Chapter 186 Prove the Hodge Conjecture!

After asking for leave from Deligne, Xu Chuan got up and walked out of the dormitory.

Before officially entering the unknown field of Hodge's conjecture, he still has a lot of work to do. Whether it is in life or mathematics.

Solving the Hodge's conjecture is like the first time humans sailed in the vast ocean. No one knows whether there are other lands in the unknown ocean, and no one knows whether they can successfully reach another coastline.

The only thing he has is a small boat that has just been built.

And after entering the unknown ocean, whether this boat will be overturned by wind and waves, whether it will sink to the bottom of the sea, or whether it will hit the reef and be stuck and unable to move, Xu Chuan doesn't know.

But despite this, he still wants to try.

Because even if it only sails out ten meters, it is a great breakthrough.

After purchasing a batch of daily necessities in the store, Xu Chuan borrowed a batch of manuscripts and materials about the Hodge's conjecture from the Flint Library.

Some of them he had read before, and some of them he had not read yet.

These are precious knowledge left by predecessors, and some of them cannot be found on the Internet at all. Because they are just some ideas and original theories of a mathematician, and they have not yet taken shape.

These things, whether he had seen them or not, were very useful for him to launch a charge against the Hodge conjecture.

However, when borrowing these things, he encountered a big trouble.

The person who managed the Flint Library was an old man who looked unkempt. This old man with messy hair like a bird's nest was a top expert in the preservation of paper materials, but he was also extremely stubborn.

And this stubborn old man was always unwilling to lend so many documents to others, thinking that he was likely to damage or lose these precious manuscripts.

In order to obtain this batch of materials, Xu Chuan spent a day in the Flint Library, and the final effort was just to get the other party to agree to put them together and read them in the library.

But for Xu Chuan, proving the Hodge conjecture in the library was not a very reliable way.

Although it was very quiet here, people came and went every day.

There was no way, and in the end he could only find David Xiu, the dean of Princeton's School of Mathematics, and made a series of guarantees, and learned some preservation methods of paper materials, and even signed a guarantee letter, before he reluctantly let the other party agree.

With a lot of information, Xu Chuan returned to the dormitory.

In fact, he would have taken good care of these things even without the reminder from the old German.

But now, in addition to preserving them well, the greater value of these materials is to play their role in the Hodge conjecture.

I believe that the mathematicians who created these knowledge must have thought so.

For a scholar, no one wants to see the knowledge they created shelved. If a piece of knowledge cannot be circulated and used, it has no value for knowledge.

After handling the preparations before entering the Hodge conjecture, Xu Chuan locked himself in the dormitory again.

Time passed like this. In the blink of an eye, the golden autumn of October arrived. The sugar maple, sycamore and other trees outside the Rockefeller Residential College began to show a hint of golden color. Occasionally, a few fallen leaves slowly fell with the wind.

In dormitory No. 306, a figure stood in front of the window, looking at the sycamore trees outside that were full of sycamore fruits.

The sunrise in the early morning was bright in the dark blue clouds. The golden and dark green leaves outside the window were intertwined, and the heavy sycamore fruits were embedded in them.

Looking at the scenery outside the window, Xu Chuan had a smile on his face.

Autumn is the season of harvest.

Although the research on the Hodge conjecture was not as smooth as he expected, he was always confident about the final result.

Two months later, in the unknown ocean of the Hodge conjecture, he finally found a coastline in front of him.

That's the new world!

Looking at the scenery outside the window, Xu Chuan turned back to the table with a smile.

Although the Hodge conjecture has not been perfectly solved, he has seen the horizon where the coast intersects and the new world standing in the sky.

All that remains is to row his boat over.

Picking up the ballpoint pen on the table, Xu Chuan continued writing where he had not finished before:

“.Let V be an algebraic variety in complex projective space, Vˊ be the set of regular points of V. The L2-de Rham cohomology group on Vˊ with respect to the Fubini-Study metric is isomorphic to the cross cohomology group of V.”

“If Y is a closed subalgebraic variety of X defined on k with codimension j, we have the standard mapping: Tr : H2(nj)(Yk k， Q`)(n j)→ Q`where (n j) is the n j-order Tate twist Q`(n j).

This mapping is similar to the restricted mapping: H2(nj)(Xk k， Q`)(n j)→ H2(nj)(Y， Q`)(n j)”

“.”

“According to Poincare’s duality theorem: Hom(H2(nj)(Xk k， Q`)(n j), Q`)= H2j (Xk k， Q`)(j)“

Time passed by little by little under his pen, and Xu Chuan devoted himself to the final breakthrough.

Finally, the pen in his hand suddenly turned.

“.Based on the mapping Tr, the restricted mapping and Poincare, the duality theorem is compatible with the action of Gal(k/k), so the action of Gal(k/k) on the cohomology class defined by Y is also trivial. Then Aj (X) is the Q vector space generated by the cohomology class of the closed subalgebraic variety defined on k in H2j (Xk k, Q`)(j) with codimension j of X”

“When i≤n/2, Ai (X)∩ The quadratic form x→(1)iLr2i(x.x) on ker(Ln2i+1) is positive definite. "

"From this, it can be obtained that on non-singular complex projective algebraic varieties, any Hodge class is a rational linear combination of algebraic closed chain classes."

"That is, the Hodge conjecture holds!"

After the ballpoint pen in his hand tapped the last dot on the white manuscript paper, Xu Chuan breathed a sigh of relief, threw the ballpoint pen aside, leaned back, leaned back on the chair and stared at the ceiling in a daze.

When the last character fell on the manuscript paper, what surged in his heart was not excitement, happiness, satisfaction or sense of accomplishment.

Instead, it was accompanied by some unbelievable confusion.

It took more than four months, starting from the manuscript left to him by Professor Mirzakhani, to the solution of the problem of 'irreducible decomposition of differential algebraic varieties', to the improvement of algebraic variety and group mapping tools, and finally the solution of the Hodge conjecture.

On this road, he experienced too much.

After staring at the ceiling for a long time, Xu Chuan finally came back to his senses and his eyes fell on the manuscript on the desk in front of him.

After going through all the manuscripts and confirming that this was really his own work, he finally showed a bright smile on his face, as bright as the sunlight coming in from the window.

If nothing unexpected happened, he succeeded.

He successfully solved the century-old problem of the Hodge conjecture.

This is the most important breakthrough in the problems related to the Hodge conjecture since the mathematician Lefschetz proved the Hodge conjecture of the (1,1) class in 1924.

Although he doesn't know whether it can stand the test of other mathematicians and time.

But in any case, he has taken another big step in mathematics.

After completing the paper proving the Hodge conjecture, Xu Chuan spent some time going through the things on the manuscript again and perfecting some other details.

After processing these, he began to organize them into a notebook.

Then he prepared to make it public.

For the proof of any mathematical conjecture, the prover is not qualified to give an evaluation of whether it is correct or not.

Only by making it fully public, and undergoing peer review and the test of time, can we determine whether it has really succeeded.

After a whole week, Xu Chuan finally entered all the nearly 100 pages of manuscript paper in his hand into the computer.

Of these hundreds of pages of proof, more than one-third of the length is the explanation and demonstration of the algebraic cluster and group mapping tools for solving the Hodge conjecture, and another one-third of the length is the theoretical framework for the Hodge conjecture and the algebraic cluster and group mapping tools.

The rest is the proof process of the Hodge conjecture.

For this paper, tools and frameworks are its core foundation.

If he wants, he can separate the tools and theoretical frameworks and publish them as independent papers.

Just like Peter Schultz's 'p-adic class perfect space theory'.

If these things are finally accepted by the mathematical community, they will be enough for him to win a Fields Medal.

This is not because the Fields Medal is cheap, but because mathematical tools are important to mathematics.

An excellent mathematical tool can solve more than just one problem.

Just like an axe, it can not only be used to cut down trees, but also as a carpenter's tool, process objects, and can be used as a weapon for fighting.

Similarly, the algebraic cluster and group mapping tools he constructed are not limited to the Hodge conjecture.

Many algebraic clusters and differential forms, polynomial equations, and even algebraic topology problems can be used to try.

For example, the "Bloch conjecture" that belongs to the same family of conjectures as the Hodge conjecture, the "Hodge theory of algebraic surfaces should determine whether the Chow group of zero cycles is finite-dimensional", and the conjecture that some motivated cohomology groups of finite coefficients are isomorphically mapped to the etale cohomology problem, etc.

These conjectures and problems support each other, and mathematicians continue to make progress in one or the other, trying to prove that they have led to great progress in number theory, algebra, and algebraic geometry.

If the algebraic cluster and group mapping tools can solve the Hodge conjecture, then it can at least play a part in the same type of conjectures, not to say that it can be fully adapted.

Because the Hodge conjecture is a conjecture that studies the relationship between algebraic topology and the geometry expressed by polynomial equations.

What it studies is not the most advanced mathematical knowledge, but a basic connection between the three disciplines of algebraic geometry, analysis and topology.

To solve this problem, the prover needs to have a deep understanding of mathematics in these three fields.

For most mathematicians, it is quite difficult to have in-depth research in one of the three fields of algebraic geometry, analysis and topology, let alone master all three fields.

For Xu Chuan, analysis and topology were the mathematical fields he was proficient in in his previous life, and only algebraic geometry was not within his research scope.

But in this life, he followed Deligne to study mathematics in depth, and with such a mentor, his progress in algebraic geometry was beyond imagination.

After finishing all the proof papers of the Hodge conjecture and entering them into the computer, Xu Chuan converted them into PDF format and sent them to Deligne and Witten by email.

After thinking about it, he uploaded it to the arxiv preprint website.

Although the arxiv preprint website has gradually become a place where computers occupy pits, there are still a large number of mathematicians and physicists on it.

Throwing your unpublished papers there can not only occupy pits in advance to prevent plagiarism, but also expand the influence of the paper in advance.

For proof papers of problems such as the Hodge conjecture, it will undoubtedly take a long time to complete the verification.

For example, the three-dimensional case of the Poincaré conjecture was proved by mathematician Gregory Perelman around 2003, but it was not until 2006 that the mathematical community finally confirmed that Perelman's proof solved the Poincaré conjecture.

Of course, this is also related to the fact that Perelman almost refused any award given to him and lived in seclusion.

After all, if a prover of a conjecture does not promote his own proof method and process, it is almost impossible for others to quickly understand this method.

Especially in the field of mathematics.

For a proof paper, if there is no original author to explain it and answer the confusion of other colleagues, it is very difficult for other mathematicians to thoroughly understand this paper.

In addition, for major conjectures such as the Millennium Mathematical Problem, the process of acceptance by the mathematical community is generally longer.

After all, its correctness or not is extremely important.

Just like the Riemann hypothesis, since it was proposed by mathematician Bernhard Riemann in 1859, there have been more than thousands of mathematical propositions in the literature of the mathematical community, which are based on the establishment of the Riemann hypothesis (or its generalized form).

If the Riemann hypothesis is disproven, not to mention the collapse of the mathematical building, at least the vast field of the Riemann hypothesis, from number theory to function, to analysis, to geometry, it can be said that almost the entire mathematics will have major changes.

Once the Riemann hypothesis is proved, the thousands of mathematical propositions or conjectures built around it will be promoted to theorems. The history of human mathematics will usher in an extremely vigorous development.

In fact, the review speed of the proof of a problem or conjecture depends largely on the popularity of the problem or conjecture and the extent to which the research work on the problem or conjecture has progressed in the mathematical community.

In addition, there are methods, theories and tools used to prove this problem or conjecture.

For example, when he proved the weak Weyl_Berry conjecture before, he only made some innovations in the two fields of Banach space symmetry structure theory and spectral asymptotics on connected regions with fractal boundaries, and used fractal drums to open the related counting functions.

So the proof process of the weak Weyl_Berry conjecture was quickly accepted by Professor Gowers.

When proving the Weyl_Berry conjecture, he made a breakthrough in the previous method, limiting the fractal dimension and spectrum of fractal measure of Ω through Dirichlet domain, and then supplemented by domain expansion and converting functions into subgroups and establishing connections with intermediate domains and collections.

The mathematical community was much slower to accept this method.

Even though his paper was eventually reviewed by six top bigwigs, four of whom were Fields Medal winners, and he was on the scene to answer questions throughout the process, it still took a long time to be confirmed.

And to this day, there are still not many people in the entire mathematical community who can fully understand the proof process of the Weyl_Berry conjecture.

Even though he later extended this method to the astronomical community, enhancing its importance.

As for the proof process of the Hodge conjecture in his hands now, let alone.

God knows how long it will take for the mathematical community to fully accept this paper.

One year? Three years? Five years? Or even longer?

During this long time, Xu Chuan did not want to see his paper shelved.

He hopes that more mathematicians and even physicists will get involved to expand and apply it to more and broader fields.