Chapter 132: International Congress Of Mathematicians
Who dares to say such inheritance is fake?
Lin Ran publicly mentions in conversations his time in Göttingen, stating he originated from Göttingen, and Seagull publicly admits Lin Ran is his “personally” trained student.
Even if Berlin newspapers, especially Pravda in East Berlin, frequently mock Göttingen for not recognizing talent, and local Göttingen newspapers question him, he has never wavered.
Roots in Göttingen, sooner or later he will return to Göttingen.
Seagull holds this simple idea.
Neither he nor Lin Ran denies it; who dares to say Randolph is not from Göttingen?
Since this is true, the Göttingen Mathematics Master handing over from the first half to the second half of the century is also an ironclad “fact”!
Lin Ran smiled and said: “Thanks to the professor’s guidance; without Göttingen, there would be no Randolph today.”
What is tacit understanding? This is tacit understanding.
Everyone’s unspoken tacit understanding.
The conference is held in the grand hall of the Royal Institute of Technology, with the Swedish national flag and the International Mathematical Union flag hanging above the hall. On site, there are not only mathematicians from the free world but also mathematicians from the Soviet Union camp.
Andrei, whom Lin Ran once met in Geneva, is also in attendance and will give a one-hour academic report at this Mathematician Conference.
Of course, Lin Ran will too, and Lin Ran is going first.
After Lennart Carlson’s opening speech, Lin Ran will give the opening academic report.
“Respected mathematicians, scholars, ladies, gentlemen:
Welcome to beautiful Stockholm for the 1962 International Congress of Mathematicians. I am Lennart Carlson, chairman of the International Mathematical Union, and it is a great honor to address you here and open this grand event in the mathematics community.”
After Lennart Carlson finished speaking, the dark velvet curtain behind quickly turned into several large blackboards.
“Hello everyone, this is my first time reporting in front of so many mathematicians. Some do analysis, some do geometry, some do number theory, and some are working on problems no one knows what they are.
Modern mathematics has developed to this point, with so many directions that even two different problems in the same subfield take mathematicians a long time to understand what the other is saying.
Like a tree growing upward, it keeps growing, becoming more and more lush, but the branches forking out are also increasing.
I once said mathematicians are divided into birds and frogs, but we are all searching for our own apples.
Today here, I hope to talk about some interesting content.
I know many of you are expecting me to talk about the Randolph Program, hoping I will discuss the connection between automorphic forms and Galois representations, and how to verify the complete establishment of the Randolph correspondence in higher dimensions and general cases.
Although you don’t know if I have proved it, you still hope I will share my ideas.
Of course, I very much want to share with you all, but is this not too unfriendly for mathematicians who have not studied the Randolph Program?
Not every mathematician is familiar with harmonic analysis and automorphic forms; not every mathematician is interested in my research direction.
Today, I have the honor to lecture in the conference hall to all attending mathematicians. I think I should return to the essence of mathematics and talk about some basic interesting content.
So, putting aside those complex mathematical theories, let’s go back to the initial, most primitive joy.”
Lin Ran walked to the blackboard; his words undoubtedly ignited interest among the mathematicians present.
Indeed, as Lin Ran said, not everyone can understand what he talks about, and not everyone is interested in the Randolph Program.
Voices rose from the audience below; everyone was very curious about what Lin Ran would talk about, while also discussing what the initial, most primitive joy is.
Döblin, sitting with Seagull, asked: “Professor, what is Randolph going to talk about?”
Seagull shook his head: “I don’t know, but you can think about what your initial joy around mathematics was.”
Döblin hesitated a bit: “Is it the joy of solving problems?”
Before the mathematicians below could discuss a result, Lin Ran’s voice rang out:
“At the beginning, we learned mathematics starting from solving problems in the real world.
For example, one apple plus one apple is how many apples; ten fingers put together, how many after adding or subtracting a few.
The initial mathematics provided guidance for the real world, but gradually it became more and more abstract, so abstract that we can no longer find corresponding real problems in the real world.
It became a pure logical thinking game.
Regardless of whether it has practical significance, I just have to find the answer.
This is good, of course this is good; mathematics represents the limit of human wisdom.
You all present are explorers of the limit of humanity.
But now I still want to talk about problems related to the real world and introduce some new concepts to everyone.
Today’s topic is the Four Color Problem.”
Lin Ran drew an irregular circle behind him, then divided it into four irregular parts, filling the four parts with chalk of different colors.
“The Four Color Problem is whether any plane map can be colored with no more than four colors such that adjacent regions have different colors?” Lin Ran said.
“The theoretical framework of the Four Color Problem is based on graph theory and combinatorial mathematics, which belong to the scope of elementary mathematics. I believe everyone present can understand.
Next, let’s begin.
We regard each region on the map as a vertex in the graph.
If two regions share a common boundary, connect these two vertices with an edge in the graph.
Thus, the map coloring problem is equivalent to coloring the vertices of the graph such that adjacent vertices have different colors, with no more than four colors in total.
That is, proving that any plane graph necessarily contains certain specific subgraph structures that cannot be avoided.
Then, for each unavoidable configuration, prove that if a large graph contains such a configuration, it can be simplified, for example by removing or merging certain vertices or edges, into a smaller graph without affecting the establishment of the Four Color Theorem.
This simplifies the problem.”
Lin Ran continued: “Of course, the Four Color Problem is more than that.
We also need to introduce a graph theory technique called the discharging method. It is a new method I thought of based on Professor Kempe’s chain method and Professor Heawood’s analysis of vertex degrees and face degrees in the process of proving the Five Color Map Theorem.”
After briefly introducing the chain method and the proof of the Five Color Theorem, Lin Ran continued:
“The core idea of the discharging method can be divided into three steps:
The first is initial charge allocation; we assign an initial charge to each vertex or face in the graph.
The charge value is usually related to the degree of the vertex or the degree of the face.”
(Degree refers to the number of edges connected to the vertex; edge count refers to the number of edges on the face boundary)
“For example, a common allocation is to assign charge 6deg(v) to each vertex v, where deg(v) is the degree of the vertex.
The second is discharging rules; design a set of rules allowing charges to transfer between vertices or faces.
If a vertex has a low degree, it can borrow charge from adjacent high-degree vertices; high-degree faces distribute charge to adjacent low-degree faces.”
“Finally, analysis after charge adjustment.
After applying the discharging rules, check the final charge of each vertex or face. Through analyzing the charge distribution, we can prove that certain specific configurations in the graph, such as certain subgraphs or cycles, must exist, or certain properties must hold.”
Lin Ran finally summarized: “Finally, we just need to apply the discharging method to the Four Color Problem.
First, according to Euler’s formula for planar graphs V-E+F=2, where V is the number of vertices, E is the number of edges, F is the number of faces, we can derive that the average face degree must be less than 6.
So we can assign initial charge def(f)-6 to each face f, where def(f) is the degree of the face.
Then discharging rules allow charges to transfer between faces or between vertices and faces.
Through the discharging process, we can prove that certain specific configurations lead to negative charges. These configurations form an unavoidable set, meaning any plane graph contains at least one of them.
Then in the proof of the Four Color Theorem, we only need to find a set containing finitely many configurations via the discharging method, then further verify the reducibility of these configurations, and finally prove the Four Color Theorem.”
After Lin Ran finished speaking, everyone understood it, but like Lin Ran, they felt this work was too tedious.
It is the kind where you can find the method, but this method might take you a lifetime to compute.
“I know everyone will think the method I proposed is nonsense because the computation is too massive; human mathematicians might exhaust their lives without results.
But I want to remind everyone that now we have tools like computers.
I believe with computer assistance, we can solve this problem in a very short time, perhaps one year, perhaps two years, using computers.”
The Four Color Problem was originally solved in 1976 by mathematicians Kenneth Appel and Wolfgang Haken with the aid of an electronic computer for a complete proof.
The method they used was the one Lin Ran mentioned—discharging method.
However, compared to Lin Ran, their reputation is obviously far inferior.
Therefore, after Lin Ran proposed it, no one questioned it; those who had heard of computers pondered how to use computers to solve it, while those who hadn’t inquired what computers were.
To add, the computer Appel and Haken used to solve the Four Color Problem was IBM’s 370-168 released in 1972, totaling 1200 hours.
But that doesn’t mean the current IBM 7090 cannot solve it.
The 128KB memory of the IBM 7090 is insufficient to store all configurations and intermediate results simultaneously, but data can be processed in batches and rely on cassette tapes for storage.
Configuration data and verification results take up a lot of storage space; cassette tapes can be used to store intermediate results, ensuring data integrity during computation.
“I hope that at the Mathematician Conference four years from now, we can hear the good news that the Four Color Problem has been solved.” Lin Ran finally summarized.
Lin Ran’s academic report was like heavenly music to mathematicians familiar with computers, as if parting the fog to see the result directly.
The more they understood computers, the more they wanted to rush back to their labs or schools to prove the Four Color Problem.
No need to think of the method themselves; Lin Ran had written it very clearly.
They even didn’t want to attend the subsequent Mathematician Conference.
Whoever produces the result first proves the Four Color Problem that has troubled mathematicians for over a century.
This is Lin Ran giving out benefits.
For mathematicians unfamiliar with the Four Color Problem, this is nowhere near basic; not basic at all.
Döblin could understand what Lin Ran was saying; he was dumbfounded. Before Lin Ran returned to his seat, he turned to Seagull and said: “Professor, didn’t you remind Randolph that after completing his work, he should publish it himself? Doesn’t the Mathematician Conference require sharing one’s own ideas?
And even if sharing one’s ideas, shouldn’t it be saying one’s thinking is not that meticulous, there might be issues, some interesting parts still need improvement, and ask everyone to help think if it can be perfected?
Instead of having already thought of the solution method, contributing the solution method, letting others directly solve the problem?
This is the Four Color Problem!”
The Four Color Problem is a very easy-to-understand problem that even laypeople can grasp; such problems with both topicality and value are too rare.
Solve one and there is one less.
Moreover, the Four Color Problem has the precipitation of time, over a hundred years from now.
For such a problem, with a mature solution idea, he doesn’t use it himself; even if not using it himself, he could leave it to students or provide it to other collaborators at his school, yet Lin Ran made it public.
What is a master’s demeanor? This is a master’s demeanor, the young mathematicians present thought.
Döblin, such a head of mathematics department, felt his heart bleeding; such a solution idea was given away for free.
After all, in terms of computer use, Göttingen definitely cannot compare to universities in America.
Like New York University and Columbia University, they have cooperative laboratories with IBM on campus; what do they have to compete with?
Seagull said: “Randolph is thinking about the entire mathematics community, not just Göttingen.
Open your vision; Randolph helping mathematicians use computers to prove the Four Color Problem also credits Göttingen!”
Right, Seagull has become increasingly adept at handling Döblin’s complaints.
Accuse me uselessly; as long as Randolph is my student, the more successful he is, the more honor for Göttingen.
As long as Döblin cannot break this logic, Seagull is invincible.
“Stockholm, August 1962—At the 14th International Congress of Mathematicians (ICM) held in Stockholm, Sweden, as the most watched mathematician of this conference, the Chinese descent genius mathematician Randolph Lin did not disappoint the attendees.
Invited to give an academic report, Randolph announced his solution to the famous difficult problem in mathematics history—the Four Color Problem. This breakthrough achievement not only opened a new chapter in graph theory but also won widespread acclaim from the mathematicians present.
The Four Color Problem originated in 1852, conjecturing that any plane map needs only four colors for coloring such that adjacent regions have different colors. This conjecture troubled the mathematics community for over a century. Despite many previous attempts, no convincing rigorous proof had been given. The new method Randolph proposed at this conference detailed his profound insight using innovative mathematical methods combined with graph theory and computers. His proof shocked the attending mathematicians with its simplicity and innovation.
Conference chairman and renowned mathematician Lennart Carlson commented: Randolph’s work embodies the perfect combination of mathematics’ creativity and rigorousness, embodying that mathematics must keep pace with the times and combine with new tools.”
Lin Ran’s academic report at the opening ceremony quickly spread globally via newspapers.
Unlike the widespread doubt in the mathematics community after Appel and Haken announced solving the Four Color Problem—because it used a computer proof not accepted by mathematicians, not compiled and published until over a decade later—Lin Ran’s proposal quickly received unanimous approval, with everyone feeling his method could indeed solve the Four Color Problem.
This is the different reaction in the mathematics community to solutions proposed by a master versus non-masters for the same problem.
Just like the ABC conjecture; when Shinichi Mochizuki says he proved it, the mathematics community, unable to understand his paper, says it is in doubt without directly denying him. If an obscure mathematician claims to have proved it and pulls out a huge incomprehensible mess, academia will directly reject the manuscript.
Fame and obscurity are completely different matters.
The mathematics community is that realistic.
August 22, 1962, the last day of the 14th International Congress of Mathematicians, the award ceremony was grandly held in Stockholm Concert Hall.
This building, famous for its blue exterior walls and elegant design, welcomed hundreds of mathematicians, scholars, and guests from around the world that day to witness the conferral of the highest honor in the mathematics community.
At exactly 3 p.m., the ceremony opened amid melodious orchestral music. The orchestra played excerpts from Swedish composer Hugo Alfvén’s “First Symphony,” with solemn and powerful melodies adding ceremony to the upcoming awards.
After the music faded, conference chairman Lennart Carlson slowly walked to the podium. Dressed in a black tuxedo, he smiled and waved to the audience below.
Lennart Carlson addressed in a low and clear voice: “Ladies, gentlemen, welcome to the 1962 International Congress of Mathematicians award ceremony. Today, we not only celebrate mathematics’ brilliant achievements but also witness the peak moment of human wisdom.”
His opening remarks ignited enthusiasm throughout the hall, with warm applause resounding from the audience.
Everyone was giving Lin Ran face, out of respect for wisdom.
After a brief review of the conference’s academic highlights, the award segment officially began. Lennart Carlson announced that this year’s Fields Medal would be awarded to a “young scholar who has changed the face of mathematics with extraordinary talent and innovative spirit.”
After his words, the same name flashed in everyone’s mind:
“Randolph Lin”
As Lin Ran’s name was called, thunderous applause erupted throughout the hall.
A slim, composed young man stood up. He wore a dark gray suit with a slightly loose tie, exuding a touch of casualness.
“Professor, congratulations.” Jenny, sitting next to Lin Ran, promptly offered a cheek kiss.
Lin Ran walked slowly to the stage, each step accompanied by the audience’s gaze.
After Lin Ran reached the podium, Lennart Carlson shook his hand and handed him the gold medal. The medal was engraved with “ICM 1962” on the front and Euclid’s portrait on the back, symbolizing the eternal inheritance of mathematics. Subsequently, a gold-embossed certificate was given to him, written in Latin and English: “Awarded to Randolph Lin in recognition of his outstanding contributions to Fermat’s Theorem.”
