Top Student at Their Peak
Chapter 41 - 41 41 Hard Effort

41: Chapter 41 Hard Effort 41: Chapter 41 Hard Effort Woken up by Zhou Shuang’s yell and vaguely given some advice, Qiao Yu wanted to continue sleeping but found himself unable to sleep.

This was probably thanks to the good habits Qiao Yu had developed previously.

Unable to sleep, Qiao Yu took out his workbook and started pondering over the problems.

Qiao Yu developed a strong interest in algebra and number theory, and at night, he would read books while using the internet to find various teaching videos to deepen his understanding of modern algebra and number theory.

Qiao Yu was truly grateful for the all-powerful internet.

Online, not only could he easily find full lecture videos from top professors of Huaxia’s top universities, but he could also find complete videos from top international universities.

Yanbei, Huaqing, Shuangdan, Xijiao, and internationally, MIT, Princeton Institute for Advanced Study, Harvard…

they were really all there!

If he was patient enough, Qiao Yu even found popular science math lecture videos on number theory posted online by Fields Medal winners.

Although many were not in blackboard form, such as the Fields Medal winner, Richard E.

BORCHERDS, who started his explanation with just a pen and paper, to Qiao Yu, it was almost the same.

Moreover, Qiao Yu’s English was strong, and he only lacked some math-specific terms, which, after encountering a few times, he could integrate and understand, allowing him to absorb those experts’ most complete explanations directly, without being troubled by poor subtitle translations, hence his rapid progress.

This is also why he entered a learning state at night and unknowingly studied until three or four in the morning, experiencing the joy of learning once again with the high level of difficulty.

Can’t sleep, can’t sleep at all.

During the day, it was a bit troublesome to watch videos on his phone in class, so he simply used it to work on problems instead.

Qiao Yu copied many algebra and number theory problems from the internet into his workbook, and luckily, the finals track of the Little Alibaba World Mathematics Competition also included algebra and number theory.

Moreover, the problems were extremely ingenious, so naturally, Qiao Yu noted them down as well.

Studying and preparing for the competition went hand in hand.

Understanding the distribution of prime numbers and solving large prime factorization problems were part of his mid- and long-term financial planning dreams.

Receiving the school’s awards and the Little Alibaba competition’s prizes were his short-term plans.

People have to survive first before considering the future.

If possible, it’s best to live more comfortably.

Money can provide adequate nutrition and maintain a good mood, so it is very important.

On one hand, Qiao Yu took out his workbook, while on the other hand, Zhou Shuang, who wasn’t yet in a learning state, silently pondered over Qiao Yu’s recent words and stuck his head over.

There was no way around it, he now had an extremely curious mind about everything Qiao Yu did.

Then he saw what seemed like a math problem, ‘seemed like’ because he couldn’t understand the problem statement that Qiao Yu had copied.

The most frustrating part was that the statement, when taken apart, consisted of words he recognized, apart from a few strange letters, but when put together, gave him the feeling of reading a cultivation manual in a fantasy novel.

“Is this a math problem?

What does an ideal mean?” Zhou Shuang couldn’t help but ask.

The problem seemed too abstract, what was meant by an ideal I?

What did it mean for an ideal to be closed?

Wasn’t it in elementary school where the language teacher said an ideal was an individual’s plan and yearning for the future?

How could it be closed?

“Yes, it’s a math problem.

This ideal is not the ideal in language, it’s a concept within ring theory.

You can understand an ideal as a special subset of a ring.”

“What’s ring theory?”

“Haven’t heard of it, right?”

“Nope.”

“Then have you heard of Linear Algebra?

Ideal is similar to a subspace in a vector space in Linear Algebra.

This stuff, you will definitely encounter it when you go to college.”

“A big universe enveloping a small universe?” Although Zhou Shuang hadn’t heard of Linear Algebra, he felt he understood subspace.

Fantasy novels often contained such settings, where the protagonist discovers, after ascending from their original world, that the universe they were in was just a branch of a bigger universe.

To make progress, they had to continue leveling up by repeating the tasks done in smaller worlds.

Qiao Yu glanced sideways at Zhou Shuang and then nodded affirmatively, stating that this understanding was really spot on!

“So, why does this math problem seem like a fantasy novel?

Can it really be solved?” Zhou Shuang asked again, like a curious baby.

“You haven’t seen the original problem.

The original statement wasn’t like this, it was even more abstract.

This is my analysis after reviewing the original problem.

There definitely is a solution because the conditions are very clear; the ideal I is closed, implying that when scaling variables x and y, the degree of the polynomial remains unchanged.

Given that the dimension of the quotient ring is 6, it indicates there are 6 independent quotient ring generators.

Combined with other conditions, it’s known that these ideals have specific algebraic geometric structures.

Along with the dimensional condition and scaling invariance, you can deduce that the number of ideals I is finite.

See, once you have the idea, this problem isn’t that hard, right?”

Qiao Yu casually explained to Zhou Shuang, a clear case of ‘chicken talking to duck’.

He knew Zhou Shuang certainly didn’t understand, in truth, he was persuading this guy to back off when faced with difficulty.

Ring theory, group theory, and the like—they weren’t taught by middle school teachers either.

He got to know about ring theory due to studying statistics, where he encountered homological statistics and needed algebraic topology to analyze data structures, which included ring structures and homology groups.

Moreover, many results in algebraic topology are based on ring theory.

Similarly, because it involves algebraic topology, Qiao Yu also researched group theory.

After all, one of the classic concepts in algebraic topology is the fundamental group, which describes the looping nature of space via paths, and it is indeed a group.

Yes, back then, just to find a solution to the lottery problem, Qiao Yu spent over two years frantically absorbing all kinds of mathematical knowledge online, attempting to find a design flaw in the mathematics of the lottery as a path to becoming rich.

Yet, he found out that Huaxia’s lottery had no flaw at all for mathematicians, which was a significant blow to Qiao Yu.

Of course, it wasn’t without benefit.

It solidified Qiao Yu’s resolve never to touch anything with significant gambling nature.

Like betting, or stock trading…

The point is that one should not aim too high.

Qiao Yu felt that trying your best yet failing to reach a self-set goal can sometimes be cruel, especially in Star City, where explicit rules prevent middle school students from repeating a grade.

Given Zhou Shuang’s exhibited learning ability and knowledge reserves, wanting to pass by simply trying hard in the last month was indeed quite difficult.

Allowing that would indeed be disrespectful to children who studied hard every day and never dared to slack off.

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