Scholar's Advanced Technological System - Chapter 1127
Chapter 1127: New Ideas on Hodge Conjecture
At the beginning of the year, before Lu Zhou poached Chen Yang from the Yan University mathematics center, Chen Yang had already begun researching the Hodge conjecture.
Lu Zhou still remembered seeing Chen Yang researching the hyperelliptic curve analysis method on a blackboard. Chen Yang used a very clever mathematical tool to improve this method, which was originally designed to solve Riemann’s hypothesis. This meant the hyperelliptic curve analysis method could be applied to singular complex algebraic clusters, as well as geometrical problems that were defined on the sub-clusters.
This left a good impression on Lu Zhou, which caused Lu Zhou to poach him from the Yan University mathematics center.
Almost a year had gone by since then, and there had yet to be any significant progress on the Hodge conjecture. Not to mention that Lu Zhou was busy with unifying algebra and geometry; he had totally forgotten about this.
“Come, let’s talk about it in my office.”
Lu Zhou brought Chen Yang to his office and took out a whiteboard. He then gave Chen Yang a marker.
Without wasting any time, Chen Yang pondered for a second and then drew a circle on the whiteboard. He marked it as S and wrote down a line of expressions.
“For a compact and boundless surface S, the Gaussian curvature K can be Lebesgue integrated over its entire surface.”
Chen Yang wrote as he continued speaking.
“We all know that a surface can contain more than one measurement, so I tried to change the measurement metric of S. The corresponding Gaussian curvature K also changes, but the integral value stays the same. The measurement metric has nothing to do with the Euler characteristic X(S) of the surface. Using this property, we can—”
Lu Zhou looked at the calculations on the whiteboard and raised his eyebrows with interest.
“Gauss-Bonnet theorem?”
Chen Yang stopped writing and nodded.
“Correct.”
He wrote down the Gauss-Bonnet theorem.
When Lu Zhou saw this, he started to get even more intrigued.
In fact, he already had a rough idea of what Chen Yang wanted to do.
According to the properties of high-dimensional Riemann manifolds M, the Gaussian curvature could be generalized to a sectional curvature, while its value could be determined by the tensor of the Riemann curvature. The integral function was a complicated algebraic formula composed of the curvature tensor and the Gauss-Bonnet integral.
As for its integral over the entire manifold, that was determined by the Euler characteristic number X(M).
By using these properties, the Hodge theory could be extended to non-compact manifolds.
These new profound mathematical relationships were found by Professor Shiing Shen Chern, one of the famous applications of the Gauss-Bonnet theorem.
By combining this with Sir Atiyah’s L2 cohomology method, this conjecture might actually be solved.
Of course, it would require more in-depth research to find a complete proof.
Lu Zhou nodded with satisfaction.
Not bad.
Not bad at all.
Without them knowing it, a crowd of people had formed behind Chen Yang.
People in the office began watching closely ever since he started writing on the whiteboard.
Li Mo looked at the equations on the whiteboard and said, “Is this the legendary…”
He Changwen looked at the kid and frowned. He said, “The legendary what? Finish your sentence.”
Li Mo looked at him strangely.
“The Hodge conjecture! Obviously.”
He Changwen: “…”
How is that obvious?!
Well, I guess it is kind of obvious.
He Changwen couldn’t help but lie to himself.
Yeah, for sure, I definitely would have recognized it.
Chen Yang stopped writing on the whiteboard, and he began to think.
Obviously, he was only halfway through this pathway. He had yet to think about where to go from here.
Professor Perelman suddenly spoke.
“This is quite an interesting pathway.”
Chen Yang looked at Perelman and asked, “When did you get here?”
“When you were about halfway done… I was coming to find Professor Lu.” Perelman paused for a second and said, “… Can I use the pen?”
Without hesitating, Chen Yang handed over the marker.
Perelman took the marker and contemplated it for a while. He then began to write down a few lines of expressions.
“Since there is also a unified algebraic geometry theory, the proof to formula 3 is trivial.
“My suggestion for the later part of the proof is that we can map the compact manifold M to its general covered manifold and get a complete non-compact manifold M.
“According to Atiyah’s theorem, if we can prove that all but the middle L2 homology group is zero under the sectional curvature condition…”
He quickly wrote down a simple yet beautiful equation.
[H^n(M)6≠{0}, when q≠n, H^q(M)={0}]
Chen Yang’s pupils shrank when he saw this.
He had a moment of realization as he spoke with excitement.
“This is how we can prove the Hodge conjecture!”
But here was the problem.
How could they prove that, under the sectional curvature, the L2 homology group was zero?
The conversation abruptly stopped.
After the initial excitement, the two people fell into silence.
In the end, they looked at Lu Zhou.
Lu Zhou noticed them looking at him. He blinked and spoke with a smile.
“I think your ideas are all pretty good… Even though I haven’t carefully researched this area, my intuition tells me that there’s an 80% chance this pathway will work.”
He paused for a second and continued, “This pathway is very interesting, why don’t you guys research together?”
They seemed to understand what Lu Zhou was trying to get at.
Perelman frowned and spoke.
“Are you not joining? This is an interesting problem.”
In fact, it was more than interesting.
The Hodge conjecture was a combination of the three major areas of mathematical analysis, namely topology, algebra, and geometry.
As a Millennium Prize Problem, there was no doubt about its difficulty.
To Perelman’s surprise, Lu Zhou didn’t look interested at all.
Lu Zhou: “I am interested, but I have some work to do at the ILHCRC, so I might not have any time to research mathematics.”
Perelman looked disappointed.
“That’s unfortunate.”
“Even though I can’t work on this myself, I can vouch for Professor Chen,” Lu Zhou said as he patted Chen Yang on the shoulder. He said, “He’s an excellent scholar, I’m sure you know that already. Anyway, if you two work together, I’m sure you’ll be able to solve this problem.”
Perelman disagreed with Lu Zhou’s statement about being able to solve the problem. He looked at Chen Yang and didn’t say anything. He nodded, as a signal of approving Chen Yang as his partner.
These two were both taciturn people.
Lu Zhou cleared his throat and spoke to Perelman.
“Speaking of which, is it fine for you to stay here? The unified theory of algebra and geometry has already been completed.”
“No problem.” Perelman shook his head and said, “I’ve already called my mother. She said I should do what I want to do. She doesn’t mind. I have some unfinished business here… I want to solve Hodge conjecture before going back.”
Lu Zhou was surprised that Perelman wanted to stay. He obviously couldn’t be happier, so he spoke with a smile.
“Then you can stay in your apartment, I’ll apply for an extension for you.”
Perelman nodded and spoke.
“Thank you.”
