Expectations for Xiao Yi

"A paper?"

Hearing Academician Hu's request, Xiao Yi's eyes lit up. As a high school student, he hadn't encountered anything like a research paper before.

Well... the only "papers" he'd encountered were the argumentative essays in Chinese Language class.

And now, Academician Hu was sending him one and asking him to look it over. It immediately piqued his interest.

"Okay, I'll take a look."

He then downloaded the paper Academician Hu had sent and began to read.

First, he saw information about the author, Zeng Ang. He was a graduate student in the School of Mathematical Sciences at USTC. His advisor wasn't Hu Guangde, but someone else.

Of course, Xiao Yi didn't dwell on this information. After all, prying into others' details wasn't a good habit. He then flipped to the next page.

"There is an inseparable relationship between the convergence of Riemann integrals and Lebesgue integrability. First, the theorem on the convergence of Riemann integrals is rigorously proved, then... finally, an application is given."

This section was the abstract, primarily explaining the paper's content.

Then he entered the main body.

Despite only having studied for a few days, Xiao Yi felt no difficulty reading the main text. Essentially, everything mentioned in the paper had been covered in his books. Even for concepts he hadn't seen before, he could understand them with a little mental deduction.

This paper was likely written by the graduate student Zeng Ang as practice. Its depth wasn't particularly high; at least, the theoretical derivations in the first half were flawless.

Soon, Xiao Yi reached the final part of the paper, the application of the previously derived conclusions.

However, after only a few glances, his brow furrowed slightly.

He had discovered an error in the application process.

"Using Dirichlet's Test, based on the current conditions, one cannot directly determine the convergence of the Riemann integral (R)."

He couldn't help but wonder if he was mistaken. After all, this paper was written by a graduate student, someone who could potentially be his senior in the future. How could he have made a mistake?

There must be an explanation or a patch later on.

So, he continued reading. However, after turning another page, he found that this graduate student senior had directly concluded, "In summary, the conclusion holds."

Below that was the conclusion.

"This..."

Xiao Yi was stumped.

He flipped back and reread it. The conclusion remained the same: it was incorrect!

However, this paper was assigned by Academician Hu. Although Zeng Ang's advisor wasn't Academician Hu, there was certainly some connection.

Would it be inappropriate to directly point out the error?

But soon, he stopped agonizing over it. Mathematics is a science where "right is right, and wrong is wrong."

Thus, he finally spoke, "Academician Hu, I've finished reading it."

Hu Guangde smiled and said, "Good, what are your thoughts?"

"Okay," Xiao Yi nodded. "The first three parts of this paper are correct, with no errors. The derivations should be relatively basic content from 'Real Analysis,' though it does involve some research to some extent."

"However, in the fourth part, the application of the derivation, there is an error."

"In the section using Dirichlet's Test to determine the convergence of the Riemann integral (R), there is an error. Under the currently known conditions, this judgment cannot be made directly."

Hearing Xiao Yi's words, Hu Guangde's brow twitched, and he then said, "Then, how do you think this error should be rectified?"

Xiao Yi was taken aback. He hadn't expected Hu Guangde to ask how to fix the error instead of how he discovered it.

However, he immediately replied, "It needs to be explained from the beginning."

"Let α ∈ R, and f(x) = sin(1/x)/x^α, for x ∈ (0, 1]. Then the following conclusions hold: (i) When α ≤ 0, f ∈ R[0, 1] and f ∈ L[0, 1]..."

Xiao Yi began his explanation. On the other side of the video call, besides Hu Guangde's smile growing brighter, several other professors, including Liu Bin, had their mouths agape.

This paper wasn't much for them, professors who supervised master's and doctoral students. But the problem was, the person on the other end was a sophomore who had just finished reading Real Analysis, and hadn't even spent two days on it!

After less than two days of study, he could grasp Real Analysis, a subject that usually requires ten readings, to this extent. Not only that, but he could also point out errors in a graduate student's paper after reading it?

He recalled that this graduate student, Zeng Ang, was seemingly a student of Academician Hu's student.

In other words, he was Hu Guangde's grand-disciple, and his abilities were certainly there.

After their astonishment, the professors exchanged glances. Finally, all their gazes focused on Liu Bin.

Old Liu, Old Liu, where on earth did you find a student like this?

Liu Bin wore a look of sorrow.

He had initially intended to bring Xiao Yi to their school to temper the arrogance of the other geniuses in the Gifted Class. But he had never intended to also temper the arrogance of them, the mathematics professors.

His confidence was about to disappear.

He then thought about the 40 books he had given Xiao Yi. At this guy's reading speed, could these many books even last a month?

At most, it would be the particularly difficult books like Algebraic Geometry and Algebraic Topology, and perhaps those few English textbooks, that might make Xiao Yi spend a little more time.

Truly...

A despair-inducing gap.

"… for the integral ∫2πk+π, 2πk |sin t/t| dt, we perform a variable substitution x=… We can easily prove the correctness of (16), thus f L[0, 1], and finally, according to the Cauchy Criterion, (iv) is proven."

"Proof complete."

Xiao Yi's voice came from the phone, indicating he had corrected the error in the paper.

"Very good."

Finally, Hu Guangde nodded, even giving a thumbs-up. If Xiao Yi were in front of him, he might have started applauding.

"That was very well explained, and it's precisely why I asked you to review this paper."

"This error isn't particularly obvious in the paper. In section 3.2, when discussing Riemann Integrals and Lebesgue Integrability, it creates an illusion that the missing condition is included, but in reality, it is not."

"Mistakes like this can be quite insidious, even for experienced individuals."

"Only those who truly master this section of Real Analysis can avoid such pitfalls."

"Furthermore, your use of the Cauchy Criterion at the end made this part even more brilliant."

The academician made his final comment: "In short, I am now looking forward to seeing you in our School of Mathematical Sciences at USTC."