Top Student at Their Peak
Chapter 290 - 119 Congratulations, You’ve Already Seen Some Wonderful Scenery_2

Chapter 290: Chapter 119 Congratulations, You’ve Already Seen Some Wonderful Scenery_2

As he spoke, Qiao Yu picked up a piece of chalk and wrote an equation on the blackboard: "f(x,y,z)=z^2−x^3y^2+sin(xyz)".

After writing, Qiao Yu stepped back, silently calculating in his mind for a moment, and then continued saying, "I believe everyone can already see that there exists a local spine limit structure at a certain position near the point (0,0,0) for this equation."

"Yes, its conjugate relation is manifested in the singular points on the algebraic variety, denoted as P1 and P2, respectively possessing local spine singular point structures, and their local geometric properties influence each other through a nonlinear homological map.

Obviously, this means that the local module structure of singular point P1 will depend on the local properties of another distant singular point P2. Note that this conjugate relation cannot be inferred through simple local geometric observation..."

At this point, Qiao Yu’s voice abruptly stopped...

The audience fell into a similar silence, but their reactions varied.

Some had already furrowed their brows and picked up pen and paper, starting calculations on the drafts they carried; some were still listening attentively; others remained bewildered, watching the situation unfold.

However, on the stage, Professor Everton was staring intently at the equation Qiao Yu had written, watching with great interest.

As for Pan Jingyuan among the audience, his brows furrowed the deepest, being the person most familiar with that series of papers at the scene, he vaguely guessed the general idea of Qiao Yu’s thoughts, but he hadn’t figured out what method Qiao Yu would use to break the situation.

However, he quickly reacted, giving a glance to Yuan Zhengxin, who was also seriously observing Qiao Yu, then pulled out his phone...

The old man might not take out the onsite recording, so he simply recorded a segment himself first.

...

At this moment, Qiao Yu’s brain was also thinking rapidly.

Although he found the key point, he still needed to design an algebraic variety background based on the framework constructed in the five-series papers.

Suddenly, with just a direction in mind, hastily designing this background was a test of his improvisational skills. Luckily, it would also take some time for the audience below to calculate the equation he proposed.

Qiao Yu couldn’t be bothered with how others viewed him now; after all, no one was rushing him, so he pondered silently.

After thinking for a whole five minutes, Qiao Yu, standing in front of the blackboard, suddenly picked up the chalk again. This time he didn’t speak, but directly began writing on the blackboard.

"Consider a higher-dimensional algebraic variety X, defined in the following form: X={(x,y,z,w)∈A4|z^2−x^3y^2+sin(xyz)+w^5=0}."

After writing, Qiao Yu stepped back again, started silently calculating rapidly in his mind, and after another minute he spoke: "According to the previous explanation, everyone should realize that the algebraic variety X has such a singular point near (0,0,0,0), and it has a conjugate relation with the distant point of the algebraic variety through the variable w.

More specifically, P1/P2 are two singular points with the same structure. Yes, that’s right! Next, we need to consider the spine structure in the p-adic framework mentioned earlier by Professor Everton.

Observe around the singular points P1 and P2 of X, the local homological algebra structures manifest as two. Firstly, near P1, the flatness and projectivity of the local p-adic module M1 extend to the distant P2 through the Spine, so the module near P1, which appears flat, no longer maintains its projectivity there.

Secondly, due to their conjugate structure, singular points P1 and P2 influence each other through a nonlinear homological algebra relationship, causing abnormal behavior in the Ext group near the Spine. That is, the local module structure near P1 cannot be correctly globalized, undoubtedly disrupting the local-global equivalence.

That’s all I can think of for now. Of course, this is just the first step, further analysis of the local module structure is required, followed by introducing High-Order Category Theory to derive the failure of the functor.

If you have read the first paper by Danis and Professor Sam on proving the Geometric Langlands Conjecture, you should know that if we can prove that the derived functor from local to global does not meet the consistency in the homology category, namely: RHomC(M1,M2)≃LHomC(M1,M2)

Then it is sufficient to prove that the local-global equivalence in the Ambidexterity theorem fails under this background. The theorem’s assumption of local homological algebra flatness condition may likely not hold with the existence of such a structure."

After speaking, Qiao Yu put down the chalk, clapped his hands, looked carefully at the two formulas he left on the blackboard, and then turned around, facing the professors in the audience for the first time, including Professor Everton standing beside him.

Then Qiao Yu belatedly realized that everyone at the scene, yes, everyone, was in an extremely odd state.

This included his grandmaster, Mr. Yuan, and Professor Everton on the stage.

It’s not exactly as if they were petrified, but their expressions varied while seemingly frozen like a snapshot.

This reminded him of a very old Japanese television show, where the protagonist had the ability to stop time and was the only one able to move...

Of course, what came to Qiao Yu’s mind was "Super Dinosaur Flash Project", but it doesn’t mean that people with skewed thoughts are unhealthy, since there are too many shows of that kind...

In any case, the reactions of the crowd made Qiao Yu somewhat uneasy, so he couldn’t help but clear his throat twice: "Ahem, um... Professor Everton, should I step down now?"

"Oh... uh... alright... you may step down for now, child, by the way, if I’m not mistaken, you’re Professor Yuan’s grandson, aren’t you Qiao Yu?" asked Professor Everton, who had come to his senses.