Top Student at Their Peak-Chapter 202 - 104 Mathematics is Not That Simple... But It’s Not Hard Either!

Zhang Shuwen hesitated for a moment, then chose to stand up, walked to Qiao Yu’s side, casually wiped off the last of the board notes, and began his on-the-spot explanation.

"The Riemann-Roch theorem is a fundamental theorem in Algebraic Geometry used to describe the dimensions of certain functions or forms on algebraic curves. Specifically, the Riemann-Roch theorem applies to any divisor D on an algebraic curve X, and the theorem states the dimension of the function space L(D) associated with the divisor D on the algebraic curve.

Its specific statement is ℓ(D) = deg(D) + 1 − g + ℓ(K−D). It has two parts that complement each other, describing the balance relationship between the divisor D and the remaining part K−D. However, there are special cases, when the degree of D is large enough, then ℓ(K−D) is zero, so in this case, ℓ(D) = deg(D) + 1 − g, do you understand what this means?"

"When the degree of D is large enough, the dimension has a linear relationship with the degree." Qiao Yu immediately replied.

"So when D is zero..."

"ℓ(0) = 1 − g + ℓ(K)... Oh, Professor Zhang, I understand what you mean... so this part of the proof actually doesn’t need to be that complicated, because the deficiency g(X) can be directly derived from the Riemann-Roch theorem, hmm, then this part of the proof is not as troublesome... let me think..."

After saying this, Qiao Yu picked up the chalk and started writing on the other side of the blackboard.

"That means when constructing functions... well, dimQH1(Cp) is the dimension of the quantized homology group, um, depending on the deficiency g of the curve and the quantum operator Q... This part can obtain the dimension of H1(Cp) through calculating the Canonical Factor...

So the decomposed dimension relationship directly is dimQH1(Cp) = g⋅f(Q), Professor Zhang, do you think this part of the derivation is correct?"

Zhang Shuwen took a deep breath, kept his expression undisturbed, and then nodded.

"Great, then the next step will be easier to prove... After deriving the dimension of the homology group, the larger the dimension of the quantized homology group, the higher the geometrical complexity of the curve, and the number of rational points on the curve will be limited, plus the Jacobian can further affect the number of rational points...

The deficiency is one of the core geometric invariants that cannot be simplified, so is #C(K) ≤ f(g, Jac(Cp))? Whew, no, looking at it this way, I feel like this method might really be able to derive the formula for the constant C?"

Qiao Yu murmured subconsciously.

Indeed, Chen Zhuoyang, who was listening in the audience, was stunned when he heard Qiao Yu say this.

Although he was similarly shocked by Qiao Yu’s insight, wasn’t everyone truly not angry upon hearing this?

Accepting a 45-minute seminar with something you’re not even one hundred percent confident you can prove?

Just seeing that nobody in the conference room cared, Chen Zhuoyang naturally couldn’t say anything.

Meanwhile, on stage, Professor Zhang snorted coldly and said, "It’s still early; I believe you can prove it, and even get the formula you want! But are those truly useful?! You at least have to simplify it to #C(K) ≤ f(g) for it to have meaning!

Introducing Peter Schultz’s theory is acceptable; as long as the proof process is within the logical framework, no matter how complex or abstract it may be, it’s fine, but you must limit all complexity to the intermediate steps of the proof!

The final result must be as simplified as possible! Otherwise, even if you prove the constant C and derive the result, inserting all those set constants into the formula, imagine how complicated the final formula would be? How would others use it?

True mathematics pursues complexity of thought and simplicity of result; only a concise result is truly useful and elegant as a mathematical tool! Too many constants or parameters will only increase the difficulty of understanding and calculation, and even if research is done, it’s still trash! Mathematics is not as simple as you think!"

...

Zhang Shuwen’s tone was extremely stern, yet Tian Yanzhen sitting there seemed to be in a pleasant mood.

Robert Green finally couldn’t hold back and leaned over to ask, "Professor Tian, what is Professor Zhang saying to that kid?"

Qiao Yu had used English when presenting his ideas, but when Zhang Shuwen went up to guide Qiao Yu, he started using Chinese.

"He’s educating Qiao Yu not to become conceited, warning the kid that what he proposed is just a thought for now, still far from yielding results, and emphasizing the importance of simplifying mathematical conclusions," Tian Yanzhen explained with a smile.

"Oh! My God, is Zhang really that strict with his requirements? Doesn’t he know that this kid is only fifteen? Fifteen! He can actually understand Schultz’s theory and even imagine such creative ideas, and yet Zhang still thinks it’s not enough? Is he crazy? I even think this is a research direction that holds a lot of promise for the future." freёnovelkiss.com

Robert Green said in confusion, clearly seeing from the perspective of this New York University professor that Zhang Shuwen was too harsh, and his expectations for Qiao Yu were excessively demanding.

"Yes, this is also why I insisted on holding this seminar; I also think it’s a very promising possibility. But at present, it’s still a bit difficult for this kid to complete this problem independently. So I am actually very grateful to Professor Zhang, because at least he is teaching Qiao Yu the principle that mathematically performing subtraction can sometimes be more difficult than the proof process itself."

Tian Yanzhen explained, with a hint of a smile at the corner of his mouth, helping to explain Zhang Shuwen’s actions.

"Although Zhang’s reasoning is very true, there is no need to use such a stern tone, it’s not fair to a fifteen-year-old kid."

Robert Green still couldn’t understand, after all, if Qiao Yu were his student, he would never be so unfeeling.

Although he couldn’t understand what Zhang Shuwen said, he could tell that the tone was even cold.