Volume 1: Student Life Chapter 457: In-depth Research
Without any surprise, Chen Zhou's attention was all drawn to the scanned copy of the manuscript in front of him.
Professor Artin is indeed the man who completed the transition from linear associative algebra to associative rings.
Looking at his manuscripts on abstract algebra research, Chen Zhou could appreciate the power of this man's mathematical thinking.
This is something I have never felt in Professor Artin.
Mathematical thinking and mathematical habits can easily influence a person.
Especially people like Chen Zhou who are good at learning and changing themselves.
Chen Zhou subconsciously learned Professor Ating's mathematical thinking and habits from these scanned manuscripts.
"The reciprocity law discovered by means of the theory of class domains, which applies to more general cases, is Artin's reciprocity law..."
"Given a number field over Q whose Galois group is commutative, the Artin reciprocity law assigns an L-function to any one-dimensional representation of this Galois group, and asserts that these L-functions are equal to some Dirichlet L-functions..."
Chen Zhou watched, learned and thought .
The pen in my hand, along with my thoughts, left lines of text and mathematical symbols on the draft paper.
"Here the Dirichlet L function, which is an analogy of the Riemann zeta function, is expressed by the Dirichlet character..."
"And Artin's reciprocity law consists of the exact connection between these two L-functions..."
"Given a noncommutative Galois group and its high-dimensional representation, we can still define some natural matching L-functions, namely Artin L-functions..."
As his thoughts wandered, Chen Zhou felt more and more that something seemed to be wrong.
According to Professor Artin's thinking on this unresolved problem, it can soon be extended to a major proposition.
Moreover, this is not just an ordinary big proposition. It is something that Chen Zhou has just sorted out and has led the development of mathematics.
This thing is the Langlands Program.
In mathematics, there are only a handful of results that can be called programs.
There are roughly only three: Erlangen Program, Hilbert Program and Langlands Program.
The Erlangen Program and the Hilbert Program are products of the second half of the 19th century to the early 20th century. They have produced important works in the history of mathematics and have influenced related fields of mathematics for a long time.
The Langlands Program has been influencing research in mathematics-related fields since its inception until today.
As for why Chen Zhou felt something was wrong, it was because the Langlands Program was based on Artin's L-function and, after in-depth research, extended his conjecture to the function domain to obtain more complete content.
Moreover, he now has a tendency to conduct research in the direction of the Langlands Program.
But now is not the time to stop.
Chen Zhou himself didn't want to stop there.
"When we find a suitable generalization of the Dirichlet L-function, it will be possible to generalize Artin's reciprocity law..."
"The connection between holomorphic functions, that is, holomorphic automorphic forms, defined in the upper half-complex plane and satisfying certain functional equations and Dirichlet's L-function..."
"The self-modifying cusp representation is a certain type of infinite-dimensional irreducible representation of the general linear group GLn on the Q-Adair ring..."
"If it is generalized to the representation of self-defining cusps..."
The pen in Chen Zhou's hand kept rubbing against the draft paper, leaving lines of text and mathematical symbols.
As the research on this sub-topic continued to deepen, Chen Zhou became more and more confused, and the number of problems that needed to be solved also increased.
At this time, the sky outside the window had already darkened.
After Chen Zhou returned to the dormitory, except for the time he entered the system space, he was immersed in research on the subject.
I have to say that Professor Artin’s manuscripts have a certain magic.
If this thing had come before, Chen Zhou would have just thought it was a bunch of scribbles.
But now, these ghostly drawings are extremely attractive.
"In this case, Langlands' reciprocity conjecture will be derived sooner or later, right?"
Chen Zhou paused for a moment with the pen in his hand.
He glanced at the contents on the draft paper and sorted them out silently in his mind.
After finishing the sorting, Chen Zhou's pen fell on the draft paper again.
Regardless of whether it is the reciprocity conjecture in the Langlands program, this topic must be done.
We can't stop at this step, can we?
To hesitate is to lose.
Time is ticking away.
It was not until after ten o'clock in the evening that Chen Zhou put down his pen and stretched.
He sat at his desk for nearly twelve hours.
It was just after 9 a.m. when we returned to Princeton from Providence.
But at this moment, the sun and the moon have changed, and it is already past ten o'clock in the evening.
After boiling a kettle of hot water, Chen Zhou planned to make some instant noodles.
This instant noodles was bought earlier, and there is only the last bucket left.
It’s a good time to finish them now and go out to restock tomorrow.
Chen Zhou has already decided to adjust his study plan after eating.
Let’s try to solve the sub-topic sent by Professor Atin first.
Then, using this as a fulcrum, or opportunity, we can conduct more in-depth research on algebraic geometry.
Therefore, through the study of algebraic geometry, we can improve the current distribution deconstruction method.
Finally, he solved the Goldbach conjecture that had troubled him for so long but had made little progress.
Of course, the overall learning rhythm here is still intertwined with the glue ball topic in physics.
Chen Zhou has always felt that this kind of cross-disciplinary learning will help improve each subject.
Moreover, it is easier to inspire thinking inspiration in the subject.
"Tsk tsk tsk... Instant noodles are still delicious!"
"The pickled cabbage is so sour and refreshing!"
Chen Zhou finished the instant noodles in the bucket with a slurp.
I opened the window to let in some fresh air and then started to clean up the mess.
In the past, Chen Zhou was reluctant to add ham sausage to his instant noodles, let alone boiled eggs.
But now he is a millionaire after all.
It’s not too much to add a sausage and a boiled egg, right?
This must correspond to your identity and temperament.
Chen Zhou sat at the desk again, with a hint of expectation in his eyes and a more determined look.
A mathematician should perhaps stick to one field and always strive for it.
It's like a professional in the workplace who strives for a career he is familiar with all his life in a field.
Because stepping into other fields always requires taking certain risks and learning more.
But even if you study hard and work diligently, you may still achieve nothing in the end.
This is also the reason why many people only make plans and work hard in areas they are familiar with.
But Chen Zhou is different. In the field of analytic number theory, he is almost reaching the ceiling.
If you want to make a breakthrough, you must enter other fields of mathematics.
Moreover, from the very beginning, Chen Zhou hoped to use knowledge from other fields to enrich his distributed deconstruction method.
What's more, I want to win more math awards and gain more linguistics experience points.
Then we certainly cannot just stay in analytic number theory.
Furthermore, upgrading Mathematics from Lv7 to Lv8 requires 500,000 Natural Science Experience Points.
I still don’t know what it’s like to upgrade from Lv8 to Lv9.
Chen Zhou also has to prepare in advance for the next path in his mathematics building.
Now the most suitable and ideal algebraic geometry has become Chen Zhou's next stop.
"Every Artin L-function from the finite-dimensional representation of the Galois group of a given number field is equivalent to some L-function from the representation of a self-morphic cusp..."
"If we want to establish a one-to-one correspondence, we must consider an appropriate extension of the Galois group, namely the Weil-Deligne group..."
As Chen Zhou immersed himself in the draft paper on the desk again, the dormitory became quiet again.
Apart from the faint smell of pickled cabbage, it also tells us that the owner of this place has just finished eating instant noodles.
All that was left was the sound of the pen tip rubbing against the draft paper, and the occasional sound of the mouse wheel sliding.
The group in the Galois group written by Chen Zhou is a relatively simple algebraic structure with only one operation.
It is a basic structure that can be used to build many other algebraic systems.
And a Galois group is a group that is associated with a certain type of field extension.
This is also an important concept of Galois theory.
As for the domain extension, it comes from polynomials.
The study of field extensions and polynomials through Galois groups is called Galois theory.
This was knowledge that Chen Zhou was not very familiar with.
Because of the content of abstract algebra, he only learned the basics.
Apart from the contents in abstract algebra textbooks and some literature, Chen Zhou did not have any deep understanding.
So this is one of the reasons why Chen Zhou was attracted to this knowledge.
The poorer it is, the more we desire it.
The difference between Chen Zhou and others is that his foundation is so solid.
He has a deep memory for these mathematical terms and algebraic symbols.
It will not be an obstacle to his study and research at all.
You know, even a genius like Schultz has a special cabinet where he keeps documents about mathematical codes, symbols and nouns for reference at any time.
This shows that these basic contents are complicated and not easy to remember.
In fact, people with a relatively low level of mathematical ability feel that the literature of modern mathematicians is like incomprehensible.
The biggest reason is those piles of scribble-like mathematical symbols.
I have no idea what these symbols mean or where they came from.
Not to mention the entire document connected together.
The night gradually deepened.
Chen Zhou still sat upright at his desk.
The pen in his hand is still working on his favorite A4 draft paper.
At least Chen Zhou didn't plan to sleep until the question at hand was solved.
He didn't know exactly what time it would arrive.
"Let ρ:Gal(-Q/F)→GL(m,C) be a finite-dimensional Galois representation, where F is an algebraic number field, then L(s,ρ)=p∏det(1-ρ(Frp)Np^(-s))^(-1)=(n=1→∞)∑λρ(n)/n^s..."
Finally, Chen Zhou turned off the lights and went to bed a little after 2:30 in the morning.
The next morning, the alarm clock woke Chen Zhou up on time.
After reaching out to turn off the alarm clock, Chen Zhou only lay down for a minute longer before getting up, putting on his clothes and getting out of bed.
It's already the end of November and the weather has officially entered the rhythm of winter.
The desire to not get up became more and more serious.
However, good living habits always urge Chen Zhou to maintain his habit.
After a quick wash, Chen Zhou went out and started his morning run.
Even in Providence , it was only the morning after I got drunk that I slept a little longer and didn't go for a morning run.
At other times, Chen Zhou always maintains the habit of morning jogging.
So, in the cold winter, let Chen Zhou warm himself by running.
What Chen Zhou didn't expect was.
Senior sister Nott, who was originally thought to be a hard worker, was actually jogging in the morning.
And she seemed to be waiting for him intentionally.
Chen Zhou couldn't help but shake his head. It seemed that this senior student still hadn't given up.
When Chen Zhou passed by Nott, Nott took the initiative to say: "After finishing today's race, I will be one day closer to the one-year deadline!"
Chen Zhou didn't comment, just smiled at her.
Chen Zhou didn't know what expression the other party would have if he knew that he was studying the topic of "Linear Representation of Artin's L-Function of Galois Group".
Maybe they will be more eager to recruit me into the group?
By the way, if this senior knew that Professor Deligne was also trying to get her to join, I wonder what kind of expression would she have?
When Chen Zhou thought about this, he felt that he was really popular.
Regardless of whether you are a man or a woman, old or young, you all have your own ideas. Oh my~
He did not keep the same running rhythm as Nott, but followed the tacit habit that he and Yang Yiyi had formed over a long period of time.
Chen Zhou maintained his own pace and completed today's morning run.
Chen Zhou had breakfast at a Chinese restaurant.
Two meat buns and a bowl of tofu pudding.
It can be considered a standard mixed breakfast.
After returning to the dormitory, Chen Zhou rested for a while before sitting at the desk again.
Yesterday was about mathematics, so this morning, let’s start with physics.
After obtaining the moments L0 and L1 with practical operational value using the QCD summation rule in the peculiar quantum number glueball.
Chen Zhou began to calculate the mass of the glue ball with strange quantum numbers.
According to Chen Zhou's calculation results, there are two 0-gelballs with masses of 3.81GeV and 4.33GeV respectively.
In addition, with regard to the possible existence of glue balls at these two energies, Chen Zhou also theoretically analyzed their possible production and decay properties.
This result also shows that it is very likely to detect the 0-glueball state in colliders that are currently in operation or being planned.
There are far more facilities in operation or planning here than the SLAC National Accelerator Laboratory in the United States, including China's High Energy Physics Laboratory.
And according to Chen Zhou's understanding, the Belle experimental group of KEK in Japan has begun to prepare to search for the peculiar quantum number glueball of 0--glueball state in the decay of bottom quark-event.
In this regard, Chen Zhou is not sure whether Japanese physicists can find it successfully.
But at least before it is found, every country has a chance.
In a sense, this also means a competition for a Nobel Prize in Physics.
After sorting out all the theoretical knowledge about rubber balls, Chen Zhou began to look for experimental literature.
This is his usual method.
Maybe the sample is biased, and the bias may even be large.
But with a sufficient number of samples, this deviation can be gradually reduced.
Until eventually it can be completely ignored.
In addition, the Standard Model of particle physics has been established for more than 40 years and has undergone a large number of experimental tests.
The correctness of its description of the microscopic world, at least below the TeV energy scale, is beyond doubt.
Therefore, these experimental research papers and documents are absolutely of reference value.
On another level, SLAC National Accelerator Laboratory also has North America's first international network.
The scientific literature database here is sufficient to provide a large number of experimental literature samples.
What's more, Chen Zhou also has a deviation calibration artifact.
Based on past experience, the performance in this regard for wrong questions is simply amazing.
And this is exactly where the collection of wrong questions plays the greatest role.
Under the guidance of the wrong question collection, the sample will be best analyzed!
