Probably at the beginning of the year, before Lu Zhou hired Chen Yang from the Mathematics Center of Yanda, Professor Chen was already studying the Hodge conjecture.
It was precisely because of this beautiful operation that Lu Zhou could not help but fall in love with talents and hired him from the Mathematics Center of Yanda to Jinling.
While writing, Chen Yang continued.
Staring at the calculations on the whiteboard, Zimmer's eyes shone and he whispered excitedly: "Is this the legendary -"
Obviously, he was only halfway through this line of thinking, and he didn’t have a good idea of how to proceed further.
"That's such a shame."
It was precisely because of this beautiful operation that Lu Zhou could not help but fall in love with talents and hired him from the Mathematics Center of Yanda to Jinling.
Lu Zhou still remembers that he was studying his hyperelliptic curve analysis method on the blackboard, and used a very clever method to improve this mathematical tool originally designed for the quasi-Riemann hypothesis and directly apply it to the analysis of Africa. Research on the algebraic topology of singular complex algebraic varieties and the geometric correlation problems expressed by polynomial equations defining sub-varieties.
"Aren't you going to participate? This is an interesting problem."
At some point, a circle of people stood behind Chen Yang.
Looking back at Professor Perelman, Chen Yang was slightly stunned and said with some surprise.
Of course, how to prove it specifically requires in-depth study.
"Let's go to my office and talk."
Probably at the beginning of the year, before Lu Zhou hired Chen Yang from the Mathematics Center of Yanda, Professor Chen was already studying the Hodge conjecture.
Finally, they all looked at Lu Zhou in unison.
It was precisely because of this beautiful operation that Lu Zhou could not help but fall in love with talents and hired him from the Mathematics Center of Yanda to Jinling.
After a brief moment of excitement, the two of them fell into silence.
Both of them are the type who don't talk much and don't communicate much.
"sorry to bother you."
"Although I'm afraid I can't spare the time to help, I solemnly recommend Professor Chen to you," Lu Zhou patted Chen Yang on the shoulder and said with a smile, "He is an excellent scholar. Regarding his ability, I believe You also understand, so I won’t brag too much. In short, if you cooperate, I believe we will be able to solve this problem. "
"No problem," Perelman shook his head. "My mother and I have already had a phone call. She asked me to do what I want to do without paying too much attention to her. I do have something else I want to do. Done, I plan to stay here... for a while longer, and then go back after solving Hodge's conjecture."
It has been almost a year now, and there is still no progress on the topic of Hodge's conjecture. In addition, Lu Zhou has been busy with the unified theory of algebra and geometry some time ago, so that Lu Zhou almost forgot about it.
"Let's go to my office and talk."
"According to Attiyah's theorem, if we can prove that under the condition of cross-section curvature, except for the L2 homology group in the middle, the rest are zero... "
Obviously, he was only halfway through this line of thinking, and he didn’t have a good idea of how to proceed further.
"Let's go to my office and talk."
Zimmer gave him a strange look.
How to prove that under the condition of cross-section curvature, all the others are zero except the L2 homology group in the middle
"Let's go to my office and talk."
Using these properties, Hodge theory can be extended to complete noncompact manifolds.
[H^n(M)6≠0｝, and when q≠n, H^q(M)={0}]
These profound mathematical meanings were obtained by Professor Chen Shengshen, which is the mathematical connotation of the famous Gauss–Bonnet–Chen formula.
The pen in his hand stopped, Chen Yang nodded and said.
Probably at the beginning of the year, before Lu Zhou hired Chen Yang from the Mathematics Center of Yanda, Professor Chen was already studying the Hodge conjecture.
Zimmer gave him a strange look.
"Gauss-Bonnet formula?"
"... My suggestion is that for the subsequent proofs, we can lift the problem of compact manifold M to its universal covering manifold to obtain a complete non-compact manifold M."
After taking Chen Yang to his office, Lu Zhou went to the corner to help him drag a whiteboard and handed his marker to his hand.
The conversation suddenly stopped here.
Perelman nodded and thanked him.
Without wasting time on politeness, after taking over the pen, Chen Yang stood in front of the whiteboard and thought for a moment. He first drew a circle on the whiteboard, then marked S next to it, and wrote a line of expressions.
The pen on the whiteboard stopped, and Chen Yang fell into deep thought.
Both of them are the type who don't talk much and don't communicate much.
Using these properties, Hodge theory can be extended to complete noncompact manifolds.
At some point, a circle of people stood behind Chen Yang.
Lu Zhou still remembers that he was studying his hyperelliptic curve analysis method on the blackboard, and used a very clever method to improve this mathematical tool originally designed for the quasi-Riemann hypothesis and directly apply it to the analysis of Africa. Research on the algebraic topology of singular complex algebraic varieties and the geometric correlation problems expressed by polynomial equations defining sub-varieties.
"This idea seems interesting."
Staring at the calculations on the whiteboard, Zimmer's eyes shone and he whispered excitedly: "Is this the legendary -"
"... For a compact and edgeless surface S, its Gauss curvature K can be integrated over the entire surface."
Well, if you look carefully, he should be able to see it.
According to the properties of the high-dimensional Riemannian manifold M, Gauss curvature can be generalized to cross-sectional curvature, and its value can be determined by the tensor of Riemannian curvature. As for its integrand, it is a very complex algebraic formula composed of curvature tensors—that is, the Gauss-Bonnet integrand.
While writing, Chen Yang continued.
"That's such a shame."
Perelman nodded and thanked him.
"Although I'm afraid I can't spare the time to help, I solemnly recommend Professor Chen to you," Lu Zhou patted Chen Yang on the shoulder and said with a smile, "He is an excellent scholar. Regarding his ability, I believe You also understand, so I won’t brag too much. In short, if you cooperate, I believe we will be able to solve this problem. "
"That's such a shame."
"As we all know, a surface does not necessarily have only one metric, so I tried to change the metric of S. After changing the metric, the corresponding Gauss curvature K will also change, but the integral value is different from that of the surface. The measurement has nothing to do with it, but is only related to the Euler characteristic number X(S) of the surface. Using this property, we can-"
But upon closer inspection, it seems that is indeed the case.
Staring at the calculations on the whiteboard, Zimmer's eyes shone and he whispered excitedly: "Is this the legendary -"
Well, if you look carefully, he should be able to see it.
"Hodge's conjecture! It's obvious."
"As we all know, a surface does not necessarily have only one metric, so I tried to change the metric of S. After changing the metric, the corresponding Gauss curvature K will also change, but the integral value is different from that of the surface. The measurement has nothing to do with it, but is only related to the Euler characteristic number X(S) of the surface. Using this property, we can-"
Looking at the calculations on the whiteboard, Lu Zhou raised his eyebrows slightly and said with interest.
Perelman nodded and thanked him.
Taking the pen from Professor Chen, Perelman stood in front of the whiteboard and pondered for a moment, then blanked a few lines under his calculations and continued writing.
Thinking of this, Lu Zhou nodded approvingly.
"... For a compact and edgeless surface S, its Gauss curvature K can be integrated over the entire surface."
"This idea seems interesting."
"I think your ideas are good... Although I have not studied this topic carefully, my intuition tells me that if I follow this path, I will probably gain something."
"Gauss-Bonnet formula?"
Looking at the calculations on the whiteboard, Lu Zhou raised his eyebrows slightly and said with interest.
How to prove that under the condition of cross-section curvature, all the others are zero except the L2 homology group in the middle
"When did you come here?"
In fact, he had probably guessed what Chen Yang was planning to do.
The pen in his hand stopped, Chen Yang nodded and said.
"That's such a shame."
As for the difficulty, as a millennium problem, there is no doubt about it.
"According to Attiyah's theorem, if we can prove that under the condition of cross-section curvature, except for the L2 homology group in the middle, the rest are zero... "
"Exactly."
After a pause, he continued.
Probably at the beginning of the year, before Lu Zhou hired Chen Yang from the Mathematics Center of Yanda, Professor Chen was already studying the Hodge conjecture.
Professor Perelman, who had been standing silently next to him when he arrived at the office, suddenly spoke.
Thinking of this, Lu Zhou nodded approvingly.
After that, he wrote down the Gauss-Bonnet formula.
Lu Zhou: "...Although I am very interested, there are still a lot of things waiting for me to deal with at IMCRC. I'm afraid I don't have any more time to allocate to mathematics recently."
"... My suggestion is that for the subsequent proofs, we can lift the problem of compact manifold M to its universal covering manifold to obtain a complete non-compact manifold M."
Professor Perelman, who had been standing silently next to him when he arrived at the office, suddenly spoke.
"Speaking of which, is it okay for you to stay here? The unified theory of algebra and geometry has been completed."
Seeing this finishing touch, the look of interest on Lu Zhou's face became more intense.
"Let's go to my office and talk."
"No problem," Perelman shook his head. "My mother and I have already had a phone call. She asked me to do what I want to do without paying too much attention to her. I do have something else I want to do. Done, I plan to stay here... for a while longer, and then go back after solving Hodge's conjecture."
"That's such a shame."
More than interesting.
In fact, he had probably guessed what Chen Yang was planning to do.
"I think your ideas are good... Although I have not studied this topic carefully, my intuition tells me that if I follow this path, I will probably gain something."
He Changwen: "..."
"About when you were halfway through writing... I originally came to see Professor Lu, but I didn't expect to find something unexpected here," after a pause, Perelman continued, "... Can you use your pen for me?"
The moment he saw this line of calculation, Chen Yang's pupils shrank slightly.
Lu Zhou: "...Although I am very interested, there are still a lot of things waiting for me to deal with at IMCRC. I'm afraid I don't have any more time to allocate to mathematics recently."
How to prove that under the condition of cross-section curvature, all the others are zero except the L2 homology group in the middle
Staring at the calculations on the whiteboard, Zimmer's eyes shone and he whispered excitedly: "Is this the legendary -"
According to the properties of the high-dimensional Riemannian manifold M, Gauss curvature can be generalized to cross-sectional curvature, and its value can be determined by the tensor of Riemannian curvature. As for its integrand, it is a very complex algebraic formula composed of curvature tensors—that is, the Gauss-Bonnet integrand.
"According to Attiyah's theorem, if we can prove that under the condition of cross-section curvature, except for the L2 homology group in the middle, the rest are zero... "
Finally, they all looked at Lu Zhou in unison.
As for its integral over the entire manifold, it is determined by the Euler indicator number X(M) of this manifold.
Perelman frowned and asked hesitantly.
Lu Zhou still remembers that he was studying his hyperelliptic curve analysis method on the blackboard, and used a very clever method to improve this mathematical tool originally designed for the quasi-Riemann hypothesis and directly apply it to the analysis of Africa. Research on the algebraic topology of singular complex algebraic varieties and the geometric correlation problems expressed by polynomial equations defining sub-varieties.
"About when you were halfway through writing... I originally came to see Professor Lu, but I didn't expect to find something unexpected here," after a pause, Perelman continued, "... Can you use your pen for me?"
Thinking of this, Lu Zhou nodded approvingly.
Zimmer gave him a strange look.
"That's such a shame."
Seeing that his junior brother only spoke half of what he said, He Changwen frowned and said in a low voice: "What is it? Don't tell me."
Using these properties, Hodge theory can be extended to complete noncompact manifolds.
Obviously, he was only halfway through this line of thinking, and he didn’t have a good idea of how to proceed further.
What surprised Perelman was that Lu Zhou didn't show any strong interest.
"When did you come here?"
These profound mathematical meanings were obtained by Professor Chen Shengshen, which is the mathematical connotation of the famous Gauss–Bonnet–Chen formula.
Lu Zhou still remembers that he was studying his hyperelliptic curve analysis method on the blackboard, and used a very clever method to improve this mathematical tool originally designed for the quasi-Riemann hypothesis and directly apply it to the analysis of Africa. Research on the algebraic topology of singular complex algebraic varieties and the geometric correlation problems expressed by polynomial equations defining sub-varieties.
These profound mathematical meanings were obtained by Professor Chen Shengshen, which is the mathematical connotation of the famous Gauss–Bonnet–Chen formula.
It has been almost a year now, and there is still no progress on the topic of Hodge's conjecture. In addition, Lu Zhou has been busy with the unified theory of algebra and geometry some time ago, so that Lu Zhou almost forgot about it.
Staring at the calculations on the whiteboard, Zimmer's eyes shone and he whispered excitedly: "Is this the legendary -"
After a brief moment of excitement, the two of them fell into silence.
Combined with Sir Attiyah's L2 cohomology method, if we continue along this line of thinking, we might be able to prove this conjecture.
"Let's go to my office and talk."
Perelman nodded and thanked him.
Finally, they all looked at Lu Zhou in unison.
Finally, they all looked at Lu Zhou in unison.
"Let's go to my office and talk."
Using these properties, Hodge theory can be extended to complete noncompact manifolds.
"Since there is a ready-made unified theory of algebraic geometry that can be used, I will omit the proof of equation (3)."
After a pause, he continued.
Of course, how to prove it specifically requires in-depth study.
More than interesting.
Seeing that his junior brother only spoke half of what he said, He Changwen frowned and said in a low voice: "What is it? Don't tell me."
"No problem," Perelman shook his head. "My mother and I have already had a phone call. She asked me to do what I want to do without paying too much attention to her. I do have something else I want to do. Done, I plan to stay here... for a while longer, and then go back after solving Hodge's conjecture."
Thinking of this, He Changwen couldn't help but comfort himself in his heart.
Probably at the beginning of the year, before Lu Zhou hired Chen Yang from the Mathematics Center of Yanda, Professor Chen was already studying the Hodge conjecture.
Without any hesitation, Chen Yang decisively gave up the marker in his hand.
Thinking of this, Lu Zhou nodded approvingly.
Then here comes the problem.
How is this so obvious? !
The conversation suddenly stopped here.
Zimmer gave him a strange look.
It has been almost a year now, and there is still no progress on the topic of Hodge's conjecture. In addition, Lu Zhou has been busy with the unified theory of algebra and geometry some time ago, so that Lu Zhou almost forgot about it.
Noticing the two people looking at him, Lu Zhou, who had not said a word from beginning to end, suddenly blinked and said with a smile.
After Lu Zhou cleared his throat, he looked at Perelman and continued.
Obviously, he was only halfway through this line of thinking, and he didn’t have a good idea of how to proceed further.
Seconds.
But upon closer inspection, it seems that is indeed the case.
Finally, they all looked at Lu Zhou in unison.
After taking Chen Yang to his office, Lu Zhou went to the corner to help him drag a whiteboard and handed his marker to his hand.
Perelman nodded and thanked him.
"Speaking of which, is it okay for you to stay here? The unified theory of algebra and geometry has been completed."
It's really wonderful.
Hodge's conjecture can be said to be a concentrated expression of the abstract features in the development of modern mathematics. It studies the intrinsic relationship between the three major branches of mathematics - analysis, topology, and algebraic geometry.
"Since there is a ready-made unified theory of algebraic geometry that can be used, I will omit the proof of equation (3)."
At some point, a circle of people stood behind Chen Yang.
Finally, they all looked at Lu Zhou in unison.
Lu Zhou still remembers that he was studying his hyperelliptic curve analysis method on the blackboard, and used a very clever method to improve this mathematical tool originally designed for the quasi-Riemann hypothesis and directly apply it to the analysis of Africa. Research on the algebraic topology of singular complex algebraic varieties and the geometric correlation problems expressed by polynomial equations defining sub-varieties.
"This idea seems interesting."
As early as when he started writing on the blackboard, people in the office noticed this.
Using these properties, Hodge theory can be extended to complete noncompact manifolds.
"According to Attiyah's theorem, if we can prove that under the condition of cross-section curvature, except for the L2 homology group in the middle, the rest are zero... "
Although he was surprised that Professor Perelman would choose to stay, Lu Zhou would naturally not refuse such a good thing and immediately said with a smile.
Looking at the calculations on the whiteboard, Lu Zhou raised his eyebrows slightly and said with interest.
Staring at the calculations on the whiteboard, Zimmer's eyes shone and he whispered excitedly: "Is this the legendary -"
It felt like Lu Zhou saw something, but he didn't explain it clearly.
While writing, Chen Yang continued.
"...We can get the proof of Hodge's conjecture!"
Obviously, he was only halfway through this line of thinking, and he didn’t have a good idea of how to proceed further.
Seeing that his junior brother only spoke half of what he said, He Changwen frowned and said in a low voice: "What is it? Don't tell me."
Lu Zhou: "...Although I am very interested, there are still a lot of things waiting for me to deal with at IMCRC. I'm afraid I don't have any more time to allocate to mathematics recently."
Thinking of this, He Changwen couldn't help but comfort himself in his heart.
The pen in his hand stopped, Chen Yang nodded and said.
Zimmer gave him a strange look.
Thinking of this, Lu Zhou nodded approvingly.
"Although I'm afraid I can't spare the time to help, I solemnly recommend Professor Chen to you," Lu Zhou patted Chen Yang on the shoulder and said with a smile, "He is an excellent scholar. Regarding his ability, I believe You also understand, so I won’t brag too much. In short, if you cooperate, I believe we will be able to solve this problem. "
Combined with Sir Attiyah's L2 cohomology method, if we continue along this line of thinking, we might be able to prove this conjecture.
"That's such a shame."
In fact, he had probably guessed what Chen Yang was planning to do.
"Hodge's conjecture! It's obvious."
[H^n(M)6≠0｝, and when q≠n, H^q(M)={0}]
It was precisely because of this beautiful operation that Lu Zhou could not help but fall in love with talents and hired him from the Mathematics Center of Yanda to Jinling.
He Changwen: "..."
Staring at the calculations on the whiteboard, Zimmer's eyes shone and he whispered excitedly: "Is this the legendary -"
As early as when he started writing on the blackboard, people in the office noticed this.
"According to Attiyah's theorem, if we can prove that under the condition of cross-section curvature, except for the L2 homology group in the middle, the rest are zero... "
After that, he wrote down the Gauss-Bonnet formula.
Hodge's conjecture can be said to be a concentrated expression of the abstract features in the development of modern mathematics. It studies the intrinsic relationship between the three major branches of mathematics - analysis, topology, and algebraic geometry.
How is this so obvious? !
The conversation suddenly stopped here.
"Although I'm afraid I can't spare the time to help, I solemnly recommend Professor Chen to you," Lu Zhou patted Chen Yang on the shoulder and said with a smile, "He is an excellent scholar. Regarding his ability, I believe You also understand, so I won’t brag too much. In short, if you cooperate, I believe we will be able to solve this problem. "
"...We can get the proof of Hodge's conjecture!"
After a pause, he continued.
While writing, Chen Yang continued.
But upon closer inspection, it seems that is indeed the case.
"That's such a shame."
Both of them are the type who don't talk much and don't communicate much.
Both of them are the type who don't talk much and don't communicate much.
How to prove that under the condition of cross-section curvature, all the others are zero except the L2 homology group in the middle
According to the properties of the high-dimensional Riemannian manifold M, Gauss curvature can be generalized to cross-sectional curvature, and its value can be determined by the tensor of Riemannian curvature. As for its integrand, it is a very complex algebraic formula composed of curvature tensors—that is, the Gauss-Bonnet integrand.
Probably at the beginning of the year, before Lu Zhou hired Chen Yang from the Mathematics Center of Yanda, Professor Chen was already studying the Hodge conjecture.
Thinking of this, Lu Zhou nodded approvingly.
Thinking of this, He Changwen couldn't help but comfort himself in his heart.
"Hodge's conjecture! It's obvious."
Taking the pen from Professor Chen, Perelman stood in front of the whiteboard and pondered for a moment, then blanked a few lines under his calculations and continued writing.
Noticing the two people looking at him, Lu Zhou, who had not said a word from beginning to end, suddenly blinked and said with a smile.
Of course, how to prove it specifically requires in-depth study.
Lu Zhou: "...Although I am very interested, there are still a lot of things waiting for me to deal with at IMCRC. I'm afraid I don't have any more time to allocate to mathematics recently."
According to the properties of the high-dimensional Riemannian manifold M, Gauss curvature can be generalized to cross-sectional curvature, and its value can be determined by the tensor of Riemannian curvature. As for its integrand, it is a very complex algebraic formula composed of curvature tensors—that is, the Gauss-Bonnet integrand.
Well, if you look carefully, he should be able to see it.
As for its integral over the entire manifold, it is determined by the Euler indicator number X(M) of this manifold.
How to prove that under the condition of cross-section curvature, all the others are zero except the L2 homology group in the middle
Noticing the two people looking at him, Lu Zhou, who had not said a word from beginning to end, suddenly blinked and said with a smile.
Lu Zhou still remembers that he was studying his hyperelliptic curve analysis method on the blackboard, and used a very clever method to improve this mathematical tool originally designed for the quasi-Riemann hypothesis and directly apply it to the analysis of Africa. Research on the algebraic topology of singular complex algebraic varieties and the geometric correlation problems expressed by polynomial equations defining sub-varieties.
It has been almost a year now, and there is still no progress on the topic of Hodge's conjecture. In addition, Lu Zhou has been busy with the unified theory of algebra and geometry some time ago, so that Lu Zhou almost forgot about it.
The pen on the whiteboard stopped, and Chen Yang fell into deep thought.
Seeing that his junior brother only spoke half of what he said, He Changwen frowned and said in a low voice: "What is it? Don't tell me."
Thinking of this, He Changwen couldn't help but comfort himself in his heart.
At some point, a circle of people stood behind Chen Yang.
These profound mathematical meanings were obtained by Professor Chen Shengshen, which is the mathematical connotation of the famous Gauss–Bonnet–Chen formula.
In fact, he had probably guessed what Chen Yang was planning to do.
The moment he saw this line of calculation, Chen Yang's pupils shrank slightly.
Obviously, he was only halfway through this line of thinking, and he didn’t have a good idea of how to proceed further.
According to the properties of the high-dimensional Riemannian manifold M, Gauss curvature can be generalized to cross-sectional curvature, and its value can be determined by the tensor of Riemannian curvature. As for its integrand, it is a very complex algebraic formula composed of curvature tensors—that is, the Gauss-Bonnet integrand.
After taking Chen Yang to his office, Lu Zhou went to the corner to help him drag a whiteboard and handed his marker to his hand.
It felt like Lu Zhou saw something, but he didn't explain it clearly.
"This idea seems interesting."
Professor Perelman, who had been standing silently next to him when he arrived at the office, suddenly spoke.
"...We can get the proof of Hodge's conjecture!"
It was precisely because of this beautiful operation that Lu Zhou could not help but fall in love with talents and hired him from the Mathematics Center of Yanda to Jinling.
"As we all know, a surface does not necessarily have only one metric, so I tried to change the metric of S. After changing the metric, the corresponding Gauss curvature K will also change, but the integral value is different from that of the surface. The measurement has nothing to do with it, but is only related to the Euler characteristic number X(S) of the surface. Using this property, we can-"
After Lu Zhou cleared his throat, he looked at Perelman and continued.
The pen in his hand stopped, Chen Yang nodded and said.
Seeing that his junior brother only spoke half of what he said, He Changwen frowned and said in a low voice: "What is it? Don't tell me."
Lu Zhou: "...Although I am very interested, there are still a lot of things waiting for me to deal with at IMCRC. I'm afraid I don't have any more time to allocate to mathematics recently."
Seeing this finishing touch, the look of interest on Lu Zhou's face became more intense.
"This idea seems interesting."
Zimmer gave him a strange look.
It was precisely because of this beautiful operation that Lu Zhou could not help but fall in love with talents and hired him from the Mathematics Center of Yanda to Jinling.
What surprised Perelman was that Lu Zhou didn't show any strong interest.
Seconds.
How is this so obvious? !
"Gauss-Bonnet formula?"
"Exactly."
Looking back at Professor Perelman, Chen Yang was slightly stunned and said with some surprise.
Taking the pen from Professor Chen, Perelman stood in front of the whiteboard and pondered for a moment, then blanked a few lines under his calculations and continued writing.
Lu Zhou: "...Although I am very interested, there are still a lot of things waiting for me to deal with at IMCRC. I'm afraid I don't have any more time to allocate to mathematics recently."
"When did you come here?"
"Let's go to my office and talk."
"... My suggestion is that for the subsequent proofs, we can lift the problem of compact manifold M to its universal covering manifold to obtain a complete non-compact manifold M."
Seconds.
It has been almost a year now, and there is still no progress on the topic of Hodge's conjecture. In addition, Lu Zhou has been busy with the unified theory of algebra and geometry some time ago, so that Lu Zhou almost forgot about it.
As he spoke, the pen in his hand shook slightly, and he quickly wrote a concise and beautiful line of calculations.
More than interesting.
Although he expressed doubts about this absolute statement, after taking a look at Professor Chen, Perelman didn't say anything. He just nodded, seemingly recognizing his partner.
Zimmer gave him a strange look.
"About when you were halfway through writing... I originally came to see Professor Lu, but I didn't expect to find something unexpected here," after a pause, Perelman continued, "... Can you use your pen for me?"
Seeing this finishing touch, the look of interest on Lu Zhou's face became more intense.
A hint of understanding suddenly appeared on his face, and he said with suppressed excitement.
Without any hesitation, Chen Yang decisively gave up the marker in his hand.
"Gauss-Bonnet formula?"
"I think your ideas are good... Although I have not studied this topic carefully, my intuition tells me that if I follow this path, I will probably gain something."
Taking the pen from Professor Chen, Perelman stood in front of the whiteboard and pondered for a moment, then blanked a few lines under his calculations and continued writing.
Finally, they all looked at Lu Zhou in unison.
How is this so obvious? !
A hint of understanding suddenly appeared on his face, and he said with suppressed excitement.
Combined with Sir Attiyah's L2 cohomology method, if we continue along this line of thinking, we might be able to prove this conjecture.
Finally, they all looked at Lu Zhou in unison.
After taking Chen Yang to his office, Lu Zhou went to the corner to help him drag a whiteboard and handed his marker to his hand.
Seeing that his junior brother only spoke half of what he said, He Changwen frowned and said in a low voice: "What is it? Don't tell me."
Taking the pen from Professor Chen, Perelman stood in front of the whiteboard and pondered for a moment, then blanked a few lines under his calculations and continued writing.
"Since there is a ready-made unified theory of algebraic geometry that can be used, I will omit the proof of equation (3)."
Without wasting time on politeness, after taking over the pen, Chen Yang stood in front of the whiteboard and thought for a moment. He first drew a circle on the whiteboard, then marked S next to it, and wrote a line of expressions.
Without wasting time on politeness, after taking over the pen, Chen Yang stood in front of the whiteboard and thought for a moment. He first drew a circle on the whiteboard, then marked S next to it, and wrote a line of expressions.
How to prove that under the condition of cross-section curvature, all the others are zero except the L2 homology group in the middle
"Speaking of which, is it okay for you to stay here? The unified theory of algebra and geometry has been completed."
"... My suggestion is that for the subsequent proofs, we can lift the problem of compact manifold M to its universal covering manifold to obtain a complete non-compact manifold M."
As early as when he started writing on the blackboard, people in the office noticed this.
"No problem," Perelman shook his head. "My mother and I have already had a phone call. She asked me to do what I want to do without paying too much attention to her. I do have something else I want to do. Done, I plan to stay here... for a while longer, and then go back after solving Hodge's conjecture."
A hint of understanding suddenly appeared on his face, and he said with suppressed excitement.
Using these properties, Hodge theory can be extended to complete noncompact manifolds.
"Gauss-Bonnet formula?"
The moment he saw this line of calculation, Chen Yang's pupils shrank slightly.
"According to Attiyah's theorem, if we can prove that under the condition of cross-section curvature, except for the L2 homology group in the middle, the rest are zero... "
After Lu Zhou cleared his throat, he looked at Perelman and continued.
Seconds.
As he spoke, the pen in his hand shook slightly, and he quickly wrote a concise and beautiful line of calculations.
[H^n(M)6≠0｝, and when q≠n, H^q(M)={0}]
At some point, a circle of people stood behind Chen Yang.
Upon hearing the news, Perelman had a look of regret on his face.
At some point, a circle of people stood behind Chen Yang.
It's really wonderful.
Well, if you look carefully, he should be able to see it.
"As we all know, a surface does not necessarily have only one metric, so I tried to change the metric of S. After changing the metric, the corresponding Gauss curvature K will also change, but the integral value is different from that of the surface. The measurement has nothing to do with it, but is only related to the Euler characteristic number X(S) of the surface. Using this property, we can-"
[H^n(M)6≠0｝, and when q≠n, H^q(M)={0}]
Thinking of this, He Changwen couldn't help but comfort himself in his heart.
Combined with Sir Attiyah's L2 cohomology method, if we continue along this line of thinking, we might be able to prove this conjecture.
"Hodge's conjecture! It's obvious."
He Changwen: "..."
The moment he saw this line of calculation, Chen Yang's pupils shrank slightly.
"No problem," Perelman shook his head. "My mother and I have already had a phone call. She asked me to do what I want to do without paying too much attention to her. I do have something else I want to do. Done, I plan to stay here... for a while longer, and then go back after solving Hodge's conjecture."
"About when you were halfway through writing... I originally came to see Professor Lu, but I didn't expect to find something unexpected here," after a pause, Perelman continued, "... Can you use your pen for me?"
Seconds.
"...We can get the proof of Hodge's conjecture!"
After a brief moment of excitement, the two of them fell into silence.
It felt like Lu Zhou saw something, but he didn't explain it clearly.
"Since there is a ready-made unified theory of algebraic geometry that can be used, I will omit the proof of equation (3)."
What surprised Perelman was that Lu Zhou didn't show any strong interest.
A hint of understanding suddenly appeared on his face, and he said with suppressed excitement.
"Hodge's conjecture! It's obvious."
Hodge's conjecture can be said to be a concentrated expression of the abstract features in the development of modern mathematics. It studies the intrinsic relationship between the three major branches of mathematics - analysis, topology, and algebraic geometry.
As for the difficulty, as a millennium problem, there is no doubt about it.
As for its integral over the entire manifold, it is determined by the Euler indicator number X(M) of this manifold.
Noticing the two people looking at him, Lu Zhou, who had not said a word from beginning to end, suddenly blinked and said with a smile.
Noticing the two people looking at him, Lu Zhou, who had not said a word from beginning to end, suddenly blinked and said with a smile.
It felt like Lu Zhou saw something, but he didn't explain it clearly.
Of course, how to prove it specifically requires in-depth study.
"...We can get the proof of Hodge's conjecture!"
It's really wonderful.
Staring at the calculations on the whiteboard, Zimmer's eyes shone and he whispered excitedly: "Is this the legendary -"
A hint of understanding suddenly appeared on his face, and he said with suppressed excitement.
But upon closer inspection, it seems that is indeed the case.
"Let's go to my office and talk."
Then here comes the problem.
Finally, they all looked at Lu Zhou in unison.
Upon hearing the news, Perelman had a look of regret on his face.
Then here comes the problem.
Noticing the two people looking at him, Lu Zhou, who had not said a word from beginning to end, suddenly blinked and said with a smile.
"Since there is a ready-made unified theory of algebraic geometry that can be used, I will omit the proof of equation (3)."
[H^n(M)6≠0｝, and when q≠n, H^q(M)={0}]
Professor Perelman, who had been standing silently next to him when he arrived at the office, suddenly spoke.
Staring at the calculations on the whiteboard, Zimmer's eyes shone and he whispered excitedly: "Is this the legendary -"
How to prove that under the condition of cross-section curvature, all the others are zero except the L2 homology group in the middle
Seeing that his junior brother only spoke half of what he said, He Changwen frowned and said in a low voice: "What is it? Don't tell me."
After a brief moment of excitement, the two of them fell into silence.
Noticing the two people looking at him, Lu Zhou, who had not said a word from beginning to end, suddenly blinked and said with a smile.
Then here comes the problem.
As for the difficulty, as a millennium problem, there is no doubt about it.
At some point, a circle of people stood behind Chen Yang.
But upon closer inspection, it seems that is indeed the case.
As for its integral over the entire manifold, it is determined by the Euler indicator number X(M) of this manifold.
The conversation suddenly stopped here.
As he spoke, the pen in his hand shook slightly, and he quickly wrote a concise and beautiful line of calculations.
"... For a compact and edgeless surface S, its Gauss curvature K can be integrated over the entire surface."
Without any hesitation, Chen Yang decisively gave up the marker in his hand.
After a brief moment of excitement, the two of them fell into silence.
"Gauss-Bonnet formula?"
"When did you come here?"
"According to Attiyah's theorem, if we can prove that under the condition of cross-section curvature, except for the L2 homology group in the middle, the rest are zero... "
"... My suggestion is that for the subsequent proofs, we can lift the problem of compact manifold M to its universal covering manifold to obtain a complete non-compact manifold M."
"That's such a shame."
How is this so obvious? !
Finally, they all looked at Lu Zhou in unison.
As for its integral over the entire manifold, it is determined by the Euler indicator number X(M) of this manifold.
Lu Zhou: "...Although I am very interested, there are still a lot of things waiting for me to deal with at IMCRC. I'm afraid I don't have any more time to allocate to mathematics recently."
Staring at the calculations on the whiteboard, Zimmer's eyes shone and he whispered excitedly: "Is this the legendary -"
The moment he saw this line of calculation, Chen Yang's pupils shrank slightly.
Noticing the two people looking at him, Lu Zhou, who had not said a word from beginning to end, suddenly blinked and said with a smile.
Well, if you look carefully, he should be able to see it.
The moment he saw this line of calculation, Chen Yang's pupils shrank slightly.
[H^n(M)6≠0｝, and when q≠n, H^q(M)={0}]
The pen in his hand stopped, Chen Yang nodded and said.
"Hodge's conjecture! It's obvious."
Zimmer gave him a strange look.
After that, he wrote down the Gauss-Bonnet formula.
"I think your ideas are good... Although I have not studied this topic carefully, my intuition tells me that if I follow this path, I will probably gain something."
Although he was surprised that Professor Perelman would choose to stay, Lu Zhou would naturally not refuse such a good thing and immediately said with a smile.
Finally, they all looked at Lu Zhou in unison.
But upon closer inspection, it seems that is indeed the case.
"sorry to bother you."
"When did you come here?"
After a pause, he continued.
"About when you were halfway through writing... I originally came to see Professor Lu, but I didn't expect to find something unexpected here," after a pause, Perelman continued, "... Can you use your pen for me?"
It was precisely because of this beautiful operation that Lu Zhou could not help but fall in love with talents and hired him from the Mathematics Center of Yanda to Jinling.
"Exactly."
"About when you were halfway through writing... I originally came to see Professor Lu, but I didn't expect to find something unexpected here," after a pause, Perelman continued, "... Can you use your pen for me?"
"This idea is very interesting. My suggestion is that you might as well study this topic together."
"About when you were halfway through writing... I originally came to see Professor Lu, but I didn't expect to find something unexpected here," after a pause, Perelman continued, "... Can you use your pen for me?"
Professor Perelman, who had been standing silently next to him when he arrived at the office, suddenly spoke.
As he spoke, the pen in his hand shook slightly, and he quickly wrote a concise and beautiful line of calculations.
It has been almost a year now, and there is still no progress on the topic of Hodge's conjecture. In addition, Lu Zhou has been busy with the unified theory of algebra and geometry some time ago, so that Lu Zhou almost forgot about it.
How to prove that under the condition of cross-section curvature, all the others are zero except the L2 homology group in the middle
It felt like Lu Zhou saw something, but he didn't explain it clearly.
Of course, how to prove it specifically requires in-depth study.
Upon hearing the news, Perelman had a look of regret on his face.
Obviously, he was only halfway through this line of thinking, and he didn’t have a good idea of how to proceed further.
What surprised Perelman was that Lu Zhou didn't show any strong interest.
Lu Zhou still remembers that he was studying his hyperelliptic curve analysis method on the blackboard, and used a very clever method to improve this mathematical tool originally designed for the quasi-Riemann hypothesis and directly apply it to the analysis of Africa. Research on the algebraic topology of singular complex algebraic varieties and the geometric correlation problems expressed by polynomial equations defining sub-varieties.
"Although I'm afraid I can't spare the time to help, I solemnly recommend Professor Chen to you," Lu Zhou patted Chen Yang on the shoulder and said with a smile, "He is an excellent scholar. Regarding his ability, I believe You also understand, so I won’t brag too much. In short, if you cooperate, I believe we will be able to solve this problem. "
"I think your ideas are good... Although I have not studied this topic carefully, my intuition tells me that if I follow this path, I will probably gain something."
In fact, he had probably guessed what Chen Yang was planning to do.
Perelman frowned and asked hesitantly.
"sorry to bother you."
After a pause, he continued.
"Aren't you going to participate? This is an interesting problem."
Perelman frowned and asked hesitantly.
It was precisely because of this beautiful operation that Lu Zhou could not help but fall in love with talents and hired him from the Mathematics Center of Yanda to Jinling.
More than interesting.
He Changwen: "..."
He Changwen: "..."
More than interesting.
Hodge's conjecture can be said to be a concentrated expression of the abstract features in the development of modern mathematics. It studies the intrinsic relationship between the three major branches of mathematics - analysis, topology, and algebraic geometry.
Without wasting time on politeness, after taking over the pen, Chen Yang stood in front of the whiteboard and thought for a moment. He first drew a circle on the whiteboard, then marked S next to it, and wrote a line of expressions.
Probably at the beginning of the year, before Lu Zhou hired Chen Yang from the Mathematics Center of Yanda, Professor Chen was already studying the Hodge conjecture.
Professor Perelman, who had been standing silently next to him when he arrived at the office, suddenly spoke.
Thinking of this, Lu Zhou nodded approvingly.
Although he expressed doubts about this absolute statement, after taking a look at Professor Chen, Perelman didn't say anything. He just nodded, seemingly recognizing his partner.
Hodge's conjecture can be said to be a concentrated expression of the abstract features in the development of modern mathematics. It studies the intrinsic relationship between the three major branches of mathematics - analysis, topology, and algebraic geometry.
As for its integral over the entire manifold, it is determined by the Euler indicator number X(M) of this manifold.
Thinking of this, He Changwen couldn't help but comfort himself in his heart.
As for the difficulty, as a millennium problem, there is no doubt about it.
The pen in his hand stopped, Chen Yang nodded and said.
"Since there is a ready-made unified theory of algebraic geometry that can be used, I will omit the proof of equation (3)."
As early as when he started writing on the blackboard, people in the office noticed this.
Upon hearing the news, Perelman had a look of regret on his face.
Seeing this finishing touch, the look of interest on Lu Zhou's face became more intense.
Although he expressed doubts about this absolute statement, after taking a look at Professor Chen, Perelman didn't say anything. He just nodded, seemingly recognizing his partner.
"This idea is very interesting. My suggestion is that you might as well study this topic together."
Hodge's conjecture can be said to be a concentrated expression of the abstract features in the development of modern mathematics. It studies the intrinsic relationship between the three major branches of mathematics - analysis, topology, and algebraic geometry.
What surprised Perelman was that Lu Zhou didn't show any strong interest.
How is this so obvious? !
"When did you come here?"
Combined with Sir Attiyah's L2 cohomology method, if we continue along this line of thinking, we might be able to prove this conjecture.
As he spoke, the pen in his hand shook slightly, and he quickly wrote a concise and beautiful line of calculations.
Lu Zhou: "...Although I am very interested, there are still a lot of things waiting for me to deal with at IMCRC. I'm afraid I don't have any more time to allocate to mathematics recently."
It was precisely because of this beautiful operation that Lu Zhou could not help but fall in love with talents and hired him from the Mathematics Center of Yanda to Jinling.
"Exactly."
Seeing this finishing touch, the look of interest on Lu Zhou's face became more intense.
"Speaking of which, is it okay for you to stay here? The unified theory of algebra and geometry has been completed."
"This idea seems interesting."
Upon hearing the news, Perelman had a look of regret on his face.
Finally, they all looked at Lu Zhou in unison.
Upon hearing the news, Perelman had a look of regret on his face.
Without any hesitation, Chen Yang decisively gave up the marker in his hand.
Without wasting time on politeness, after taking over the pen, Chen Yang stood in front of the whiteboard and thought for a moment. He first drew a circle on the whiteboard, then marked S next to it, and wrote a line of expressions.
"That's such a shame."
Then here comes the problem.
Although he expressed doubts about this absolute statement, after taking a look at Professor Chen, Perelman didn't say anything. He just nodded, seemingly recognizing his partner.
It was precisely because of this beautiful operation that Lu Zhou could not help but fall in love with talents and hired him from the Mathematics Center of Yanda to Jinling.
"No problem," Perelman shook his head. "My mother and I have already had a phone call. She asked me to do what I want to do without paying too much attention to her. I do have something else I want to do. Done, I plan to stay here... for a while longer, and then go back after solving Hodge's conjecture."
A hint of understanding suddenly appeared on his face, and he said with suppressed excitement.
While writing, Chen Yang continued.
Looking back at Professor Perelman, Chen Yang was slightly stunned and said with some surprise.
"Although I'm afraid I can't spare the time to help, I solemnly recommend Professor Chen to you," Lu Zhou patted Chen Yang on the shoulder and said with a smile, "He is an excellent scholar. Regarding his ability, I believe You also understand, so I won’t brag too much. In short, if you cooperate, I believe we will be able to solve this problem. "
After a pause, he continued.
Taking the pen from Professor Chen, Perelman stood in front of the whiteboard and pondered for a moment, then blanked a few lines under his calculations and continued writing.
Perelman frowned and asked hesitantly.
While writing, Chen Yang continued.
Of course, how to prove it specifically requires in-depth study.
As he spoke, the pen in his hand shook slightly, and he quickly wrote a concise and beautiful line of calculations.
Although he expressed doubts about this absolute statement, after taking a look at Professor Chen, Perelman didn't say anything. He just nodded, seemingly recognizing his partner.
Using these properties, Hodge theory can be extended to complete noncompact manifolds.
Although he expressed doubts about this absolute statement, after taking a look at Professor Chen, Perelman didn't say anything. He just nodded, seemingly recognizing his partner.
It's really wonderful.
"No problem," Perelman shook his head. "My mother and I have already had a phone call. She asked me to do what I want to do without paying too much attention to her. I do have something else I want to do. Done, I plan to stay here... for a while longer, and then go back after solving Hodge's conjecture."
"No problem," Perelman shook his head. "My mother and I have already had a phone call. She asked me to do what I want to do without paying too much attention to her. I do have something else I want to do. Done, I plan to stay here... for a while longer, and then go back after solving Hodge's conjecture."
Professor Perelman, who had been standing silently next to him when he arrived at the office, suddenly spoke.
Staring at the calculations on the whiteboard, Zimmer's eyes shone and he whispered excitedly: "Is this the legendary -"
Both of them are the type who don't talk much and don't communicate much.
But upon closer inspection, it seems that is indeed the case.
"Let's go to my office and talk."
After Lu Zhou cleared his throat, he looked at Perelman and continued.
Staring at the calculations on the whiteboard, Zimmer's eyes shone and he whispered excitedly: "Is this the legendary -"
"About when you were halfway through writing... I originally came to see Professor Lu, but I didn't expect to find something unexpected here," after a pause, Perelman continued, "... Can you use your pen for me?"
"That's such a shame."
"Speaking of which, is it okay for you to stay here? The unified theory of algebra and geometry has been completed."
Both of them are the type who don't talk much and don't communicate much.
Well, if you look carefully, he should be able to see it.
"No problem," Perelman shook his head. "My mother and I have already had a phone call. She asked me to do what I want to do without paying too much attention to her. I do have something else I want to do. Done, I plan to stay here... for a while longer, and then go back after solving Hodge's conjecture."
As for its integral over the entire manifold, it is determined by the Euler indicator number X(M) of this manifold.
Looking at the calculations on the whiteboard, Lu Zhou raised his eyebrows slightly and said with interest.
Well, if you look carefully, he should be able to see it.
As for the difficulty, as a millennium problem, there is no doubt about it.
Obviously, he was only halfway through this line of thinking, and he didn’t have a good idea of how to proceed further.
Probably at the beginning of the year, before Lu Zhou hired Chen Yang from the Mathematics Center of Yanda, Professor Chen was already studying the Hodge conjecture.
Although he was surprised that Professor Perelman would choose to stay, Lu Zhou would naturally not refuse such a good thing and immediately said with a smile.
Thinking of this, He Changwen couldn't help but comfort himself in his heart.
"I think your ideas are good... Although I have not studied this topic carefully, my intuition tells me that if I follow this path, I will probably gain something."
It felt like Lu Zhou saw something, but he didn't explain it clearly.
Seconds.
According to the properties of the high-dimensional Riemannian manifold M, Gauss curvature can be generalized to cross-sectional curvature, and its value can be determined by the tensor of Riemannian curvature. As for its integrand, it is a very complex algebraic formula composed of curvature tensors—that is, the Gauss-Bonnet integrand.
"I think your ideas are good... Although I have not studied this topic carefully, my intuition tells me that if I follow this path, I will probably gain something."
He Changwen: "..."
It felt like Lu Zhou saw something, but he didn't explain it clearly.
"Then you should stay in the original apartment. I will help you apply for an extension of the apartment's use time."
Combined with Sir Attiyah's L2 cohomology method, if we continue along this line of thinking, we might be able to prove this conjecture.
After a pause, he continued.
"...We can get the proof of Hodge's conjecture!"
It was precisely because of this beautiful operation that Lu Zhou could not help but fall in love with talents and hired him from the Mathematics Center of Yanda to Jinling.
"As we all know, a surface does not necessarily have only one metric, so I tried to change the metric of S. After changing the metric, the corresponding Gauss curvature K will also change, but the integral value is different from that of the surface. The measurement has nothing to do with it, but is only related to the Euler characteristic number X(S) of the surface. Using this property, we can-"
"This idea seems interesting."
At some point, a circle of people stood behind Chen Yang.
Perelman nodded and thanked him.
But upon closer inspection, it seems that is indeed the case.
Professor Perelman, who had been standing silently next to him when he arrived at the office, suddenly spoke.
"Gauss-Bonnet formula?"
"sorry to bother you."
Perelman nodded and thanked him.
"... For a compact and edgeless surface S, its Gauss curvature K can be integrated over the entire surface."
What surprised Perelman was that Lu Zhou didn't show any strong interest.
The moment he saw this line of calculation, Chen Yang's pupils shrank slightly.
As early as when he started writing on the blackboard, people in the office noticed this.
Perelman nodded and thanked him.