201: Chapter 103: The Proud Youngster, Brandishing His Power with Vigor_6 201: Chapter 103: The Proud Youngster, Brandishing His Power with Vigor_6 Right, I would like to especially thank Professor Robert for the inspiration from today’s lecture, and my ntor Professor Tian Yanzhen for his guidance.
Just as Director Tian ntioned earlier, after reading the paper by Professor Schulz and Professor Robert recently, I suddenly ca up with this bold idea.”
As soon as Qiao Yu finished speaking, almost everyone picked up the report on the table.
It was so rudintary, and everyone only arrived at the conference room a few minutes earlier, busy with small talk, and indeed, no one had taken a serious look at it.
However, Zhang Shuwen sitting next to Tian Yanzhen, along with Professor Robert, had already picked up the simple booklet and started leafing through it.
After Qiao Yu finished his opening remarks, he cut to the chase.
“My idea is to derive an upper bound expression for rational points on an algebraic curve utilizing the complete space theory constructed by Professor Peter Schulz, combined with modular forms theory, -adic geotry, and quantized homology category.
To achieve this, we first need to consider the geotric background of the curve X, particularly its genus g(X).
Genus is an important topological invariant that represents the geotric complexity of the curve.
For curves with genus g>1, Faltings’ theorem tells us the number of rational points is finite.
But this isn’t enough, because we all hope to obtain a specific upper bound.
Geotric analysis suggests that as the genus increases, the complexity of the algebraic curve increases, which ans the number of rational points relatively decreases.
So my initial conjecture is: N(X) ≤ C(g).
Then I will testify this result from several assumptions.
Although I believe this result is correct, I cannot yet prove a specific formula for the constant C, but I’ve thought of several very interesting thods to derive the result of the constant C.
These thods have yet to be proven, so I hope the professors present can provide so inspiration.
First, we introduce the moduli space, where X is an algebraic curve with genus g, and its moduli space Mg paraterizes all curves with genus g.
Since modular forms are closely related to moduli spaces, I understand them as functions defined on the moduli space that provide geotric constraints on the complexity of the curve.
Assuming the level of modular forms is k, we further assu there exists a constant A1, such that: N(X) ≤ C1(g,k) = A1⋅g⋅k^α…”
In the audience, all the professors in the conference room had already adopted a serious deanor, their expressions starting to grow solemn.
The only two whose expressions didn’t really change were Tian Yanzhen and Xue Song.
This was sothing Chen Zhuoyang, sitting at the back, could attest to.
He wasn’t very interested in what Qiao Yu was presenting, so he focused more on the expressions of the ntors and those two academic giants opposite him.
Obviously, Director Tian was quite relaxed, quietly watching Qiao Yu write on the board.
The two giants beside him had markedly furrowed their brows, and one had even picked up a pen to jot down notes alongside the manuscript…
Chen Zhuoyang felt like he was losing his mind…
Seriously, is everyone actually taking this seriously?
So it’s not Director Tian trying to force his junior disciple onto the stage; everyone actually accepts these kinds of unproven things?
Indeed, when Chen Zhuoyang learned about the seminar this afternoon, he truly thought Director Tian just wanted his junior disciple to et everyone face-to-face.
After all, Director Tian also ntioned that these were just Qiao Yu’s ideas…
What kind of practice takes ideas this seriously?
Chen Zhuoyang even felt that Director Tian was too hasty, given that this junior disciple is only fifteen!
Although he could participate in the CMO and win first place, proving he’s mastered high school knowledge, who knows if he has even been exposed to university knowledge, let alone understanding research!
He even felt that it would be dreaming for Qiao Yu to understand Peter Fields’ papers.
But judging by the professors’ expressions now, it was clear everyone genuinely began to think deeply…
No way, is the junior disciple really defying all odds?
What made him even more desperate was that Qiao Yu on stage showed not the slightest stage fright but grew more excited as many professors began seriously reviewing his board writings, and wait, Professor Robert even took out his phone to take pictures of his notes…
“…At this step, we can introduce the -adic number fields along with Professor Schulz’s homology theory.
We know that for each pri number P, the properties of the etale homology group can constrain the local distribution of rational points on the curve.
Therefore, according to Schulz’s -adic Hodge theory, the following inequality can be derived: N(X) ≤ C2(g,p) = A2⋅g2⋅log(p).
An important point here is that a core trait of Schulz’s -adic Hodge theory is its completeness.
So if the inequality we derive holds, we could start from the properties of the curve in the local domain and derive global geotric constraints, hence needing to verify whether this inequality holds true.
To this end, under Director Tian’s guidance, I thought of a thod by introducing a quantum homology category…”
For half an hour, Chen Zhuoyang only felt like he was sitting on pins and needles.
Because within the whole room, only the two students of Director Tian were present, one was speaking eloquently at the front, and the other couldn’t even understand what his junior was saying…
The conference room was awfully quiet too, with no murmuring at all—everyone was intently focused on Qiao Yu’s board writings.
Including the three mathematical giants in the room, whom most professors could only look up to.
Finally, Qiao Yu finished speaking…
“That concludes my complete thought process.
The issue is that I still cannot handle the constants in the setup entirely nor provide a complete logical proof for the specific tools.
But I believe this should be a new research direction, as once we derive the result for the constant C, it would an we can directly predict the upper bound of rational points for the relevant curve.”
When Qiao Yu’s voice finally dissipated in the air, Chen Zhuoyang finally breathed a sigh of relief, feeling a bit better.
However, the re-silenced conference room made him tense again.
Seriously, professors, aren’t you planning to say sothing?
One by one, you’re all adults— don’t give that incredulous look at the junior disciple, alright?
He’s only fifteen, now should be the ti for him to receive so tough educational setbacks!
Everybody should be sharply critiquing his ideas right now!
Chen Zhuoyang thought fiercely in his head, but when he saw Director Tian across from him first raise his hand to start applauding, he could only promptly cooperate and join in the applause too…
“Clap, clap, clap…”
The scattered applause seed to awaken the professors, and the conference room was imdiately filled with applause.
Fortunately, there weren’t many people, so after just a few dozen seconds, the applause subsided.
Then Chen Zhuoyang finally heard the lodious sound he was hoping for.
“I have a question, Qiao Yu, in your third section, why not directly use the Riemann-Roch theorem?” Chen Zhuoyang glanced at the stern-faced Zhang Shuwen across from him, indeed, a major professor is awe-inspiring!
“Huh?
What’s the Riemann-Roch theorem?” Qiao Yu asked inquisitively.
Everyone had different reactions.
Qiao Yu, standing there, seed unaffected, but his nominal minor ntor Professor Xue was mortified, his face turning beet red.
As for the other professors, including Robert Green, they all seed perplexed, probably unable to fathom how this kid who elaborated for half an hour on algebraic curves reportedly didn’t know this important theorem in algebraic and complex geotry.
Tian Yanzhen remained unfazed, speaking warmly: “Professor Zhang, as I ntioned earlier, Qiao Yu is only fifteen years old, a prodigy I discovered through the CMO, and he has yet to receive a complete undergraduate education, so his mathematical knowledge reserve is relatively fragnted.
Please guide him on the spot.”
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