After heading downstairs, Li Dong stopped by a nearby supermarket.
He picked out two bottles of what looked to be decent-quality liquor and also bought two boxes of imported cherries.
He got on a bus and made his way to the Sunshine Garden District.
Although Li Dong had joked about squeezing Old Yang for all he was worth, he knew the truth in his heart.
Old Yang had been very good to him, a kindness he had never forgotten.
If Old Yang hadn’t bent the so-called rules of professional ethics to recommend him to Sixth Middle School, he might still have good grades now, but he definitely wouldn’t have had such a broadened perspective.
To put it bluntly, Seventh Middle School didn’t even have the qualifications to register for the Qiu Prize.
So even though the school had only given them a two-day break, Li Dong felt this was a visit he had to make.
Li Dong stood before the door to apartment 406 and knocked gently.
"Coming, coming!"
Old Yang’s voice came from inside.
The door opened. Old Yang, wearing a sweater, saw Li Dong standing outside and immediately broke into a smile.
"You kid, couldn’t you stay home a little longer? What’s the rush?"
Old Yang looked at the bags, big and small, in Li Dong’s hands and said out of habit.
"You should have just come, what’s the point of buying all this stuff? You’re still a student, you don’t have..."
He was only halfway through his sentence when he remembered that this kid was richer than him now. He dropped the pretense and smilingly took the gifts inside.
In the living room, Old Yang’s wife was on the sofa watching TV with a girl in a middle school uniform.
When she saw Li Dong enter, she smiled and nodded.
"Hello, Ma’am! Happy New Year!" Li Dong greeted her politely.
"Ah, Xiaodong, you’re here! Come, sit down, sit down. I’ll go wash some fruit for you," his wife said warmly.
Old Yang pointed to the girl on the sofa who was secretly sizing Li Dong up. "This is my daughter, Ningning."
"Ningning, hurry and say hi to your big brother. This is the one Dad’s always telling you about at home, Brother Li Dong, the one who took first place in the national physics competition."
Yang Ning was just a middle schooler, after all. Being introduced like that by her dad, she blushed slightly and murmured quietly.
"Hello, Brother Li Dong."
"Hi, Ningning," Li Dong replied with a smile.
After a few more pleasantries, Old Yang spoke.
"Alright, you two can keep watching TV. Li Dong, come to the study with me."
Entering the study, Old Yang pulled out a chair for Li Dong to sit, then sat down opposite him.
He carefully sized up the young man before him.
He was standing taller and straighter than before, his eyes were clear, and he exuded an air of confidence.
"Attaboy," Old Yang said, nodding in satisfaction.
"You’ve got a great spirit about you."
"It seems sending you to Sixth Middle School was the right move. The environment there really is a better fit for you."
Li Dong said, "That’s all thanks to you, Teacher Yang. Otherwise, who knows where I’d be right now."
Hearing this, Old Yang was clearly taken aback for a moment.
He looked at Li Dong with some surprise.
"Oh? A trip to Sixth Middle School didn’t just boost your spirits, your EQ seems to have gone up too? You were never this smooth-tongued before, kid."
"Hehe, people have to grow up sometime."
Old Yang nodded in satisfaction and asked casually, "The pace for third-years at Sixth Middle School must be intense, right? How many days off did you get for the Spring Festival?"
"Just two," Li Dong said helplessly. "I have to be back at school for evening self-study tomorrow afternoon."
Hearing this answer, Old Yang gave a wry smile.
"Sigh, that’s how all these super high schools are nowadays. They’d love to squeeze every last second out of their students." Old Yang habitually prepared to offer some motivational platitudes.
"But hey, it’s just for these last few months, no matter how tough it gets."
"Once you get through the college entrance exams and get to university, you can do whatever you want..."
Halfway through his sentence, Old Yang’s voice trailed off.
He looked at Li Dong and suddenly realized that his usual spiel for placating ordinary high school students didn’t seem quite right for the kid in front of him.
For the average high school student, what is university?
It’s the end of a sea of suffering and the beginning of freedom.
Once in university, most students just play games, date, and join clubs.
For exams, their motto is ’just pass and you’re golden.’ As long as they don’t fail, scrape together enough credits, and get their diploma, they consider the four years a success.
Those with a little more ambition might go to the library to snag a seat and study for the grad school entrance exams. But a large portion of them are just going with the flow, caught up in the modern academic rat race, not because they genuinely love learning.
But Li Dong was different.
In Old Yang’s mind, the current high school education system, or rather, the entire college entrance exam selection mechanism, was like an iron cage, locking Li Dong down tight.
Forcing him to memorize classical Chinese texts and fixed English phrases every day was, in fact, a huge waste of Li Dong’s talent for scientific research.
Once Li Dong entered a top-tier university...
The cage would be broken. At that point, the name Li Dong would likely be known far beyond the student circles of a small place like Jiangcheng.
Old Yang swallowed the rest of his words and changed the subject.
"I heard you’re tackling both physics and biology now. You haven’t let your math slip, have you?"
The moment Li Dong heard this, he let out a long sigh.
’Teacher Yang, oh, Teacher Yang, I really came here today purely out of gratitude. Why must you insist on handing me the shears yourself?’
Seeing Li Dong sigh, Old Yang asked with some concern.
"What’s wrong? Stuck on math? Hit a bottleneck? Where’s the problem?"
Li Dong looked up, his eyes burning with a thirst for knowledge!
Stared at by such a gaze, Old Yang’s heart skipped a beat.
’Oh no, did this kid set me up?’
’His New Year’s visit was just a pretense. He really came to get some "tutoring" from me, didn’t he?’
While Old Yang was still stunned, Li Dong had already started speaking.
"Teacher Yang, I ran into a problem in physics before. The teachers at Sixth Middle School said it was a ’knowledge island’—that the different fields of knowledge are disconnected and lack bridges."
"Lately, whenever I have free time, I’ve been looking at things like Real-Valued Function, Functional Analysis, and Algebraic Geometry, and I’ve found that I seem to have the same problem in mathematics."
This was a genuine problem Li Dong had been facing recently.
"For example, I understand the basics of group theory and I understand mappings in topology, but when I try to combine them to understand some deeper conjectures, I always feel like there’s a bridge missing in the middle that I just can’t cross."
"How exactly do you build a bridge in mathematics?"
After hearing Li Dong’s dilemma, Old Yang leaned back in his chair and let out a hearty laugh.
The laughter left Li Dong completely baffled.
’What’s with Teacher Yang? I’ve hit a bottleneck, so why is he laughing so happily?’
Old Yang laughed for a good while before stopping.
He looked at Li Dong, a long-forgotten sense of accomplishment as a teacher welling up inside him.
’You rascal, so you’re not some omnipotent monster after all! It still takes me to set you straight!’
Why was Old Yang so happy?
Because before this, his impression of Li Dong was simply too absurd.
It seemed like no matter how difficult the problem or how advanced the university-level concept, the kid could master it as soon as it was put in front of him.
Old Yang had once suspected that he, a master’s graduate from Capital University, might no longer have anything to teach him in mathematics.
But now, knowing that there were things Li Dong didn’t know, that he had systematic gaps in his thinking, Old Yang felt even more gratified.
In academia, the scary part isn’t not knowing something; it’s being ignorant and blindly assuming you know everything!
Old Yang’s smile vanished as he said, "Li Dong, in physics, knowledge islands do exist, and you absolutely must build bridges."
"That’s because the essence of physics is the exploration of the objective truths of the universe! God set the rules in stone. Classical Mechanics, electromagnetism, thermodynamics, quantum mechanics... the distances between these islands are fixed."
As he spoke, Old Yang drew a few circles on a blank piece of paper on the desk.
"If you want to walk from the island of electromagnetism to the island of optics, you must have the bridge of Maxwell’s Equations! This bridge isn’t something you can conjure from your imagination; it’s a law that already exists in the universe. You merely discover it."
At this point, Old Yang straightened his back, a look of disdain for physics on his face.
"But! Mathematics is different!"
"Mathematics has islands too, but its bridges are different from those in physics!"
Old Yang’s eyes were full of reverence for mathematics.
"Mathematics is a product of pure Logic! Therefore, in the world of mathematics, you can build a bridge wherever you want! You can cross it however you want!"
"To put it bluntly, mathematics is a tool."
"When the island of algebra can’t solve a problem, you can use the bricks of geometry to build a bridge."
"If you feel two islands are too far apart..."
"You can even just smash the two islands together!"
"Take Euler, for example. When he was lost in algebra, he used a single formula, e^(iπ) + 1 = 0, to forcibly drag the natural logarithm base, the imaginary unit, pi, and the basic integers together from three completely unrelated islands, creating the most beautiful bridge in the history of mathematics!"
Old Yang just looked at Li Dong.
"So, don’t go looking for inherent bridges in mathematics! That’s just rote learning!"
"When you encounter a problem, you have to create that bridge yourself!"
After Old Yang finished his speech, Li Dong understood.
’That’s right!’
’Why was I trying to brute-force my way through existing theorem connections, like I was studying physics?’
’The boundaries of mathematics are man-made! As long as the logic is self-consistent, I can forcibly construct a new spatial mapping just to solve a single problem or a single model.’
Seeing Li Dong’s eyes grow brighter and brighter, Old Yang felt a deep sense of satisfaction.
’This is what they call "a great teacher produces a brilliant student"!’
Old Yang took a sip of his tea and, in a magnanimous mood, casually added.
"So? Figured it out? If there’s anything else you don’t understand, just ask. Your teacher will answer everything for you today."
However, the moment the words left his mouth, Old Yang regretted them.
Because he saw Li Dong nodding frantically.
"Yes, yes, yes! There’s more, there’s more!"
Li Dong looked at the powered-on computer on Old Yang’s desk.
"Teacher Yang, may I use your computer?"
’It’s the New Year, not a tutoring session!’ Old Yang was a little speechless.
But he had already made the offer, so he could only nod helplessly.
"Go ahead, go ahead."
Li Dong immediately pulled the keyboard over and quickly typed in a URL.
’Who taught you to memorize URLs including the subdomain?’ Old Yang watched from the side, his heart pounding.
After typing in the long, complex URL, Li Dong hit Enter.
An ugly, English-language webpage popped up.
Duke Mathematical Journal.
The content within was the very same top-tier paper that Old Yang had opened to teach him about paper formatting.
On Local-Global Compatibility for Automorphic Representations of GL2 in the Case of Ramification Index ≤ 2
Authors: Jiang Yubai, Zhou Shenzhi.

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