Chapter 820 The key to the Poincaré conjecture
Perelman did not simply rewrite the steps Chang Haonan originally sent him -
For his level For a mathematician, doing this kind of thing is somewhat demeaning.
But based on that proof method, some optimizations have been made.
“In order to further reflect the beauty of this proof, we first introduce a concept: Τ-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 in front of him in a dumbfounded way. I wrote several pages of this notebook densely, shook my sore wrist, and asked in a low voice to my girlfriend next to me.
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 eyes into the distance...
I found that indeed, Chang Haonan, who had just been sitting and listening, now didn't know where to take it out. A notebook, writing and drawing in it.
"Hiss..."
There was another gasping sound.
Then there was a sound of turning pages.
Finally, there is the rustling sound when the paper and pen are rubbed...
However, if someone close to Chang Haonan takes a look, they will find that Chang Haonan is actually writing on the paper. is not what is on the blackboard.
Instead, I drew 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 three-dimensional space has never been overcome.
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 equation generated by the action of a finite group distance quotient space, then it seems to be diffeomorphic to a three-dimensional compact manifold..."
Chang Haonan's ears have gradually stopped hearing Perelman's voice:
"It seems that we cannot directly draw this conclusion."
He frowned slightly:
"But if you add a limiting condition...to make the Ricci curvature of this manifold non-negative..."
"..."
< br>Off 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 walked to the next one as usual.
But this time, he didn’t start writing immediately.
But he raised his hand to wipe the sweat from his forehead.
He has been speaking continuously on the 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.
So, it takes a lot more effort than a simple lecture.
Fortunately, the staff nearby had already prepared and took this opportunity to quickly put a glass of warm water on the small table——
If you are a Chinese scholar, this link will usually be straightforward Hot tea was served, but considering that foreigners might not be used to this step and would 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.
The eyes focused on the bottom of the first blackboard.
Since this was the first time that this method was systematically sorted out, there were some details that even Perelman himself failed to notice at the first time.
There is an inequality.
R≥(-v)[lg(-v)+lg(1+t)-3]
Originally, he just regarded 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 goes to infinity at all times, 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.
Yes, 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.
The eyes suddenly met with Perelman's.
Although the two didn't say a word to each other, they both saw one thing from their eyes - the other person and themselves were thinking about the same thing.
Two top scholars in the field of differential geometry finally reached 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 of using Ricci flow is more feasible than the direct topological method.
Therefore.
This is probably 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.
We also need to solve the problem that the general initial metric causes the Ricci flow to produce singular points.
But.
These are all 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 march 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 "walking a hundred miles." Or half ninety."
Of course, neither Perelman nor Chang Haonan would agree to use the combination of the two people's 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 did not participate 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 instead of You will see, at least not for a short time, that one of these humble inequalities will have a historic impact on the entire world of mathematics.
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 open his brows.
Although Perelman has not finished writing the entire derivation process at this time, he has already thought of the remaining steps.
Compared with 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...so that 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 hands folded, looking at the blackboard in front of him, without speaking.
The lecture hall that had just been quiet gradually began to be filled with whispers.
"As expected of Professor Tian... he can get the results faster than me."
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 have been standing there for two hours, and I have already pushed halfway. I am just relying on my energy to take advantage..."
Tian Gangzhuo waved his hand.
Having said that, he was still relatively happy——
Although the result definitely belongs to Perelman, and perhaps to Chang Haonan who was mentioned just now, the fact that I was able to deduce it myself before reading the proof process at least shows that I have not fallen too far behind in terms of ability...
Of course, he doesn’t know yet at this time.
Perelman stopped because he had seen a road leading to a higher mountain.
In comparison, these things on the blackboard are not worth mentioning at all...
"Please give me a few pieces of paper, thank you."
After a few minutes of silence, Perelman's first words were not to continue introducing his ideas, but to ask for paper and pen from the floor manager.
"Huh?"
"Why do you need these all of a sudden..."
"Is there something wrong with the derivation process?"
"No way... ...I just vaguely heard that several teachers in front of me have already introduced this idea, and I understand it and feel that it is quite smooth..."
"Is there a possibility that even we can understand it, but it is more likely that there is something wrong?"
"Don't..."
"..."< br>
The strange behavior made the scene agitated again.
But Perelman turned a deaf ear.
He took the paper and pen, sat in front of the temporary desk, lowered his head and started his calculations...
(End of Chapter)