Chapter 414

"That's all I want to say, the derivation process is here, you can understand it after reading it a few more times at your level, and I don't bother to talk about it a few more times. Cheng Nuo stretched his waist and stood up from his seat, "Chali, I'll withdraw first, and if you encounter this kind of thing in the future, just send the question directly, you guys are still quite far from the experimental building." ”

He raised his wrist to look at the time, and pointed out the door, "It's not too early, if there is nothing else, I'll go." ”

"That...... Wait!" When Cheng Nuo walked a few steps to the door, Milo, whose face was uncertain, called out to Cheng Nuo.

"I'm sorry! I shouldn't have questioned your strength just now. Milo apologized to Cheng Nuo for bending over.

Cheng Nuo waved his hand indifferently, "Anla, Anla, I'm used to it, and I don't mean to hate you!"

"Believe it or not, but I really didn't take it to heart. Cheng Nuo turned around and waved his hand with his back to everyone, "Let's go!"

A day's leisure, Cheng Nuo is not such a wanton waste.

............

Next.

Cheng Nuo carried his bag and came to a café on the campus of the Massachusetts Institute of Technology, ordered a cup of coffee, and started today's work while drinking it leisurely.

According to the schedule of the project, today is the first formal meeting of the research group on geometric cohomology.

Although the research group is of a very high level, consisting of a full professor, two associate professors, and a graduate student, the main content of the discussion is nothing more than the overall framework of the research, plus the specific division of research tasks.

Putting on the headphones, Cheng Nuo and the three professors at Princeton started talking.

Cheng Nuo: "Hey, hey, can you hear clearly?"

Miller: "Hahaha, Tom, your voice sounds so young!"

Cheng Nuo didn't answer and asked, "Is the team leader there?"

Hatch: "Wait a minute, the boss is answering the phone, right away." ”

Tens of seconds later......

Byrne: "Tom, sorry for the wait. Now that everyone is here, let's stop chatting and get straight to the point!"

Miller: "OK!"

Hatch: "OK!"

Cheng Nuo: "OK." ”

After a few seconds of silence, there were a few light coughs, and then Professor Bourne's voice sounded, "We all know that the proposal of Cheng Nuo's theorem directly links the algebraic clusters and complex algebraic clusters in geometry. At the same time, the cohomology method, which only exists in the topological space, has the possibility of being applied to clusters and generalizations. ”

"I have to say that the proposal of Cheng Nuo's theorem has too great an impact on our geometric community. and the young man named Cheng Nuo, even I admire it. I'd love to ask him if I could. ”

Cheng Nuo couldn't help blushing a little when he heard Bourne's praise of Cheng Nuo in the headphones.

I'm ...... now Is it that powerful?

Fortunately, Professor Bourne quickly ended this meaningless praise and continued to say solemnly, "The purpose of our project is to derive the cohomology theorem for algebraic clusters on the basis of Chengno's theorem, and then ...... through the cohomology theorem."

Professor Bourne's way of speaking seems to be very similar to that of a Chinese-style leader, and it is obviously just a few words, concise and concise, and Professor Bourne has added to it for nearly half an hour.

Fortunately, this is an audio conference, and Cheng Nuo can still wander. As for Miller and Hatch, who are now in front of Professor Bourne, I'm afraid it's very uncomfortable.

"I'll start with that. Now, let's put together the overall framework of this topic. Professor Byrne finally put an end to his ramblings.

The atmosphere fell silent again.

Professor Miller broke the awkward atmosphere, "Tom, why don't you say a few words?"

"Ah, me?" Cheng Nuo was stunned for a moment, he just thought it was Miller who wanted to say it first?

He straightened out his thoughts, "Then I'll give you my opinion." ”

"We all know that cohomology is a positive variable functor in the category of topological space, that is, it does not change the direction of the arrow. A range of axioms including Excision Lemma are satisfied. There is a descending operation on a chain complex, such as the boundary operation: dn:Cn→Cn-1. After two boundary operations, we get 0:dn-1*dn:Cn→Cn-2=0."

“...... Let X be the d-dimensional smooth projective cluster on Fq, and the convention É=X-Fq, on the projective cluster X, we can define Fx, F^2x, F^3x,...... The set of Fq^n points X (Fq^n) on the projective cluster X happens to be the set of fixed points of the automorphic F^nx:X→χ!"

"Then how do you calculate the number of fixed point sets on the projective cluster?" Before Cheng Nuo finished speaking, Professor Miller couldn't help but ask.

Cheng Nuo smiled and said slowly: "Lefschetz's immovable point theorem!"

Miller: "Lefschetz's fixed point theorem?"

Cheng Nuo accentuated his tone, "Yes, it's Lefschetz's immovable point theorem!"

"Let X be a tight differential real manifold, f:X→ X is a differential map, and a fixed point of f is a point x\in X such that f(x)=x .For a fixed point x of X, the differential df_{x} is a linear transformation of the tangent space T_{x}X.A fixed point x is said to be non-degenerate, if 1-df(x) is reversible.

Cheng Nuo said this almost without thinking.

"That's right, I really didn't react for a moment just now!" said Miller's sudden voice.

Professor Byrne continued, "My starting point is similar to that of Mr. Tom. Taking advantage of the basic properties of cohomology groups in topology, a smooth algebraic projective cluster is constructed, and the fixed point set is used to cut in. ”

Then, Professor Byrne told Cheng Nuo and the others about his thoughts.

Similar.

Except for some specific details that are indistinguishable from the good and the bad, the general content is the same.

Professor Miller is a major student in topology, and although he does not know as much about geometry as the other three, as a well-known young mathematician in the field of topology, he naturally has a deep understanding of the cohomology group of topology.

But even he, after Cheng Nuo's explanation, could not mention any flaws in this plan.

Professor Hatch had no objection either.

Byrne immediately clapped, "In that case, let's follow this one of me and Tom." As for those different details, it will be up to you whose scheme is simpler and whose one you use. ”

Once the entry point is finalized, the rest is simple.

While there are some things that can't be done with anything until the result is actually calculated, building a framework that contains only bones doesn't require anything so precise.

It took a morning, and after Cheng Nuolian drank three cups of coffee, the framework was finally built, and the task was assigned.

In the eyes of Miller and Bourne, Cheng Nuo was naturally regarded as a mathematician of the same level as them, so he did not take advantage of the assignment of tasks.

Looking at the items in his task list, Cheng Nuo moved his fingers.