CHAPTER 294: CHAPTER 119 CONGRATULATIONS, YOU’VE ALREADY SEEN SOME WONDERFUL SCENERY_6

Not taking offense, Tian Yanzhen smiled and asked, "Oh? Which part do you disagree with?"

Qiao Yu spoke assertively, "You said that all participants of this conference are top experts in the relevant fields. Let me put it this way, I explained the problem so clearly to them, yet they still thought I had wild imagination. This is enough to show that they are half-baked and definitely not top experts!"

With one statement, Tian Yanzhen was rendered silent.

He suddenly felt that specifically calling this kid over was a complete waste of time.

This mindset is countless times better than when I was young.

Wow, he hasn’t even presented the complete proof process, and when questioned, those people are considered half-baked. This confidence definitely surpasses ninety-nine percent of peers. At least in terms of not blindly worshiping authority, this kid is already standing at the pinnacle of the world pyramid.

However, having said that, Tian Yanzhen genuinely liked Qiao Yu’s assertive character, thinking "If not me, who?" And this personality indeed suits the future prospect of carrying the banner of the Yanbei School.

The only problem might be that when this kid grows up and he steps down, if there are any conflicting opinions in the future, he probably won’t be able to influence the kid’s decisions. But it doesn’t matter.

As long as Qiao Yu can win the Fields Medal before turning thirty, the future flag of Huaxia’s mathematics community will probably be carried by this kid, and he will just watch by then. He hopes that Qiao Yu can surprise him and elevate Huaxia’s voice in the world mathematics community.

If Qiao Yu can do this, helping him vent his frustration, the trifles at home really don’t matter.

"Well, well, well said. There are still two days for this seminar, and if necessary, we can extend it by a day. The last day is up to you, arrange the time as you see fit. I was just hoping earlier, but now it’s a demand. In any case, you need to present a complete piece before the seminar, silencing those professors who oppose you. No problem, right?"

Tian Yanzhen straightforwardly made the demand.

This kid is not modest to himself, so he has no need to be too courteous.

Qiao Yu patted his chest and said, "Don’t worry, Director Tian. I already have an idea. I just need to provide an accurate definition of this local deformation module structure and then prove it.

You should know, Professor Everton even defined the special singular point I conceived as ’Qiao Dian.’ Now, I just need to supplement the duality proof process of ’Qiao Dian.’ It’s simple and should be done by tomorrow night at the latest."

Tian Yanzhen glanced at Qiao Yu and said, "Alright, as long as you think there won’t be any unexpected problems in the proof process, I certainly believe in you. Once the proof process is completed, contact me anytime. Send it to me for a review first."

"No problem, Director Tian! Then I’ll go first. To let you see my proof process sooner, I’m racing against time."

"Okay, go ahead. Oh, tomorrow morning is Professor Tan’s lecture, you don’t need to attend. If necessary, I’ll have him give you a private lesson. As for the lecture by Professor Phelps in the afternoon, you should go, it’s a rare opportunity."

"Understood, Director Tian."

...

Returning to his room, Qiao Yu sat in front of the computer and directly opened LaTeX.

What follows is the proof.

The most important part of the paper is proving the nonlinear dual spine structure, which means providing a full proof that two distant singular points P1 and P2 on the algebraic variety each have a local spine singularity and influence each other through nonlinear cohomology mapping.

On the way back, Qiao Yu already figured out how to prove it.

The first step is naturally local structure analysis, essentially defining local equations to describe the geometric structure of singular points, then calculating the Jacobian matrix to examine the nature of the singular points and using blow-up and analytic decomposition methods to study their spine structures.

These are all conventional methods that Qiao Yu doesn’t need to think too much about.

The focus is on constructing the nonlinear cohomology mapping.

The true consideration for Qiao Yu is which tool to choose to calculate the local cohomology of P1 and P2, which is likely the only difficulty. Different cohomology theory tools directly affect the solvability when handling this content.

After careful consideration, Qiao Yu decided to handle this issue using a combination of Sheaf Cohomology and Grothendieck Local Cohomology.

Sheaf Cohomology captures the local geometric and topological information of algebraic varieties more conveniently, while Grothendieck Local Cohomology provides deeper tools for handling local rings and algebraic structures, thus further analyzing the properties of local algebraic rings at singular points, revealing the subtle algebraic structure of the algebraic variety at singular points.

This should be the simplest method of associating the local cohomological dimension of the singular point with the properties of local rings through cohomology mapping.

Qiao Yu’s aim is precisely to use a simple and direct method to shut the mouths of those so-called senior professors who think there’s a problem with his reasoning.

In fact, using Sheaf Cohomology combined with De Rham Cohomology could also yield the same conclusion. However, De Rham Cohomology, when dealing with analytic singularities or algebraic varieties, provides a differential geometry perspective, which complicates problem-solving.

There’s no need to show off with analytic geometry here. Moreover, Qiao Yu feels his analytic geometry isn’t strong, and if there’s any loophole in the proof process using De Rham Cohomology, both Director Tian and his grandmaster might feel humiliated...

After all, both his mentor and grandmaster are big names in the field of differential geometry.

All in all, once this problem is solved, more than half the proof process is completed. The rest is just following the steps, and as long as such points exist, the functor derived from High-Order Category Theory must fail.

When the derived functor is non-equivalent, all conclusions naturally collapse.

The real challenge still lies in how to reconstruct the Ambidexterity Theorem, enabling this key theorem to be effective again in the proof process of the Geometric Langlands Conjecture.

At this stage, Qiao Yu plans to solve this problem himself but won’t tell the other side...

This chapter is updated by freew(e)bnovel.(c)om