Top Student at Their Peak-Chapter 197 - 103: Proud Youth, Commanding with Vigor_2

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Qiao Yu didn’t think much about it, just felt that what Old Xue said made a lot of sense.

So he decided not to join the fun, and instead followed Old Xue back to his room to start working on his computer.

"What do you want for lunch? I can go get it for you." Seeing Qiao Yu start working, Xue Song asked.

He increasingly felt like a nanny, but fortunately, in a couple of days, the PhD student he’s supervising will be arriving at the university. ƒrēenovelkiss.com

"Anything’s fine, just a takeaway. I’m not picky, just make sure there’s plenty of meat," said Qiao Yu.

"Then I’ll get you two drumsticks?"

"Sure!"

Xue Song pursed his lips and then left. Shortly after, there was a knock on the door, and without looking up, Qiao Yu said, "Come in."

The door opened and Tian Yanzhen walked in. Qiao Yu, busy as he was, turned his head and quickly said, "Director Tian."

"Hmm, preparing for it?"

"Yes!"

"Let me take a look."

"Please have a seat."

"Change this part; when you haven’t completed the proof yet, the wording needs to be more precise. Change it to, based on geometric intuition, it can be conjectured that there exists a constant C dependent on both the geometric and arithmetic properties of curve X such that the number of rational points on the curve N(X)≤C."

"Oh."

"And here, your description that the homology category QH(Cp) is an enhanced homology category... doesn’t emphasize its difference from the general understanding of homology categories. I’ve thought carefully about your idea.

If you want to better analyze the local homological behavior of curves in the p-adic complete space, you might introduce a quantized homology category. If you introduce quantum characteristics at the homology level, perhaps you can capture subtle local changes in the geometric structure?"

"What? Quantization? But it has nothing to do with quantum physics, right?"

"I’m talking about mathematical quantization. In fields like topology and algebraic geometry, quantization refers to the process of discretizing or enhancing classical structures into more complex structures, which is usually non-commutative."

Seeing that Qiao Yu still didn’t quite understand, Tian Yanzhen picked up paper and pen from the table, saying, "Time is limited, so I’ll use the example of geometric quantization in symplectic geometry to explain.’

First, we need to choose a polarization in the phase space; you can think of it as determining a direction or coordinate in classical phase space to simplify the complexity of the problem. Choosing a polarization can be considered as choosing a way of decomposition so that part of the coordinates are used to describe quantum states, and the momentum is turned into a differential operator acting on these quantum states.

Then, using the polarization condition to construct a Hilbert space, which can be viewed as a sort of function space over the classical phase space. This function space contains all possible quantum states, i.e., wave functions, and its structure depends on the symplectic structure of the classical phase space and the result of the polarization choice."

As Tian Yanzhen spoke, he started writing out a concrete example.

"Look, if the phase space of a single harmonic oscillator is composed of position q and momentum p, forming a plane (q, p). The symplectic form can be written as ω = dq∧dp. Now we want to quantize this plane into a Hilbert space, first choosing the polarization as ∂/∂p=0..."

Qiao Yu listened quietly to the mentor’s explanation, asking questions whenever he didn’t understand, and after ten minutes, he suddenly had a flash of insight.

"Oh, I get it, my Q can represent the quantization invariant, wait, let me think, I need a quantized homology category to decompose the homology group of the curve so that I can use quantization to explain the behavior of rational points in the local quantum structure, right, Director Tian?"

"Hmm..."

"Right, right, that’s it, let me use the pen," Qiao Yu said as he swiftly took the pen from Director Tian’s hand and completed the first formula he had been pondering the previous night on the draft paper.

Tian Yanzhen looked at the series of formulas Qiao Yu had written and said with an unchanged expression, "What about the proof process?"

"First, Q is confirmed to be the quantum operator acting on the homology group of the curve, and the first step is to construct a quantum homology category, first decompose H, build new quantum states, and then use the dimensions of the quantum states to describe the homology of the curve.

The second step is to find the relationship between the quantized homology group and rational points. Here, it’s quite clear that the dimension of the homology group is directly related to the deficiency g of the curve. The larger the deficiency, the more complex the geometric complexity of the curve, and the fewer rational points there are.

At this point, adding Q can lead to dimQH1(Cp) = f(g, Q), which makes the changes in the local geometric structure more sensitive, further limiting the number of rational points.

Then, impose restrictions on the rational points via the Jacobian, which is the method that Professor Robert used in today’s lecture. We can modify it, putting it into the complete space. According to previous research, the higher the degree of the Jacobian, the fewer rational points can be distributed on the curve.

Finally, construct this function. The left portion of the function represents the dimension of the quantumized homology group, which depends on the deficiency g of the curve and the quantum operator Q, and the right portion reflects the geometric structure and rational point restrictions of the curve.

You’re really amazing, Director Tian! With just a few casual pointers, you’ve helped me take a big step towards proving the existence of this constant C!"

Qiao Yu sincerely thanked him.

Tian Yanzhen silently watched Qiao Yu writing out the proof process quickly on the draft paper.