My IQ has been increasing year by year.

Chapter 306 A Question

In Lecture Hall 1 of the Harvard Science Center, the derivation on the podium had been going on for more than an hour.

Zhu Xiping stood in front of the huge dark green blackboard, his white chalk constantly tapping and moving across the surface, leaving behind a series of dense partial differential equations.

"Here, we need to handle the extreme points of the Ricci flow."

Zhu Xiping's voice was a little hoarse. He didn't use a microphone, but simply raised his voice slightly so that people in the back row could hear him.

He turned around and drew a circle on the blackboard with his chalk-dust-covered finger, encircling a complex measurement parameter.

"As the curvature tensor approaches this region, the local geometry undergoes drastic changes. Following the previous steps, we introduce a smoothing function here to allow for a smooth transition..."

He turned around, picked up the chalk, and prepared to write down the convergence condition of the smooth function.

"Professor Zhu, please wait a moment."

A voice from the third row in the audience suddenly interrupted him.

The previously dense sound of note-taking in the meeting room suddenly stopped, and hundreds of people turned their gaze in the direction of the sound.

Professor Taubes pointed to the preprinted lecture notes spread out in front of him.

Zhu Xiping stopped writing with the chalk and turned his head.

"Professor Taubes, do you have any questions?" Zhu Xiping asked.

Taubes stood up, pushed the lecture notes forward, and frowned slightly.

"Regarding the smooth transition you just mentioned."

Taubes speaks slowly, but his pronunciation is very strong.

"At this extreme point, the curvature is close to diverging. If you forcibly introduce a smoothing function, the continuity of the topology will definitely break here, and your convergence condition will have no foundation to support it."

A low humming sound filled the lecture hall.

Many people lowered their heads, reread the thick lecture notes in their hands, and followed Taubers's line of thought to verify the boundary.

Zhu Xiping stood on the podium, looking at Tao Beisi.

He didn't immediately refute it, but turned around and faced the blackboard again.

He stared at the metric he had just circled for a long time.

Then, Zhu Xiping picked up the blackboard eraser and erased the half of the smooth function he had just written. He then changed to a new piece of chalk and wrote two more lines of derivation next to it.

"If we slightly relax the curvature limit when performing variational calculations..."

Zhu Xiping wrote and spoke at the same time.

He stopped halfway through the second line.

The chalk hangs half an inch in front of the blackboard.

No.

If the boundaries are relaxed, the subsequent geometrical rheology will become completely chaotic, and the previous derivations will lose their meaning.

This is a dead end in pure partial differential equations. As long as we are still working within the framework of a continuous manifold, the break at this extreme point is an insurmountable obstacle.

Zhu Xiping slowly lowered his hand that was holding the chalk.

He stared at the two and a half lines of equations on the blackboard, his brows furrowed tightly.

The lecture hall fell completely silent; no one urged anyone on or spoke.

Chen Zhuo sat in the center of the first row, looking at the area on the blackboard where he was stuck.

Those complex continuous measurements and divergent curvature tensors naturally swirled in his mind.

He saw the crux of the dead end.

Having stayed up all night, his brain was in a strange state of excitement. The logic of algebraic loops and quotient space mappings was now like something he had in his intuition. He knew how to handle that extreme point.

Chen Zhuo turned his head and glanced at Zhu Xiping on the stage.

A fine layer of sweat appeared on Professor Zhu's forehead, and his temples were also damp with sweat. He continued to stare at the blackboard, the chalk in his hand unconsciously twirling between his fingertips.

Chen Zhuo placed his hand on the table.

He didn't suddenly stand up, nor did he deliberately raise his voice to attract everyone's attention.

He simply sat in his chair and slightly raised his right hand—a small movement, but noticeable enough in the front row.

In the quiet meeting room, this action immediately attracted the attention of those around.

Sitting next to him, Qiu Chengtong turned his head and glanced at him.

Zhu Xiping on the stage also noticed the commotion. He shifted his gaze from the blackboard to the first row.

"Chen Zhuo?"

Zhu Xiping looked at him, his hoarse voice carrying a hint of inquiry.

Chen Zhuo lowered his hand.

"Professor Zhu, if the continuity cannot be smoothly transitioned here, can we look at this problem from a different angle?"

Hundreds of eyes in the hall were fixed on Chen Zhuo, and Tao Beisi was also standing on his seat, looking at the young man sitting in the first row.

Chen Zhuo didn't care about the stares from those around him.

"Since continuous measurement doesn't work, let's try adding a quotient space mapping here."

Chen Zhuo looked at the extreme point on the blackboard and continued speaking.

"Using discrete algebra, if we directly truncate at the boundary where the curvature diverges, do you think the physical metrics can re-close?"

His tone was inquisitive, without any hint of aggression.

The lecture hall was silent for a few seconds.

"Quotient space mapping? Discrete truncation?"

Taubes repeated the two words to himself in the audience. He looked down at his handouts and quickly drew several symbols on the paper.

Zhu Xiping on the stage was stunned for a moment.

He looked at Chen Zhuo, quickly translating the words Chen Zhuo had said into the dead end on the blackboard.

Continuous manifold, curvature divergence, fracture.

Quotient space mapping, folded redundancy, discrete truncation.

Zhu Xiping's eyes suddenly lit up.

He didn't say anything more, turned around, and strode to the blackboard.

He picked up the blackboard eraser and wiped away the two lines of equations he had just tried and failed.

Zhu Xiping picked up the chalk again and slammed it heavily onto the blackboard.

"If we introduce commercial space here..."

As Zhu Xiping spoke, the chalk in his hand began to move rapidly across the blackboard.

He replaced the originally complex continuous measurement parameters with a new algebraic symbol. When encountering the divergent curvature extremum, he did not try to smooth the transition, but instead directly added a cutoff condition to the formula.

The sound of chalk scraping against the blackboard became more rapid.

One line, two lines, three lines...

The previously stalled derivation logic, after this truncation was added, was like a bridge being forcibly erected, the broken boundary was bypassed, and the subsequent variational equations were naturally reconnected.

After Zhu Xiping finished writing the last concluding equation, he drew a heavy circle on the blackboard with chalk.

He stopped and took a half step back.

Looking at the new, somewhat domineering derivation path on the blackboard, Zhu Xiping let out a long sigh of relief.

"It's closed."

Zhu Xiping turned around and looked at Tao Beisi below the stage.

Tao Beis stood in the third row, his eyes following Zhu Xiping's chalk. When he saw the final equation, Tao Beis's brows completely relaxed.

He nodded.

"This is a very crude method."

Taubes looked at the blackboard, a hint of sigh in his voice.

"Abandoning local continuity and forcibly truncating it at the algebraic level, but undeniably, it did work, and I have no more problems."

Taubes sat back down in his seat.

The lecture hall came alive again.

Everyone picked up their pens and quickly jotted down the new derivations on the blackboard in their handouts and on their draft paper.

Zhu Xiping stood on the podium and wiped the sweat from his forehead.

He looked at Chen Zhuo in the first row.

Zhu Xiping smiled with relief and nodded to Chen Zhuo.

Chen Zhuo also looked at Professor Zhu on the stage and nodded slightly in return, as a junior's gesture.

The crisis is over.

Zhu Xiping picked up the water cup on the lectern, took a sip to moisten his throat, and turned to continue his lecture.

Once we got past that biggest dead end, the subsequent derivation of Ricci's rheology became much smoother.

Professor Zhu Xiping's seasoned skill and profound knowledge were fully demonstrated on the blackboard. Those complex geometric manifolds were deconstructed and recombined in his hands, gradually approaching the core of the Poincaré conjecture.

Chen Zhuo sat below, listening quietly.

He would nod slightly when he heard something interesting.

When Zhu Xiping wrote down an extremely ingenious topological transformation on the blackboard, Chen Zhuo reached out, picked up a pen from the table, and wrote down the formula for the topological transformation on a piece of paper.

After finishing writing, he simply marked two geometric symbols next to it, then put down his pen and continued to listen to the lecture.

Sitting next to him, Shing-Tung Yau noticed Chen Zhuo's actions.

The old man turned his head and glanced at the equations in Chen Zhuo's notebook.

"Old Zhu's geometric intuition is still sharp."

Old Qiu lowered his voice and said with a smile.

"Very solid."

Chen Zhuo turned his head slightly to the side.

"That transformation you just did was impressive. While I can take shortcuts in algebra, I'm definitely not as experienced as Professor Zhu and his team when it comes to refining these delicate manifolds."

Looking at this humble young man, Mr. Qiu's smile deepened. He didn't say anything more and turned back to look at the blackboard.

Time slowly passed amidst the sounds of chalk and paper turning.

Several large blackboards on the podium were pushed up and pulled down, densely covered with calculations.

Finally, Zhu Xiping wrote down the last conclusion of the day.

He tossed the remaining chalk stub, no bigger than a fingernail, into the chalk box, turned around, and dusted off his hands.

That concludes today's report on extreme point handling.

Zhu Xiping's voice was completely hoarse, revealing a kind of exhaustion after overexertion, but also a sense of relief after a heavy burden was lifted.

Thank you everyone.

The audience fell silent for a second.

Immediately afterwards, enthusiastic applause rang out in the auditorium.

The scholars in the front row applauded, and the students in the back row clapped their hands vigorously. The applause was for Zhu Xiping, who was working hard on the deduction on the stage, and also for the fact that they had finally overcome that difficulty today.

Chen Zhuo sat in his seat, clapping gently with his hands, a warm smile on his face, and clapped along with everyone.

The applause lasted for a while, then gradually subsided.

As Zhu Xiping stepped down from the podium, the atmosphere in the lecture hall completely relaxed.

Everyone started tidying up the lecture notes and draft papers on the table. Some people stood up to stretch their sore shoulders, while others gathered in twos and threes to discuss the details of the derivation they had just done.

"Today's lesson was worthwhile."

As Shing-Tung Yau packed his lecture notes into his briefcase, he smiled and said to Chen Zhuo.

"You've really helped Old Zhu out of a big problem."

"I just happened to think of it."

Chen Zhuo closed the notebook in front of him and put the pen in his pocket.

"Professor Qiu, I'll head back now, I'm a little sleepy."

Chen Zhuo greeted Qiu Chengtong.

"Go ahead, drive carefully." Old Qiu nodded.

Chen Zhuo turned around.

Zhu Xiping had just stepped down from the podium, surrounded by several young scholars who were still asking questions.

Chen Zhuo didn't go over to join the fun; he stood still, looking in Professor Zhu's direction.

When Zhu Xiping finished answering a question and looked up, Chen Zhuo, sitting a few rows away, nodded slightly to him as a farewell.

Zhu Xiping saw him, smiled, nodded, and waved.