In the same way, the algebraic variety and group mapping tools he constructed are not limited to Hodge conjecture.

It can be used to try many algebraic varieties, differential forms, polynomial equations, and even difficult problems in the direction of algebraic topology.

For example, the 'Bloch conjecture', which belongs to the same family of conjectures as the Hodge conjecture, the issue of 'the Hodge theory of algebraic surfaces should determine whether the chow group of zero cycles is finite-dimensional', and some motivations for finite coefficients.

The homology group isomorphism is mapped to etale's cohomology problem guessing, etc.

These conjectures and problems supported each other, and mathematicians continued to make progress on one or the other, trying to prove that they led to huge advances in number theory, algebra, and algebraic geometry.

The algebraic variety and group mapping tools can solve the Hodge conjecture, so it cannot be said that it can be fully adapted to the same type of conjecture, but it can at least play a partial role.

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

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

To solve this problem, you need a prover who has a deep understanding of mathematics in these three fields.

For most mathematicians, it is quite difficult to conduct in-depth research in any of the three major fields of algebraic geometry, analysis, and topology, let alone be proficient in all three major fields.

For Xu Chuan, analysis and topology were the areas of mathematics he was proficient in in his previous life, but algebraic geometry was not within the scope of his research.

But he studied mathematics in depth with Deligne all his life. With such a mentor, his progress in algebraic geometry was beyond imagination.

...

After finishing all the proof papers of Hodge's conjecture and inputting them into the computer, Xu Chuan converted them into pdf format and sent them to his two tutors, Deligne and Witten, via email.

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

Although today's arxiv preprint website has gradually become a place for computers to occupy, there are still a large number of mathematicians and physicists on it.

By posting your unpublished papers, you can not only take advantage of them in advance to prevent plagiarism, but also expand the influence of your papers in advance.

For proof papers on issues such as Hodge's conjecture, it will undoubtedly take a long time to completely complete the verification.

This chapter is not finished yet, please click on the next page to continue reading the exciting content! For example, the three-dimensional case of the 'Poincare Conjecture' was proved by the mathematician Grigory Perelman around 2003, but it was not until 2006 that

, the mathematical community finally confirmed that Perelman's proof solved the Poincare conjecture.

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

After all, if a prover of a conjecture does not promote his own proof method and process, it will be 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 will be very difficult for other mathematicians to fully understand the paper.

In addition, for major conjectures such as the Millennium Mathematics Problem, the mathematical community generally takes a long time to accept them.

After all, the relationship between whether it is correct or not is extremely important.

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

as a premise.

If once the Riemann Hypothesis is proven false, let alone the collapse of the mathematics building, at least it will involve the huge field of Riemann Hypothesis, from number theory, to functions, to analysis, to geometry... It can be said that almost the entire mathematics

There will be significant changes.

Once the Riemann Hypothesis is proven, 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 to a large extent on the popularity of the problem or conjecture, and how far the research work on this 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 previously proved the weak Weyl_berry conjecture, he only made some innovations in the two fields of Banach space symmetry structure theory and spectral asymptotic on connected regions with fractal boundaries, using fractal drum pairs to associate counting functions.

Made an opening.

Therefore, the proof process of the weak Weyl_berry conjecture was quickly accepted by Professor Gowers.

When proving the Weyl_berry conjecture process, he made a breakthrough in the previous method. He used the Dirichlet domain to limit the fractal dimension of Ω and the spectrum of the fractal measure, and then supplemented it with the expansion of the domain and the conversion of the function into sub-dimensions.

Groups and establish relationships with intermediate fields and collections.

The mathematical community has been much slower to accept this method.

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

To this day, there are still not many people in the entire mathematical community who can fully understand the proof process of Weyl_berry's conjecture.

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

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

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

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

During this long period of time, Xu Chuan was not willing to see his thesis shelved.

He hopes that more mathematicians and even physicists will participate in expanding and applying it to more and wider fields.
Chapter completed!