Things Were Looking Up
Algebraic Geometry is the most popular research area in Mathematics Community today, with more than half of mathematicians devoting themselves to it. Shen Qi is by no means the only one who aspires to complete the Langlands Program or the Hodge conjecture.
The teams led by Ngô Bảo Châu at the University of Chicago and Lafog in the ENS Paris are at the forefront of the world in terms of the Langlands Program. The Mathematics Community generally believes that whoever can prove and perfect the Langlands Program first will achieve the Grand Unification that mathematicians have been dreaming of.
The Langlands Program is quite difficult to understand. Ngô Bảo Châu and Lafog have each studied it for more than ten or twenty years, and so far they have only proven one lemma and one inference in the Langlands Program.
It was based on this lemma and this deduction that Ngô Bảo Châu and Lafog were awarded the Fields Medal.
The Langlands Program establishes a system, Hodge conjecture is the most complex case.
Since the Hodge conjecture was proposed in 1950, mathematicians have been trying to prove it.
The fruitful research comes from the University of Bonn in Germany, where Peter Schultz's team proved two years ago a corollary that algebraic homology Algebraic variety on nonsingular projective complex algebra varieties.
Two years after narrowly missing out on the Fields Medal, Schultz has been working tirelessly on the Hodge conjecture ever since. His latest paper, published in Inventiones Mathematicae, has made Shen Qi a little nervous.
"In his latest research, Schultz proved the similarity of elements in cohomology classes, and it has to be said that it's a very elegant proof." Shen Qi repeatedly studied Schultz's new paper, and he felt that Schultz was on the right track and doing the right thing. "I absolutely cannot let Schultz or anyone else prove the Hodge conjecture before me. Completing the proof of the Hodge conjecture is one of my essential tasks."
However, mathematics is not a combat-based sport; the principle of give and take does not apply to mathematics.
The only effective way to beat your rivals in mathematics is to do things faster and better than them.
Mathematics is more like a sprint, with each person on their own track. You don't need to care too much about how fast or slow your opponents are. What Shen Qi needs to do is run at the fastest speed on his own track.
After the Christmas holidays, Yu Lei and Ralph returned to campus for their second semester as doctoral students.
"Guys, Schultz from the University of Bonn is catching up with us! Yes, he and his team are also working on the Hodge conjecture! We must understand that the Hodge conjecture is not our patent; anyone has the right to have ideas about it and take action! Obviously, the number and quality of teams competing with us on the Hodge conjecture far exceed those competing on the Riemann Hypothesis. After all, there are only a handful of internationally renowned number theory teams, while Algebraic Geometry teams... are simply countless!" Shen Qi and his two doctoral students' battlefield shifted from outdoors to his office.
During this special period, secrecy is essential; even a single formula in the unfinished product should not be seen by others.
Although most people in the world cannot understand the Hodge conjecture or comprehend the reasoning behind Shen Qi Team's investigation, they do have mobile phones, which have camera functions.
"Peter Schultz, two years ago he derived a corollary that algebraic homology Algebraic variety on nonsingular projective complex algebraic varieties, and just now proved the similarity of elements in homology classes. With these two achievements combined, he has already... oh no, he's almost catching up with us in the proof of the Hodge conjecture." Ralph frowned; every mathematician harbors the ambition to be the first to prove something.
"Ralph, are you scared? Are you afraid?" Shen Qi asked.
"No, we will never back down, and there is nowhere to back down."
"Very good." Shen Qi expressed his satisfaction, as he was very concerned about Ralph's mindset and condition.
Ralph is the only American in the entire team. He completed his undergraduate, master's, and doctoral studies at Princeton, making him a true Princeton University mathematician.
"This Schultz is quite capable; I heard he's Mary's ex-husband?" Yu Lei's thinking is rather divergent; he's good at approaching problems from multiple angles, analyzing and solving them comprehensively and objectively.
Shen Qi nodded. "Yes, this German madman loves math more than his wife."
"Studying Mathematical Research to the point of abandoning one's wife—it's not the fearless who's afraid of death, it's the wifeless who's scary... Schultz, he's our strongest rival." Yu Lei felt a chill. He said, "Professor Shen, I think you're more humane, which is why you won the Fields Medal, while Schultz didn't."
"Remember, we don't befriend men who don't love their wives. Okay, back to the point. I hope your papers can be published this semester, or at least pre-recorded on arVix. We need to demonstrate our capabilities." Shen Qi made specific demands of the two doctoral students, and then came the technical guidance session.
Shen Qi's thinking shares similarities with Schultz's, but also has its own unique highlights.
Shen Qi will inevitably have to cross paths with Schultz in the area of high-synchronous classes; it's always the same pattern—you're allowed to use it, but I'm not.
Schultz is a master of playing tricks, and to do something even more brilliant than Schultz, Shen Qi needs to play some unconventional tricks. The most crucial thing is how to reverse-engineer the conclusion that nonsingular projective complex algebraic Algebraic variety are a special type of complex manifold.
The reverse derivation process lasted for more than half a year. Yu Lei and Ralph were responsible for the details, while Shen Qi kept track of the direction.
In proving the Hodge conjecture, to put it simply, Schultz used a forward and forceful reasoning approach. His core logic was based on pure Algebraic Geometry, and he achieved good results.
Shen Qi adopted a combination of forward, backward, and lateral reasoning. His core logic is based on Algebraic Geometry, Mathematical Analysis, and topology, and his system framework is more complete. However, his proof process is also more brain-burning and complex, requiring a large team of basic theoretical researchers to complete this arduous task together.
Yu Lei and Ralph worked right under Shen Qi's nose. Shen Qi was very clear about the progress of the projects his two doctoral students were in charge of. He believed that he had done a pretty good job of guiding them and was a qualified doctoral supervisor.
Shen Qi was also very concerned about the progress of the research project led by Lu Guozhen and Director Wu on the Chinese side. They kept in touch and shared information.
"Director Lu, how's the progress of your Mathematical Analysis section?" Shen Qi asked during the video call.
"It's going well, so don't worry! My 973 Program is a major annual project for the institute. The dean has assigned me two elite members, and our team is in high spirits. The situation is very promising!" Lu Guozhen in the video was beaming, as if he were experiencing youthful exuberance.
Shen Qi was greatly encouraged. "That's great! Director Lu is really capable! We don't need to discuss the specific technical details over a video call, as it's impossible to discuss them clearly anyway. Director Lu, could you give me a hint: when can we do our first joint draft?"
Lu Guozhen: "Expected in July."
Shen Qi: "Great, it's close to the timeline I envisioned. I look forward to your research results!"