The Science Fiction World of Xueba

This supply mission came at the right time, just when Pang Xuelin's research was at a standstill.

The arrival of the supply ship allowed Pang Xuelin to get out of that state of ecstasy, otherwise it might not be a good thing for him if that state continued.

In the next two months, Pang Xuelin built a brand new living cabin on the side of the energy cabin according to the mission log sent by the Ares project headquarters. The new living cabin was connected to the energy cabin, and the energy cabin was then planted with tomatoes. The cabin and the potato planting cabin are connected to form a four-chamber structure.

Both the structural strength and service life of the new living cabin are much higher than the original living cabin. At least when Pang Xuelin enters and exits the airlock through the new living cabin, he no longer has to worry about an air explosion.

After that, Pang Xuelin conducted a large-scale inspection of the habitat again, and replaced some severely worn parts, such as the oxygenation system, water circulation system, air conditioning system, nickel-metal hydride battery pack, solar battery pack, and so on.

In order to facilitate Pang Xuelin's heavy physical labor, the Ares Project Headquarters also specially equipped two sets of exoskeleton equipment in this supply.

One set for EVA missions and one set for cabin missions.

Adequate material support and perfect living facilities have further improved Pang Xuelin's safety and quality of life on Mars.

After several months of busy work, life became stable again, and Pang Xuelin also turned his attention to the ABC conjecture.

After more than a year of dedicated study and research, Pang Xuelin has a relatively clear understanding of Far Abelian geometry and general Teich-Miller geometry theory.

He was almost certain that Mochizuki Shinichi's work had serious problems.

But he didn't think about finding out the detailed errors in Mochizuki's thesis, and then set off a war of words. He had a higher goal.

He wants to take advantage of the rare opportunity of being undisturbed in the past few years to create a new set of theories on the basis of Far Abelian geometry to completely solve the problem of the ABC conjecture.

This is very difficult.

In the field of mathematics, it is easy to conquer a conjecture, but it is extremely difficult to create a system.

Anyone who can create a new mathematical system is almost always a figure at the level of a grand master.

For example, Galois, who pioneered group theory, died young at the age of 21, but in any ranking of the greatest mathematicians of all time, Galois is among the top five or even the top three.

For another example, Grothendieck, the founder of modern algebraic geometry, EGA, SGA, FGA, has thousands of pages of eloquence, and is an immortal masterpiece in the history of algebraic geometry. The student Pierre Deligne completely proved the Weil conjecture, which is considered one of the most significant achievements of pure mathematics in the 20th century.

Thanks to Grothendieck's leadership, in the 1960s and 1970s, the Paris Institute for Advanced Study was recognized as the world's algebraic geometry research center, for which he also won the Fields Medal, the highest international mathematics award in 1966.

Therefore, Pang Xuelin can say that he has set up a Mount Everest in front of him. When he will be able to climb this high mountain, Pang Xuelin himself has no idea.

Even up to now, Pang Xuelin has not been able to find a suitable route into the mountain.

Pang Xuelin can only live on Mars with peace of mind while thinking.

Of course, this kind of thinking is only intermittent. When he has inspiration, he will concentrate on thinking. When he has no inspiration, he will work and live step by step. When he is free, he will listen to music and watch movies in the living cabin to relax himself. one time.

Later Pang Xuelin found a better way of thinking.

That is to wear an EVA spacesuit, take the habitat as the center, and walk around the habitat with a radius of 100 meters.

Walking alone in this desolate and lonely world,

There is a unique sense of loneliness, and it is easier to let your head go into a state of emptiness.

Especially at night, the surface of Mars is pitch black, leaving only the faint lights of the habitat cabin and the sky full of stars above.

At the moment when this complex world is hidden into darkness and only a few stars are left, Pang Xuelin can feel that in the universe of number theory, prime numbers are like shining stars, presenting a complex mathematical configuration.

He often walked for hours, until the EVA's carbon dioxide filter sounded an alarm, and he would recover and return to the habitat.

Later, he learned to behave well, and he took a spare carbon dioxide filter with him. When one was used up, he could replace it with another one at any time, and returned to the habitat when the inspiration was exhausted.

Time passed day by day.

One month, two months...

One year, two years...

The cold atmosphere of Mars sharpened Pang Xuelin's thinking. During the long journey of thought, the logic system of far-Abelian geometry gradually dissipated in Pang Xuelin's mind, replaced by a more messy, but closer to the essence of mathematical logic.

Pang Xuelin's thinking became clearer and clearer, and his logic became sharper and sharper.

Unknowingly, Pang Xuelin has lived on Mars for more than five years, and the Ares 4 mission team has also started a new journey to Mars.

On the 1468th solar day, the Ares Project Command launched an MAV 500 meters away from the habitat module. One month later, the Hermes spacecraft set sail from the geosynchronous orbit space station. The No. 1 mission team officially went to Mars, and their main task was to bring Pang Xuelin back to Earth safely.

On the 1689th solar day, there is still more than a month before the Ares IV mission team arrives at Mars.

That night, Pang Xuelin went out of the cabin again and started another journey of thought.

"The absolute Galois group Gal(Qˉ/Q) can act on all smooth algebraic curves, because each smooth algebraic curve corresponds to a polynomial whose coefficient is an algebraic number, and the absolute Galois group Gal(Qˉ/Q) is the symmetry of the algebraic number group……"

...

"The simplest nontrivial transformation in the absolute Galois group Gal(Qˉ/Q) is the complex conjugation. On the complex plane, the complex conjugation is the mirror symmetry along the real number axis, so it acts on the smooth algebraic curve, and the obtained It is also the mirror symmetry of a smooth algebraic curve. If the mirror symmetry of a smooth algebraic curve is itself, according to Bely's theorem, the complex conjugate is applied to the corresponding algebraic curve to obtain the original algebraic curve, that is to say, all coefficients are real numbers .If two smooth algebraic curves are mirror images of each other, the coefficients of their corresponding algebraic curves must be conjugate to each other, that is to say, at least some of the coefficients are imaginary numbers.”

...

"In smooth algebraic curves, there are many combination invariants, which remain unchanged under the transformation of the absolute Galois group Gal(Qˉ/Q): the number of vertices, the degree of vertices, the number of faces, the degree of faces, etc. .In addition to these seemingly simple invariants, we can also assign a group to each smooth algebraic curve, which can be called the single-valued group of smooth algebraic curves. These groups have more complex structures, but also in absolute The Galois group Gal(Qˉ/Q) remains unchanged under the transformation."

...

"Then, can the absolute Galois group Gal(Qˉ/Q) act on the Teich-Miller level? All the higher parts of the Teich-Miller level can be combined from the first two levels, and the first level provides the Elements, the second layer provides the relationship between elements. The first two layers correspond to smooth algebraic curves, and the second layer corresponds to elliptic curves that are widely used in number theory..."

...

Faintly, Pang Xuelin seemed to have caught some kind of wonderful clue.

He looked up, the starry sky above him reflected in the glass shield of his spacesuit helmet.

The prime numbers rippled layer by layer in the number field universe. Under the surface of the complex number field, a more intuitive connection between prime numbers began to appear in Pang Xuelin's eyes.

"Smooth algebraic curve itself has many symmetries. For these symmetries, it can be determined that it must come from the absolute Galois group Gal(Qˉ/Q). If we can know this, it is equivalent to describing the absolute Galois group Gal(Qˉ/Q) itself!"

There was a gap in the digitally constructed starry sky, and Pang Xuelin's eyes became brighter and brighter.

A flash of lightning flashed across his brain, illuminating the starry sky of prime numbers that was gradually becoming rich in regularity, which was hidden in the darkness.