How powerful is Professor Feng?
What conjectures did he prove, what theoretical foundation did he lay, what achievements did he ultimately make, and how was he evaluated?
Lin Chaoxi really wanted to know this.
In the following days, Lin Chaoxi still maintained the rhythm of attending classes during the day, training in the evening, writing programs at Chen Zhu's house at night, and supervising Lao Lin late at night.
When she is free, she will start searching for terms she has never heard of before based on Feng Deming's personal academic achievements in Baidu Encyclopedia.
The metric space structure of a certain manifold, scaling theory and its fractal geometry... Behind those terms are many difficult papers, which seem to be the contents of another space.
The development of relevant academic forums seven years ago was far behind what it is now. Many times, she could not find out what a certain theory or conjecture meant, and could only roughly understand what branch they belonged to.
So she didn't treat these things as serious work, but only started to check them leisurely when she was tired, more like gossiping, a kind of aimless academic gossip.
Professor Feng has made great achievements and is highly respected. Whether in academic conferences or on the Yongchuan University forum, students truly admire him.
Her fame was like a huge wave, sometimes making it hard to breathe. It seemed that even if she brought all of Lao Lin's unfinished work back to reality, compared with Professor Feng's achievements, it would be like the light of a firefly compared to the bright moon.
Lin Chaoxi could only comfort himself. It didn't matter. Old Lin's target had never been him.
During this long period of time, Pei Zhi would occasionally accompany her at night. After three weeks, Lin Chaoxi confirmed that this "occasionally" specifically refers to Wednesday nights and Saturdays.
On Wednesday, they went to Chen Zhu's house to surf the Internet together. Pei Zhi would also use the computer for a while. At ten o'clock in the evening, Pei Zhi would send her home and then leave alone.
Saturday is Lin’s cram school. After Lin Chaoxi finishes his Olympiad math class for the elementary school students in the morning, he starts to learn some graph theory-related content with the two of them.
In these days that seemed to be as plain as water but Lin Chaoxi was becoming increasingly anxious, Lao Lin's proof work made real progress.
Of course, the so-called progress meant that Lin Chaoxi finally waited until he got to the page of draft where Old Lin made the mistake.
It was a Saturday in early autumn. She was standing in front of the podium. As usual, she secretly showed Lao Lin dozens of pages of drafts. Her mind was full of v(g) and v(h) and set mathematics symbols.
On the blackboard was the model of the Seven Bridges that she had just drawn. The content that was used to prove the history of the times by Euler hundreds of years ago could now be explained to elementary school students.
"In fact, Mr. Ola's method of proof was very simple. He represented the island as a vertex and the bridge connecting the land as a line. Then the map I drew would become a simpler pattern."
Lin Chaoxi drew a shape similar to an ice cream cone in a very realistic manner.
“It seems much simpler.”
"But is it really impossible to get down all at once?"
The children's questions were heard again.
The inspiration was almost there, Lin Chaoxi said: "Can you guys give it a try?"
"How do you try?"
“Draw a route starting from the starting point, passing through each bridge, and then returning to the starting point to see if you can find that road~” The chalk moves on the vertex and the line.
"Or prove it mathematically that it's impossible," interrupted the ambitious little boy.
Lin Zhaoxi nodded.
Before she finished speaking, many students began to discuss among themselves. Some decided to try them one by one, while others wanted to find a simple, mathematical method. Lin Chaoxi said nothing more.
She went down and walked around twice, answered a few questions, and then was driven back to her seat by the children.
She could only take out the draft that she had just "stolen" and read it casually.
The autumn wind blows by, and some leaves fall outside the window, crisp and golden.
Lin Chaoxi flipped through the draft that she had already read in the morning. She felt that the problem that Old Lin had made before should be in it. She was very close, but she couldn't find it.
During this period, a group of students who decided to divide the work and count the total number of ways to move had a small quarrel over how to work. Lin Chaoxi ran over and gave them an idea.
Just as I sat down and picked up the pen, I heard the "courageous" little boy yelling, "How do you prove that Ola is wrong?"
She turned her eyes away from the draft in front of her and looked at the boy.
"How could Ola be wrong!" another person retorted.
“I think it’s possible to get it done in one go!”
"Then you can look for it. If you find the one that says 'don't go back', you've found a counterexample. Finding a counterexample can prove that Euler is wrong." said another very organized little girl.
The classroom was filled with these noises, which were noisy yet made people feel exceptionally peaceful. Lin Chaoxi did not try to stop their argument.
She turned the page and saw a few lines of proof.
The memories in my mind and the manuscript in front of me gradually overlapped, and the voices of the children in my ears became quiet.
——Found a counterexample.
It is here.
——
Unable to wait until the class was over, Lin Chaoxi could not contain his excitement and rushed directly to Lao Lin's office with his manuscript paper.
She pushed open the door, slapped the paper on the table, and held the red pen she had just used to mark the children's homework. She pointed the tip of the pen downward and completely circled a few lines of proof. Then she pushed it in front of Lao Lin and said, "There's something wrong here!"
Lin Chaoxi's heart was beating very fast. She stared at her father's briefly surprised face, then took a half step back: "I'm going back to class!"
She was well aware of how exaggerated her actions were, and now she wanted to run out the door. But before she even reached the door, she was stopped.
"Wait." Old Lin paused, "Turn around, come here, and sit down."
Lin Chaoxi leaned against the door, feeling hopeless, but had to turn around slowly.
Lao Lin and Pei Zhi in the office had no time to pay attention to her. Their expressions were very consistent. After a brief surprise, they frowned and looked at the contents she circled carefully.
Lin Chaoxi sat beside him respectfully, not daring to make a sound.
At that time, Lao Lin realized that his proof was wrong because there was a problem with the assumption. After proving that the mapping induced a certain automorphism to be g(p), he directly identified s as its subset.
She had memorized the key symbols such as c→c, aibj→b, etc., but she did not fully understand why there was a problem with this assumption. And because the proofs she had done repeatedly were full of these symbols, she did not recognize this until the child said "counterexample".
Yes, it is essentially counterexamples. When mathematicians encounter difficulties in proving a proposition, they will start looking for counterexamples to prove that it is not true. But they can easily forget about it in their daily work.
The camphor tree outside Lao Lin's office never turns yellow and remains green in autumn. Lin Chaoxi took a deep breath and heard him say, "You are right. This is wrong."
His expression inevitably showed loss, regret, a bit solemn, but also relieved. Admitting mistakes meant that the wall had appeared, all his previous efforts had gone to waste, and all assumptions had to be completely overturned. For anyone who had worked hard for a long time, this seemed extremely cruel.
Lin Chaoxi pursed her lips, and for a moment she didn't know how to help Old Lin regain his confidence and then discover the new algorithm based on this.
"How did you find out?" Lao Lin interrupted her thoughts.
"Ah? Just now the kid suddenly shouted 'counterexample', and I was wondering, if you made this assumption, did you fail to consider the empty set?
She made up a reason. The actual problem was not that simple. It was impossible for a high school student like her to discover it, and it had nothing to do with Kong Ji. But she still observed Lao Lin, hoping that her sudden reason could get away with it.
"That's a bit ridiculous." Lao Lin concluded.
"!!!" Lin Chaoxi immediately felt guilty, and she tried to refute it, "It's really like this."
"You should say: The God of Mathematics suddenly favored me." Lao Lin spoke slowly.
"The God of Mathematics suddenly favors me?"
"Yeah." Old Lin looked relaxed and calm, looking at the area she circled with a smile, "You suddenly showed me some favor as well."
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