Technology invades the modern world

Chapter 318: Throwing bricks to attract others
It was so touching.

While the media was criticizing McNamara as the worst Secretary of Defense, Lin Ran stepped forward and publicly declared that he had done the best job.

Such a stark contrast.

His claim that the Vietnam War should not be attributed to any specific person further helped him shift the blame.

Moreover, if it's just mindless boasting, only expressing opinions without providing evidence, even if Lin Ran said it, it wouldn't be enough to move McNamara too much.

The problem is that Lin Ran not only praised McNamara, but also explained the reasons very clearly, pointing out exactly where McNamara did a good job compared to his predecessors as Secretary of Defense.

These are precisely the places McNamara is proud of, and his praise is spot on.

Having a slight understanding of Chinese culture, McNamara recalled Chinese proverbs such as "High mountains and flowing water meet a kindred spirit" and "A thoroughbred horse meets a discerning eye."

I am overwhelmed with gratitude, words cannot express it.

At this moment, Lin Ran was thinking about something else.

You increased the Vietnam War from 1.6 advisors to 53.5 combat troops. If someone else had taken over, and instead of launching a spring offensive, increasing troop numbers, and de-escalating the war, how could that have worked?
As for Melvin Laird, the Secretary of Defense during Nixon's term, he reduced the number of soldiers from 53 to 2.4 during his term.

This is something Lin Ran doesn't want to see. I actually hope that more American soldiers are on the front lines of the Vietnam War.

As for why McNamara had such an important effect.

Generally speaking, it is difficult for an individual to change the overall trend.

Could it have been any other person, whether he or someone else, who was the Minister of Defense, have influenced America's strategy on the front lines of the Vietnam War and produced such a significant effect?

It really works.

Because the famous Prague Spring will take place next year, 1968.

Historically, McNamara is scheduled to leave office in February of next year, while the Prague Spring begins in January and reaches its climax in August.

Operation Danube in August, in which Soviet-led troops marched toward Prague, caused a global sensation after being reported by the media.

This further propelled the Cold War situation to its climax.

If McNamara can hold out until August, the public sentiment will reverse, and the White House will have more room to maneuver on the Vietnam War issue.

It has become possible to further expand the scale of the military.

If the Soviet Union could do that, why should we back down?

There's a reason why Lin Ran was willing to appear on Big T's show instead of Cronkite.

As a soldier who had just returned from the front lines of the Vietnam War, Big T would definitely ask questions related to the Vietnam War and would likely make sarcastic remarks about Lyndon Johnson and McNamara in the White House.

This is his chance to shine.

And this is the first show hosted by Da T, which will further boost the show's popularity.

Lin Ran's views are what caused a sensation and sparked a significant discussion and media coverage.

This is a conspiracy.

Even if we disregard the ongoing Vietnam War, replacing him with someone else would mean Lin Ran would lose a familiar face in the White House.

We've lost a close friend who could have had a profound impact on us.

The position of Minister of Defense is still very important.

McNamara can support NASA in securing funding, and some Department of Defense projects can be handed over to NASA, such as the Star Wars program. It's doubtful that someone else would have done that as Secretary of Defense.

With America already far ahead in the moon landing race, there are even fewer White House bureaucrats willing to voluntarily relinquish budgets and projects.

Therefore, from any perspective, Lin Ran hopes that McNamara can continue to serve as Secretary of Defense for as long as possible.

Compared to previous years, this year's New York Mathematicians Congress had a clear theme for discussion: Grothendieck's "Algebraic Geometry," which took seven years to compile.

Algebraic geometry began in 1960 and was compiled and published in 1967, which is also the work that established its historical status.

The reason why later generations called him the Pope of Mathematics is because of this work.

Following this, Grothendieck withdrew from the mathematics community due to the Prague Spring and the school strikes in France the following May, and has not published any more works since.

He even wrote to his students in 2010 announcing that his works were not allowed to be published, reprinted, or distributed electronically.

It can be said that the year "Algebraic Geometry" was published, the global mathematics community was discussing this book.

Compared to the original "Algebraic Geometry," Grothendieck's work is closer to the blueprint of the Grand Unified Mathematics because he learned about the Randolph Program many years earlier.

Therefore, this year the American mathematics community is also eager to hear Lin Ran's views on "Algebraic Geometry" and its advancement of the Randolph Program.

Even though everyone knows that Lin Ran focuses more of his work on NASA and the moon landing, mathematicians are still very interested in his views.

After all, Lin Ran is the creator of the Göttingen myth, and is considered to have been able to think for seven days as much as other mathematicians think for seven years.

Mathematicians believe that even just the inspiration that comes from conversation can give everyone new insights.

Of course, Lin Ran did not disappoint them. He told Fox in advance that he would be giving a lecture this year about his latest findings, the proof of the Model conjecture.

Model conjectured that curves with genus greater than 1 over algebraic number fields have only a finite number of rational points.

Okay, that's too complicated. Just explaining what a genus is is like gibberish to someone who hasn't received professional training.

Simply put, it's about "points" on a "curve".

Imagine a curve drawn using mathematical equations, such as a circle (x + y = 1) or a more complex shape.

These curves can be "simple," that is, like circles, without holes.

Or it could be "complex," like a donut or a shape with more holes.

In mathematics, "genus" is used to measure complexity: a genus of 0 or 1 is simple, while a genus greater than 1 is complex.

The core of the conjecture is that if you use rational numbers, such as integers or fractions, as coordinates to find points on these complex curves with a genus greater than 1, you can only find a finite number of points, not an infinite number.

For example, a simple curve like an ellipse may have an infinite number of rational points, but a complex curve cannot; it always has an upper limit. Why is this important?
It connects algebra, geometry, and number theory, helping mathematicians understand the deep patterns in numbers and shapes, much like proving that "infinite points don't go astray."

You can think of it this way: in the world of mathematics, some "maps" have a limited number of "waypoints" that can be reached, so it won't go on forever.

This year’s New York Mathematicians Congress was held in the auditorium of the Courant Institute for Mathematical Research at New York University, and the buzzing anticipation was louder than that of a beehive.

Since Fox released the news, all the famous mathematicians in America have gathered together.

Even those who don't conduct research in this field made thorough preparations in advance, studying the Model Conjecture and related papers beforehand to avoid misunderstanding Lin Ran's academic lectures.

There's a saying in the mathematics community that if Lin Ran continues working in the White House, sooner or later the New York Congress of Mathematicians will become more important than the quadrennial International Congress of Mathematicians.

Lin Ran walked up to the podium from the first row. Apart from the microphone and the blackboard that had been prepared in advance, there was nothing else on the podium.

He patted the microphone to make sure the sound was clear enough:

"Colleagues, I have always been a professor in the Department of Mathematics at Columbia University, but I have probably spent more time communicating with you than with the students at Columbia University. This makes me feel a little ashamed. I hope to leave the White House as soon as possible and return to academia so that I can communicate with more colleagues in the mathematics community."

Lin Ran's opening remarks caused an uproar in the audience. This was the first time Lin Ran had expressed his weariness of Washington and his desire to return to academia.

Therefore, as soon as he finished speaking, Fox in the audience immediately shouted, "Professor, Columbia welcomes you. I believe that if the president knew this news, he would probably be so happy that he couldn't sleep."

The math professors at Princeton looked rather grim, feeling that Princeton's status as a mathematical mecca was in jeopardy.

"Haha." Lin Ran didn't answer directly, but continued, "Today I will mainly talk about the proof of the Model conjecture, and I will also show multiple paths to reach the final destination."

The audience leaned forward, whispering amongst themselves.

Proving the Mordell conjecture is already quite impressive; you still need to use multiple methods.

"As expected of a professor."

"That's just the professor's style; he always manages to do things that outsiders think are impossible."

"It was worth flying all the way from Toronto."

Lin Ran wrote the number "3" on the blackboard.

"The integration paths I used all reflect the deep interaction between number theory, algebraic geometry, and high-order functions. I hope that everyone can gain some inspiration from them for the future unification of mathematics."

Looking at the focused expressions of the audience below the stage, Lin Ran continued, "First, consider a path based on the Shafalevich conjecture, although it has not been fully proven yet, but suppose we can prove the finiteness theorem for abelian varieties."

Using Pashin's technique, we can reduce the curve problem to a finite cover over a number field, thus proving the finiteness of rational points.

Here, algebraic geometry provides the foundation: using finite flat group schemes and p-separable groups, geometric objects are transformed into finite structures, avoiding the thorny arithmetic Riemann-Roch theorem.

He paused, scanned the room, and Lin Ran could already sense that most mathematicians were beginning to have difficulty understanding it.

"Secondly, I introduce an application of the Tate conjecture: by using the finiteness of end isomorphisms, I compare the homology of Jacobi varieties with the height function."

Imagine that a variant of the Siegel module acts as a bridge, comparing metric and naive height to define an upper limit for the height of a point beyond which there will be no more rational points, without violating the analytic properties of the L-function.

This embodies the fusion of Galois representations in number theory and modular spaces in geometry.

Andrew Weil raised his hand and asked, "Professor, how does this fusion avoid infinite descent? Isn't it a circular argument?"

Lin Ran smiled and said, "Good question, Andrew."

At this point, we need to draw on the idea of ​​Diophantine approximation, just like Harvey did, using height inequalities and Vita’s techniques to verify the low genus case.

This is not isolated; it is a combination of multiple methods: the L-function of number theory, the probability model theory of geometry, and the sieve method of computation. This embodies the interdisciplinary spirit shown by Grothendieck in *Algebraic Geometry*, and is not merely EGA.

Andrew still felt there was a problem, "But can your height bound be calculated efficiently? After all, the core of the Model conjecture is finiteness, not a specific number."

“Of course,” Lin Ran replied without hesitation, recalling a derivation game he had played in his spare time: “By sharpening the Bombieri refinement, I reduced the limit to a factor of log(h), making it applicable to practical testing.”

"Okay, that was the overall concept. Now let's talk about the specific technical aspects."

First, let's introduce the fusion of abelian varieties and height functions. Recall that abelian varieties are a higher-dimensional generalization of elliptic curves, and are a smooth, appropriate, and connected algebraic group scheme.

We begin with the Jacobi variety Jac(C) of curve C, a g-dimensional abelian variety that captures the point and divisor information of the curve, and introduce a new height h_F(A), a metric in Arakelov geometry defined as the Arakelov degree of the Hodge line bundle of the Néron model of the abelian variety A.

Specifically, for A” over the number field K

“Using the Zarhin trick, we transform (A× A^∨)^4 into the local polarization group, reducing it to the case of polarization 1, which is the cornerstone of the entire proof.”

Next, we prove Finiteness II: For a fixed A, the set of clusters isomorphic to A is finite. This involves p-separability and p-adic Hodge theory, calculating the height variation under isogeneity, ensuring the height set is finite, and thus deriving the isogeneity finiteness of abelian clusters.”

After explaining Faltings' proof method, Lin Ranji also provided two other paths. The first is a proof based on Diophantine approximation, and the other is a proof starting from the proof of Hodge's theorem for p-advanced numbers.

Neither of the latter two paths involves any specific technical aspects; in other words, whoever comes up with the idea first can publish a paper.

This is considered a publicly distributed benefit.

After delivering a full half-day academic report, Lin Ran said, "Ladies and gentlemen, when we trace back the proof of the Model Conjecture, from Arakelov geometry to the Galois representation of the Tet conjecture, and then to Shafarevich's finiteness and Pashin's trick, we see not only the conquest of a theorem, but also a great fusion of the fields of mathematics."

The probability theory of algebraic geometry is the cornerstone, L-functions and representation theory support number theory, and the arithmetic measure of height functions bridges the gap between the infinite and the finite.

This is not an isolated victory, but rather the convergence of different subfields: the elegance of geometry, the profundity of number theory, and the rigor of analysis, all contributing to the solution of the Mordell conjecture.

Grothendieck's "Algebraic Geometry" is excellent. He told me that countless mathematicians are contributing to the unification of mathematics from their own perspectives.

My presence here today is merely a prelude to spark discussion.

Lin Ran briefly introduced the Chinese proverb "throwing a brick to attract jade" (抛砖引玉).

He then continued, "I hope that everyone can solve more and better problems based on the concept of mathematical integration, and make a contribution to mathematical integration."

The moon landing required the combined efforts of tens of thousands of engineers; similarly, I believe that the unification of mathematics is something that cannot be achieved by one or a few mathematicians alone.

I share this sentiment with you all.

　I'll resume writing two chapters tomorrow to speed up the plot. I haven't been able to write much lately because I've been out running errands.

　　
　
(End of this chapter)