In the Society of Truth.

After instructing Li Qun and them for two hours, Huang Mingzhe looked at the night sky with sparse moons and stars.

The development of the city has not only brought light, but also brought light pollution, and the bright Milky Way when I was a child can only be seen in the wilderness.

He turned around and looked at the blackboard on the wall, which was filled with dense formulas and derivations.

Although his Huang Chaos topology was already approaching Hodge's conjecture, a kick in the door was often the most difficult step.

These days, he has focused on learning analysis, algebraic geometry, and polishing Huang's chaotic topology more sharply, but in the face of the last step of Hodge's conjecture, there is still a feeling of helplessness.

The Hodge conjecture is mainly a way and method to simplify complex geometric problems into simple geometric problems.

After simplifying some complex things, they are classified according to their same parts, so that mathematicians can generalize some responsible things.

The basic idea is to ask to what extent we can take the shape of a given object by gluing together simple geometric building blocks of increasing dimensions.

Eventually, some powerful tools have enabled mathematicians to make great strides in classifying the wide variety of objects they encounter in their research.

Unfortunately, in this promotion, the geometric starting point of the program becomes blurred.

In a sense, certain parts must be added without any geometric explanation.

The Hodge conjecture asserts that for a particularly perfect type of space called projective algebraic clusters, the components called Hodge closures are actually (rational linear) combinations of geometric components called algebraic closures.

In short, no matter how majestic and strange the palace in this world is, it can be built with building blocks.

To complete the Hodge closed chain, the premise is that the entire universe can be composed of countless geometric parts, and as long as there is one thing that cannot be composed of geometric parts, the Hodge conjecture is not valid.

In this way, the difficulty is very huge.

Huang Mingzhe stared at the blackboard contemplatively.

In fact, the most obvious application of Hodge's conjecture in daily life is finite element analysis.

Suddenly his eyes widened, finite element analysis! Inverse Finite Element Analysis! Huang Mingzhe thought of the finite element inverse analysis technique he had obtained earlier.

This is the technology that can break down everything into pieces of geometric parts.

His brain was running rapidly, and the finite element inverse analysis and Huang's chaotic topology and Hodge's conjecture were collided with the relevant knowledge bodies.

In an instant, countless knowledge gushed out, forming a new body of knowledge - [Inverse Finite Element Analysis—Geometric Algebraic Cluster Clusters and Chaotic Topological Fuzzy Clusters]

This body of knowledge does not prove the Hodge conjecture, but divides the Hodge conjecture into two parts: geometric algebraic clusters and chaotic topological fuzzy clusters.

The geometric algebraic cluster group represents the ordered computable part, while the chaotic topological fuzzy cluster group represents the fuzzy non-computable part.

The relationship between the two is like the brick and cement in a house, the part that can be expressed by geometric parts, and the other part that cannot be expressed by geometric parts, that is, chaotic topological fuzzy clusters.

However, this relationship also requires a finitely limited reference value, that is, the smallest unit of the geometric component is limited, so that an object will form a geometric algebraic cluster and a chaotic topological fuzzy cluster, or only a geometric algebraic cluster cluster.

After limiting the minimum unit, the constituent parts of the object must partially support the Hodge closed chain, and the rest is a chaotic topological fuzzy cluster.

If Huang Mingzhe can deduce the type law of fuzzy clusters in chaotic topology, he may be able to prove part of the Hodge conjecture.

Based on the mathematical rule that numbers can be infinitely small, it is deduced that things can also be infinitely small, and the existence of infinitely small things means that Hodge's conjecture has a dead end that can never be approached.

That is, the Hodge closed chain can only be established in the case of finite elements.

Huang Mingzhe's brain immediately gave him countless formulas, and then he typed quickly on his laptop.

Lines of formulas and numbers appeared on the screen, and he was deducing them frantically.

A week later.

Dead.

Huang Mingzhe stopped his slightly sore fingers, stood up and hammered his arms and shoulders.

At this time, the formulas of three chaotic topological fuzzy cluster groups have been obtained on the screen, namely, pseudo-geometry-fuzzy cluster-chaotic formula, differential geometry-fuzzy cluster-chaotic formula, and topological geometry-fuzzy cluster-chaotic formula.

Combined with the formula of finite element-geometric algebraic cluster groups, it can be proved that the Hodge conjecture holds for H^2 under finite element conditions, and the Hodge conjecture also holds for the Hodge class of degrees p, where p< n,n is the dimension of the projective algebraic cluster described above, then the Hodge conjecture also holds for the Hodge class with degrees of 2n-p.

However, all of this is true in the case of finite elements, and if it is infinitely small or infinitely large, the Hodge closed chain cannot be established.

Unless humans can prove that the number is finite, the Hodge closed chain can only be infinitely approximated, and can never form a closed chain.

Obviously, numbers are necessarily infinite, and finite numbers are illogical.

Just like pi, no matter how you calculate it, you can't get the final number, because pi is an infinite non-cyclic number, and you can only get an approximate value.

After reading the 526-page derivation process and the 12 final formulas, Huang knew that he had put an end to the Hodge conjecture.

For a moment, he felt empty in his heart, and a problem that had plagued him for months had been overcome by himself.

In addition to the excitement, there is also a sense of loneliness that has reached the top.

Sitting on the sofa, a pot of tea was steaming faintly, and Huang Mingzhe poured himself a drink.

There was a slight sound of footsteps outside the corridor, and then the wooden door slowly opened, and Li Qun carried some seafood porridge for takeout, followed by Zhu Xiping and Gao Zishang.

"Why does Zhu Yuan have time to come?"

"I heard that you have been in retreat for more than a week, so I came to see you, the difficulty of Hodge's conjecture is known to the whole world, there is no need to rush." Zhu Xiping comforted with concern.

"Thank you Zhu Yuan for your concern." Huang Mingzhe said with a smile.

"That is, the future is long."

"It's been proven."

"It's been proven, it's ......" Zhu Xiping was stunned before he finished speaking, he asked uncertainly: "Proved?" Hodge's conjecture proves it?"

"It's falsified, to be exact." After Huang Mingzhe finished speaking, he took a sip of tea.

Li Qun and Gao Zi on the side were still stunned.

Grunt! Zhu Xiping grabbed Huang Mingzhe's shoulders with trembling hands, and asked in a high-pitched and hurried tone: "Mingzhe, this joke can't be made." ”

"The proof process is in my notebook." Huang Mingzhe said, pointing to the laptop on his desk.

Zhu Xiping hurriedly ran over and opened the paper, but when he saw the 526-page proof process, his head suddenly became bigger.

Although it is only about the 12 formulas, the problem is that the derivation process here is no longer understandable to ordinary mathematicians.

Even if Zhu Xiping is asked to re-deduce it after reading it, it is estimated that the complete process cannot be deduced.

After roughly reading the introduction and conclusion, Zhu Xiping said with a wry smile: "This is not something that mortals can complete, do you plan to publish a mathematical annual?"

"Of course, this is theoretical mathematics, and it's useless to hide it." Huang Mingzhe spread his hands and smiled.