next level, and there is no hope of gaining any free attribute points.

This was the first time Chen Hui had stayed in one realm for so long. The pile of draft paper on the table seemed to be telling him that if he could not improve his mathematics level, his research would hardly make any breakthrough progress.

"Is it really only possible to rely on the skill of grinding water?"

Chen Hui hesitated for a moment, but quickly turned to the next page and continued his deduction.

Snap—Boom!
A muffled, loud bang seemed to explode right above my head, and the wind outside the window suddenly picked up, sounding like the roar of a giant beast.

In the blink of an eye, a light drizzle began to fall outside the window.

Chen Hui looked up at the window.

Thick black clouds, like mountains descending from the sky, pressed down overhead.

The branches and leaves of the large tree outside the window swayed back and forth, making a rustling sound. Chen Hui got up and closed the window.

Click!

Just as Chen Hui reached out and grabbed the glass window to push it open, a bolt of lightning, like a silver snake, cleaved through the clouds.

The next moment, large raindrops came down from the sky, and the rain became heavier and heavier. The awning outside the balcony was hit with a loud thud, and the water in the drainpipe flowed into a muddy waterfall, flowing down the wall and accumulating into small puddles on the bluestone slab.

The wind blew harder and harder, causing raindrops to pelt Chen Hui's window haphazardly, making the glass rattle incessantly as if it were about to shatter at any moment.

Suddenly, Chen Hui's eyes became unfocused as he looked out the window. The broken raindrops were no longer disordered dust; they seemed to be directed by an invisible hand full of mathematical rhythm, rapidly shuttling, colliding, combining, and decomposing in the void.

They disregard the trajectory of classical physics, depicting dazzling, higher-dimensional geometric scenes in three-dimensional space.

Lines begin to emerge—not straight Euclidean lines, but complex curves with pure transformations, gracefully twisting and extending to form the skeleton of the complex manifold.

The surface is then generated—a complex boundary that satisfies the strongly quasi-convex condition, appearing and disappearing in the particle flow, with the spectrum of the -Neumann operator □ spreading out like ripples.

Fibers fell like rain of light—countless magnetic powder filaments extended and intertwined from the main structure, constructing the prototype of the non-exchange fiber bundle that he had been pondering but had never been able to fully capture!

What's even more breathtaking is that the particle streams that make up the cluster structure spontaneously condense and arrange themselves, clearly and unmistakably outlining the core, globally controlling topological invariant—the first Chern class c1(V)! Its expression πi/2∫VTr(F) shines like an oracle in the particle stream!

"Topology is the bone...complex geometry is the soul..."

He muttered to himself absentmindedly, his whole body trembling violently with excitement, a surge of scalding heat rushing from the soles of his feet to the top of his head!

After painstakingly deducing for nearly half a year, the last boulder blocking the path to proof collapsed with a roar before this miraculous scene born of the power of heaven and earth!
Dennis's insistence on topology and his belief in complex geometry are not incompatible, but rather perfectly integrated within a higher-order, complex-topological unified framework that he has just glimpsed!

Chen's class c1(V) is the bridge that connects the bone and the soul. It not only firmly grasps the topological essence of vortex annihilation—the upper bound of energy dissipation—but also perfectly embeds itself within the regularization framework of his complex geometry!
"Hahaha! I see! I see!"

Chen Hui burst into laughter, as unrestrained as the raindrops outside the window.

He lunged at the table, grabbed the almost frozen pen, his hand trembling with extreme excitement, but his strokes were filled with a do-or-die determination and ecstatic joy.

Inspiration, like a flood bursting its banks, surged forth, overcoming all obstacles. Formulas, lemmas, and proof structures, with unprecedented clarity and power, poured from his pen and flowed onto the pristine white paper.

After an unknown amount of time, Chen Hui looked up from his draft paper. The wind and rain had stopped outside the window, and a ray of morning sun rose from his left, casting a golden ray of light through the window onto Chen Hui's desk, illuminating the title of the paper he had written on the draft paper: "Proving the smoothness of solutions to the Navier-Stokes equations through complex geometry compactification and Chern class control".

call!
With a light sigh, Chen Hui looked exhausted, but his eyes were full of joy.

During the final deduction, barrages of comments kept popping up in front of him, but he was so absorbed in the feeling of breakthrough that he didn't pay attention. Now, when he brought up the data panel, he immediately discovered that his math proficiency had increased by a full 16% in the past few hours, and his math proficiency had now reached level 5 with 37.

The improvement he made in those few hours was more than the improvement he made in the previous two months!
After tidying up the draft paper, Chen Hui turned on his computer. He didn't plan to organize and upload it immediately, but after two months of seclusion, he still had many things to take care of.

Sure enough, when I opened my email, dozens of unread emails appeared before my eyes.

He received emails from Terence Tao discussing the latest research topics, as well as from Schultz sharing his recent groundbreaking work on high-temperature superconductivity. Dennis even continued to email him to share his latest research findings.

The president of the International Mathematical Union sent him five emails to confirm matters related to the International Congress of Mathematicians.

After reading through them one by one, Chen Hui first replied to Terence Tao's introduction of his latest research project, and then explained to Schultz that he was currently in seclusion and was looking forward to his latest results.

After doing all that, Chen Hui opened the email Dennis had sent and clicked to reply.

Dear Professor Dennis,

I have completed the proof of the smoothness of the solutions to the Navier-Stokes equations. You are right; topology plays a crucial role in this proof. It is a pity that I was unable to complete the final proof with you.

The proof process is still being organized, and I will share it with you as soon as it is digitized.

Although he completed the final proof independently, he found it difficult to say how much Dennis's research on topology helped him. It was truly a pity that he would still use topology in the final proof after parting ways with Dennis.

He was even more moved by Dennis's continued sharing of his research findings with him; Dennis was a respectable elder and a true scholar.

Finally, Chen Hui replied to the email from Nakajima Kei, president of the International Mathematical Union, explaining that he had been focused on his research these past few days and had not been able to reply. He also stated that he would attend the International Congress of Mathematicians and bring his new achievement, the proof of the Navier-Stokes equations.

After finishing all that, Chen Hui snapped the computer shut, lay down on the cot next to him, and fell into a deep sleep as soon as his head hit the pillow.

My uncle and aunt have already informed me, so there's no need to worry. I can just video call them over after I've rested.

Little did he know the profound impact his emails would have on the world.

……

University of London, 8:00 AM
Dennis walked toward the lab as usual.
At over sixty years old, his diligence is still remarkable, a rarity even among mathematicians.

When you can learn something from it, learning truly is the most addictive thing in the world. If it weren't for his physical condition, Dennis wouldn't even want to go back to sleep.

"teacher."

As he entered the teaching and research office, his newly admitted student James was already there. Judging from his dark circles, James probably hadn't gone home last night, but he seemed to be in good spirits, which pleased Dennis.

Dennis nodded slightly. "How's it going? Any progress on the topology of the viscous dissipation term?"

This is a problem they have been working on recently: the viscous term that determines energy dissipation in the Navier-Stokes equations is the differential operator, whose effect depends on the local smoothness of the flow field.

Topological methods, such as closed-loop homology and braid groups, can only describe the overall topological class of vortex tubes and cannot characterize the energy dissipation rate caused by molecular-scale viscosity.

James shook his head, looking dejected.

However, even assuming that the vortex tube annihilation process corresponds to the transition [γ]+[γ]→[γ'] of the closed-chain homology group, the energy ΔE∝ν∫|ω| dV released by this process depends entirely on the local gradient of the annihilation region and has no direct quantization relationship with the global topological class [γ']. Therefore, explicit control of ΔE cannot be derived under the topological framework.

"It's okay, we're on the right track, and we'll succeed sooner or later."

Dennis smiled gently and patted the student on the shoulder. "What you need to do now is go back and get some sleep."

"Research is not something that can be done overnight. What we are trying to solve is a problem that the world's top mathematicians have not been able to solve for decades. This is a long-term battle. You must ensure that you are in good health. Only a healthy body can nurture a clear mind. Only a clear mind can grasp the truth hidden in the fog of numbers!"

Dennis was in a fairly good mood. He was very satisfied with his last disciple. He knew that proving the Navier-Stokes equation was not something that could be accomplished overnight, and he might not be able to do it in his lifetime, but his disciple had already done it!

James nodded as if he understood, and stepped out of the research room. He really needed a good night's sleep.

Dennis then returned to his seat, turned on his computer, and went through his emails every morning before starting work.

(End of this chapter)