Chapter 820 The key to Poincaré’s conjecture

Perelman did not simply rewrite the steps Chang Haonan originally sent him -

For a mathematician of his level, doing this kind of thing is somewhat demeaning.

Instead, some optimizations have been made based on that proof method.

"In order to further reflect the beauty of this proof, we first introduce a concept: t-length..."

"Do you... still... remember this?"

While Perelman was writing equations on the blackboard, a young teacher looked at the blackboard that had just been wiped clean and the notebook in front of him with several densely written pages. He shook his sore wrist and said, He asked in a low voice to his girlfriend next to him.

He is not a researcher in the direction of differential geometry. He was just a ruthless note-copying tool just now, and now...

I really can’t write anymore.

"Of course you have to memorize it. You see, even Professor Chang is lowering his head to memorize it. Are you better than him?"

Several people nearby who heard this sentence immediately cast their gaze into the distance...

I found that sure enough, Chang Haonan, who had just been sitting and listening, now took out a notebook from nowhere and was writing and drawing in it.

"hiss……"

There was another gasping sound.

Then there was the sound of turning pages.

Finally, there is the rustling sound when paper and pen are rubbed together...

However, if someone close to Chang Haonan came over and took a look, they would find that what Chang Haonan actually wrote on the paper was not what was written on the blackboard.

Instead, draw a ball with a pencil.

This is an extremely rare situation.

Because for research in the field of differential geometry, high-dimensional spaces are often easier than low-dimensional spaces.

Take the Poincaré conjecture as an example. The Poincaré conjecture in five-dimensional or even four-dimensional space has actually been proven long ago.

But the barrier of three-dimensional space has never been overcome.

And as everyone knows.

On paper, it is impossible to draw a high-dimensional space.

You can only rely on imagination or calculation.

In fact, even what Perelman is talking about on the blackboard at this moment is mainly about four-dimensional space.

However, the content he optimized on the blackboard pointed out a brand new possibility to Chang Haonan...

"If this is a free isometric quotient space generated by finite group action, then it seems to be diffeomorphic to a three-dimensional compact manifold..."

Chang Haonan could no longer hear Perelman's voice in his ears:

"It seems that we cannot directly draw this conclusion."

He frowned slightly:

"But if a restriction is added...to make the Ricci curvature of this manifold non-negative..."

"..."

Under the stage, Chang Haonan was lowering his head, immersed in his own thoughts.

On the stage, Perelman was giving his lecture as usual.

According to the plan, after comparing the three types of singular models, he will be able to draw the same conclusion as before.

After using up one blackboard again, Perelman moved on to the next one as usual.

But this time, he didn't start writing immediately.

Instead, he raised his hand to wipe the sweat from his forehead.

He has been speaking continuously on stage for nearly two hours.

My energy and stamina are indeed a little behind.

In fact, the idea on the blackboard was even thought of by him on the plane to China. He used it as the content of the lecture with the intention of introducing it and verifying it at the same time.

Therefore, it takes a lot more effort than a simple lecture.

Fortunately, the staff nearby had already prepared it, and took this opportunity to quickly put a glass of warm water on the small table——

If you are a Chinese scholar, you will usually serve hot tea directly at this stage. However, considering that foreigners may not adapt to this step and may be burned, the temperature was lowered under Tang Lintian's special care.

Perelman was not polite. He came to the table and picked up the tea cup. While drinking water, he looked at the first two blackboards that were already filled with his writing.

Suddenly, the movements of his hands stopped.

His eyes focused on the bottom of the first blackboard.

Since this was the first time that this approach had been systematically sorted out, there were some details that even Perelman himself failed to notice at the first time.

There's an inequality there.

R≥(-v)[lg(-v)+lg(1+t)-3]

Originally, he just used it as a common estimate generated during the derivation process, but looking back now, it seems that some very interesting conclusions can be obtained along this direction...

For example, when the curvature approaches infinity at a time, the smallest negative cross-sectional curvature is smaller than the largest positive cross-sectional curvature.

In other words, the three-dimensional limit solution must have a nonnegative curvature operator.

That's right, three dimensions.

Perelman didn't even have time to put down his teacup, so he turned to look at Chang Haonan sitting in the audience.

I found that the latter was concentrating on writing something with his head down.

At this time, Chang Haonan finally proved his conjecture on paper.

He raised his head.

His eyes suddenly met with Perelman's.

Although the two of them didn’t say a word, they both saw one thing in their eyes——

The other party and myself thought together.

Two top scholars in the field of differential geometry finally came to the same conclusion through relatively independent thinking.

That basically eliminates the possibility that this conclusion is wrong.

In other words, it is feasible to perform surgery on Ricci flow in three-dimensional space.

For the differential geometers of this millennium, there is a consensus.

To solve the Poincaré conjecture problem in three-dimensional space, the geometric method using Ricci flow is more feasible than the direct topological method.

therefore.

This could very well be a key.

A key to Poincaré's conjecture.

Of course, even if you do open the door with the key, there's still some work to be done.

For example, it is necessary to ensure that the appropriate neck area can be found for truncation surgery within a limited number of operations.

It is also necessary to solve the problem that the general initial metric causes the Ricci flow to produce singular points.

but.

These are details.

It can even be said that it is a problem that can be solved by just spending time.

If we say that Chang's Lemma is just the first step in a long journey for Poincaré's conjecture.

So today’s conclusion——

Perhaps it can be called the three-dimensional manifold theorem, or more simply, the Perelman-Constant theorem, which can be regarded as "a man who travels a hundred miles is half a mile".

Of course, neither Perelman nor Chang Haonan would agree to use the combination of their last names in this place.

Because if they continue in this direction, their surnames will most likely be directly named after "Poincaré".

……

At the same time, other listeners below were also taking advantage of this rare buffer period to review the contents of the notes they had just written down.

Of course, these people were not involved in the initial derivation of the process on the blackboard, so their mindset dictated that they would definitely think along the steps Perelman wrote on the blackboard without seeing that, at least in It will not be obvious for a short time that one of these humble inequalities will have a historic impact on the entire mathematical world.

However, most of them are professional mathematicians after all, so it is impossible to gain nothing...

"I seem to understand..."

Tian Gang was the first to frown.

Although Perelman has not yet finished writing the entire derivation process, he has already thought of the remaining steps.

Compared to the first solution, which is a headache, the one currently written on the blackboard is obviously much simpler and easier to understand.

"It is indeed... an extremely exquisite proof... This way, the local injective radius of the compact manifold can be directly calculated..."

"It's called the non-local collapse theorem, and it's very accurate..."

He whispered to himself, attracting the attention of several people next to him.

Soon, Tian Gang's notes were circulated.

Perelman, on the stage, still stood there with his arms folded, looking at the blackboard in front of him, without speaking.

The lecture hall that had just been quiet gradually started to hear some whispers.

"As expected of Professor Tian... he can get the results faster than I did first."

A scholar looked at the notebook in front of him, then raised his head to check the half-written content on the blackboard.

"Where... I've been standing there for two hours, and I'm halfway through pushing. It's just because of my energy that I took advantage..."

Tian Gang waved his hand.

Even so, he was still relatively happy in his heart——

Although the result definitely belongs to Perelman, and perhaps to Chang Haonan who was mentioned just now, the fact that he was able to deduce it on his own before reading the proof process at least shows that he has not fallen too far behind in terms of ability...

Of course, he didn't know it yet.

Perelman stopped because he could see a path leading to a higher mountain.

In comparison, these things on the blackboard are not worth mentioning at all...

"Please give me some paper, thank you."

After a few minutes of silence, Perelman's first words were not to continue introducing his ideas, but to ask the floor manager for pen and paper.

"Ok?"

"Why do you want these all of a sudden..."

"Is there something wrong with the derivation process?"

"No way... I just vaguely heard that the previous teachers have already introduced this idea, and I understand it and it feels pretty smooth..."

"Is there a possibility that even we can understand it, but it is more likely that there is something wrong?"

"Don't..."

"..."