"How is it, Mr. Pang, are you sure to solve this problem?"

The old tree on the side looked at Pang Xuelin with a smile.

Pang Xuelin smiled faintly and said, "I can try." ”

Of course, Pang Xuelin and Murphy are not willing to be controlled by this seemingly kind, but actually scheming old monster.

After all, whether it was looking for a high-dimensional human civilization or thinking about the safety of the Eternal Night, he had to break away from the control of the Tree Elder.

But he couldn't think of a good way out of trouble in a short period of time, so he could only stay here for the time being, and he was sorry for the old man.

As for helping Shu Lao solve some mathematical and physical problems, Pang Xuelin didn't resist much.

Living in this tree cavity for a long time, studying math problems is undoubtedly the best way to pass the time.

Shu Lao smiled: "Okay, Mr. Pang, then I won't disturb your research." But while you do your research, I hope that you and Ms. Murphy will work harder to create people, because that's your main job. ”

Pang Xuelin said lightly: "Don't worry, Elder Shu, of course I understand this." ”

"That's good."

The tree disappeared in an instant.

……

In the following time, Pang Xuelin devoted all his attention to the study of Hodge's conjecture.

Since the birth of Galois's group theory, modern mathematics has become more and more inclined to refine the abstract understanding of the nature of things.

For more than 100 years, mathematicians have continued to build deeper abstractions on top of abstractions, each level of abstraction moving further away from the world of our everyday experience.

Taking group theory as an example, our common "addition, subtraction, multiplication, and division" is abstracted into four algorithms.

The Hodge conjecture is a problem born out of the extreme abstraction system of modern mathematics.

As a highly specialized problem, it deals with objects that are so far removed from people's intuition that it is difficult to judge not only whether the conjecture itself is right or wrong, but even the formulation of the problem itself seeks to establish a real consensus.

Therefore, the definition of the Hodge conjecture must begin with the birth of geometry.

In ancient Greece, Pythagoras used the deductive method to prove that the square of the hypotenuse of a right triangle is equal to the sum of the squares of two right angles, that is, the Pythagorean theorem.

Since then, humans have begun to associate shapes with mathematics.

Two hundred years later, Euclid listed some recognized facts into definitions and axioms, and used these definitions and axioms to study the properties of various figures by the method of formal logic, thus establishing a set of geometric argumentation methods that start from axioms and definitions and demonstrate propositions to obtain theorems, forming a rigorous logical system - geometry.

After thousands of years of change, the study of shape has become more and more complex, and this is where the Hodge conjecture comes in.

Before the seventies of the seventeenth century, geometry and algebra had developed considerably, but they were two separate disciplines.

Descartes compared the geometric method and the algebraic method at that time, and he advocated that the problems of geometry should be reduced to problems in algebraic form, and the algebraic methods should be used to calculate and prove, so as to achieve the goal of finally solving geometric problems.

Following this idea, he founded what we now call "analytic geometry."

Descartes' mathematical ideas prove that if you take abstraction one step further, geometry is actually the same as algebra, geometry can be transformed into algebraic equations, and algebraic equations can also be transformed into geometric figures.

If you want to see where a certain line intersects a particular circle, you can draw the shape geometrically, or simply compare the equations algebraically.

Both methods will give the same answer.

By the 19th century, mathematicians were trying to popularize Descartes' method.

They started with some algebraic equations and defined the solutions of these equations as "geometric" objects.

Objects that are generated from algebraic equations in this way are called "algebraic clusters".

Thus, algebraic clusters are a generalization of geometric images.

Any geometric correspondence is an algebraic cluster, but there are many algebraic clusters that cannot be visualized.

However, just because a particular algebraic cluster can't be visualized, you can't do (algebraic) geometry on it.

You can do it, but it's geometry without figures.

After that, mathematicians soon discovered that more complex equations, or even systems of equations, work together to produce amazing shapes in a variety of dimensions.

Mathematicians have found a very practical way to get more complex shapes, and the basic idea is to what extent we can take the shape of a given object by gluing together simple geometric building blocks of increasing dimensions.

This technique is very useful and makes it possible to promote it in many different ways.

Through this approach, mathematicians hope to expand step by step in a variety of different ways, and eventually build a powerful set of algebraic equations or/and geometric tools that can classify various complex objects into specific simple geometric objects and their combinations.

This has allowed mathematicians to make tremendous strides in classifying the wide variety of objects they encounter in their research.

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

What simple geometric objects should you start with?

What is the combined program/sequence?

Therefore, some "non-geometric" basic modules that do not have any geometric explanations must be added.

It is based on this dilemma that in 1958, Professor Hodge, a British mathematician and chairman of the 13th International Congress of Mathematics, proposed that for the projective algebraic cluster space, on the non-singular complex projective algebraic cluster, any Hodge class can be expressed as a combination of rational linear (geometric components) of algebraic closed-chain classes.

This sentence is expressed in a popular mathematical language, which is:

Let x be a projective algebraic manifold and p a positive integer. Then set H^2p(X,Q)afg?? (X,Q) is the closed subspace of the algebra, that is, the Q vector space generated by the basic class of the algebraic subcluster of the courier number p in X. The Hodge conjecture asserts that Hodge's theory can be used to "compute" the subspace H^2p(X,Q)afg, specifically, H^2p(X,Q)afg=H^p,p∩H^2p(X,Q).

In this context, the so-called "non-singular projective algebraic cluster" refers to the "surface" of a smooth, multidimensional object generated by the solution of an algebraic equation.

……

In the real world, Perelman once talked to Pang Xuelin, and he found the inspiration to prove Hodge's conjecture in his research on Ponzi geometry.

After that, Perelman went into retreat, and Pang Xuelin didn't know where his research was going.

However, intuitively speaking, Pang Xuelin felt that there was an inseparable close relationship between Ponzi geometry theory and Hodge's conjecture.

Therefore, in the past six months, Pang Xuelin has not focused on the Hodge conjecture itself, but has been looking for some internal connections between Ponzi geometry and algebraic geometry, analysis and topology.

This kind of work is very boring, and you need to think about something that involves the nature of mathematics.

Although half a year has passed, it is far from the point where it can solve the Hodge conjecture.

However, the study of analysis and topology has given Pang Xuelin a deeper understanding of Ponzi geometry theory.

As for Shu Lao's side, at the beginning, Shu Lao was a little unconcerned about Pang Xuelin's research.

After all, in his opinion, no matter how powerful a small human individual is, it is impossible to easily solve this level of problems.

But as time passed, after reading Pang Xuelin's research notes one by one, Shu Lao was shocked and at the same time, he also began to have a trace of expectation for Pang Xuelin's research.

This morning, Pang Xuelin opened his eyes and saw a shallow smile on the corner of Murphy's mouth, sleeping comfortably on his chest.

In the past half a year, the two have been getting along day and night, and they have long been close to each other.

Especially when there was a tree elder staring at him, except for Murphy's monthly visit from relatives, the two of them had to applaud almost every night.

It's just that for some reason, half a year has passed, and Murphy's stomach still hasn't moved.

Pang Xuelin and Murphy are not in a hurry, but they are old, but they are a little anxious.

Thinking of this, a faint smile appeared on Pang Xuelin's face.

He stroked Murphy's smooth back with his hand, lowered his head, and pecked her on her red lips.

In the past six months, with Murphy by his side, Pang Xuelin's research life has not been very boring.

Every day, in addition to spending eight hours thinking about Hodge's conjecture, Pang Xuelin spent the rest of his time with Murphy.

Play a computer game from Earth in the shuttle, watch an early film and TV show, and occasionally cook together to cook Earth Age food.

Life was peaceful and warm.

Perhaps Pang Xuelin's actions just now disturbed Murphy, Murphy opened his eyes in a daze, a pair of blue eyes touched Pang Xuelin, and the girl's face bloomed with a bright smile, and she raised her head and offered a fragrant kiss.

And then this move once again caused a storm.

By the time everything calmed down, it was already after nine o'clock.

After the two of them finished dressing, they came out of the shuttle and saw that the old tree was already waiting outside.

Pang Xuelin smiled and said, "Old Shu, good morning." ”

"Good morning."

The tree elder nodded, looking at Murphy, and the eyebrows simulated by the vines furrowed again.

Murphy said: "Tree Lao, I still don't have it?" ”

Shu Lao shook his head and said, "It's strange, it's been half a year, you guys are all normal, and you haven't taken any contraceptive measures, why hasn't Murphy been pregnant yet?" ”

Pang Xuelin and Murphy glanced at each other and shrugged helplessly.

In fact, Pang Xuelin was not surprised by this.

At the beginning, the system told him that in the dimensional world, it was impossible to have children.

Therefore, when Pang Xuelin was in the dimensional world, he never took any contraceptive measures.

Then when he arrived in the world of "Interstellar", he found that after this world merged with the world of "District 9", Chrissy actually gave birth to a child for himself.

He asked the system what was going on.

The system replied: "The plot requires." ”

According to the system, in the world of "Interstellar", Pang Xuelin needs to be concerned about family affection, so some rules in the world of the ninth district have undergone strange changes, resulting in a daughter between him and Chriss.

Although Pang Xuelin vaguely felt that there were some key things hidden in the answers given to him by the system.

But Pang Xuelin also knew that as long as he didn't do things systematically in the future, he might not be able to have descendants in the dimensional world.

Even in the main world, Pang Xuelin was not sure if he could really get Yao Bingxia and Qi Xin pregnant.

After all, the bottle of genetic optimizer rewarded by the World of the Ninth Region has made Pang Xuelin and ordinary humans have a drastic difference.

At this time, Shu Lao said: "Mr. Pang, Ms. Murphy, please stretch out your hands." ”

Pang Xuelin and Murphy glanced at each other, and the two stretched out their hands one after another.

Soon, the vine stretched out and wrapped itself around Murphy's wrist, turning the tip into a needle and piercing Murphy's vein, sucking some of the blood before the vine left Murphy's wrist.

The tree elder closed his eyes, was silent for a while, and said, "There is nothing wrong with Ms. Murphy's health. ”

Then, he turned his gaze to Pang Xuelin: "Mr. Pang, it's your turn." ”

Pang Xuelin smiled faintly and said, "Elder Shu, please." ”

Soon, the vines spread over and wrapped around Pang Xuelin's wrist, and the needle slowly pierced into Pang Xuelin's body.

It's just that in the next second, the expression on Mo Lao's face suddenly changed, and the vines were instantly pulled out of Pang Xuelin's veins and shrunk into a ball, Mo Lao's face showed an incredible color, and he pointed at Pang Xuelin and trembled a little and said, "You...... You ......"

Pang Xuelin and Murphy glanced at each other, looked at Mo Lao with some surprise, and said, "Mo Lao, is there anything wrong with my blood?" ”

Old Mo had a complicated expression on his face and said, "Mr. Pang, can you take a step to speak?" ”

Pang Xuelin nodded and said to Murphy: "My dear, you go back to the shuttle first, and I'll talk to Mo Lao." ”

Murphy showed a hint of worry on his face, but he still nodded obediently and said, "Husband, you ...... Be careful......"

Pang Xuelin smiled slightly.

In the next second, he felt the scenery in front of him suddenly change, the tree hole where Murphy and the shuttle were located disappeared in front of his eyes, and he and the tree elder appeared in another tree hole space.

Pang Xuelin said: "Old Shu, if you have any questions now, you can ask." ”

The old tree on the side looked at Pang Xuelin with a complicated expression and said, "Mr. Pang, you are not an ordinary human being, are you?" Not even a member of this civilization, right? ”

Pang Xuelin smiled faintly and said, "How can you see it? ”

Shu Lao said: "The genes in your body have undergone the optimization of high-grade genetic medicines, and this kind of genetic medicine is not something that an ordinary civilization that has mastered the power of points can make...... This civilization has reached at least ......"

Suddenly, the old man seemed to realize something, and did not continue.

Pang Xuelin nodded and said lightly: "I have indeed undergone a genetic optimization transformation, and I am indeed not from this world. ”

Shu Lao's face suddenly showed a look of excitement, and he said, "Pang ...... Mr. Pang, in the past six months, the old man has neglected Mr. Pang and asked Mr. Pang for forgiveness. In addition, I also asked Mr. Pang to tell the old man how to leave this small universe, old and ...... The old man is willing to provide Mr. Pang with a trend of thousands of years, and transform more than ten terrestrial planets for Mr. Pang. ”

As he spoke, Elder Shu made a long salute to Pang Xuelin.

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