Chapter 318: Throwing A Brick To Attract Jade
It was truly too moving.
When the media were all accusing McNamara of being the worst Secretary of Defense, Lin Ran stood up and publicly declared that he did the best job.
Such a huge contrast.
Plus claiming that the Vietnam War problem should not be attributed to any specific individual, which further helped him shift the blame.
Moreover, if it were mindless flattery, just stating viewpoints without providing arguments, such praise, even if said by Lin Ran, would not be enough to move McNamara too much.
The key is that Lin Ran not only praised McNamara, but also explained the reasons very clearly, pinpointing exactly where McNamara did well compared to the previous Secretaries of Defense.
These were precisely the areas McNamara took pride in; it was praise that hit the nail on the head.
McNamara, who had a slight understanding of Chinese culture, thought of Chinese allusions in his heart: high mountains and flowing water meeting a soulmate, a thousand-mile horse meeting its Bole.
Overwhelmed with gratitude, beyond words.
At this time, Lin Ran was thinking about another issue.
You escalated the Vietnam War from 16,000 advisors to 535,000 combat troops; if someone else took over, stopped the Spring Offensive, ceased reinforcements, and downgraded the war, how could that work?
As for Melvin Laird, the Secretary of Defense during Nixon’s time, who reduced the number of soldiers from 530,000 to 24,000 during his tenure.
This was something Lin Ran did not want to see; he still hoped for more America soldiers on the Vietnam War frontline.
As for why McNamara had such a significant effect.
Generally speaking, individuals find it hard to change the tide.
Whether it was him or someone else as Secretary of Defense, could it really influence America’s strategy on the Vietnam War frontline and produce such remarkable effects?
It really could.
Because next year, 1968, the famous Prague Spring would occur.
Historically, McNamara resigned in February next year, the Prague Spring started in January, with its climax in August.
The August Operation Danube, led by the Soviet Union army marching on Prague, caused a global sensation under media reports.
Further pushing the Cold War situation to a climax.
If only McNamara could hold on until August, public opinion would reverse, and the White House would gain greater room to maneuver on the Vietnam War.
Continuing to scale up military operations would become possible.
If the Soviet Union can do that, why should we lower our flags and still our drums?
Lin Ran chose to go on the big T program instead of Cronkite for a reason.
Big T, as a soldier just back from the Vietnam War frontline, would definitely ask about Vietnam War-related issues and sarcastically mock Lyndon Johnson and McNamara at the White House.
This was his opportunity to shine.
And big T’s first hosted program itself would further boost the topic’s heat.
Lin Ran’s viewpoint would cause a sensation, sparking sufficient discussion and media propagation effect.
This was an open scheme.
If someone else took over, even without considering maintaining the Vietnam War, it would mean one less familiar face for Lin Ran at the White House.
One less familiar face who could be profoundly influenced.
The Secretary of Defense position is still very key.
McNamara could support NASA getting funds; some Department of Defense projects could be handed to NASA, like the Star Wars Program—probably not if someone else were Secretary of Defense.
Today, with America far ahead in the moon landing race, even fewer White House bureaucrats would willingly cede budget and projects.
Therefore, from any angle, Lin Ran hoped McNamara could stay in the Secretary of Defense position as long as possible, the longer the better.
Compared to previous years, this year’s New York Mathematicians Conference had a clear discussion theme: Grothendieck’s Algebraic Geometry, compiled over seven years.
Algebraic Geometry started in 1960 and was published in 1967, which is also the work that established its historical status.
Why later generations called him the Pope of Mathematics was because of this work.
After this, due to the Prague Spring and next year’s May strikes in France, Grothendieck withdrew from the mathematics community, with no more works published thereafter.
He even wrote to his students in 2010 announcing that his works must not be published or reprinted, nor disseminated electronically.
It can be said that this year, with the release of Algebraic Geometry, the global mathematics community was discussing this work.
Compared to the original spacetime’s Algebraic Geometry, because Grothendieck learned about the Randolph Program much earlier, the entire work is even closer to a blueprint for grand unification in mathematics.
So this year, the America mathematics community also wanted to hear Lin Ran’s views on Algebraic Geometry and its advancement of the Randolph Program.
Even though everyone knew Lin Ran’s main work focus was on NASA and the moon landing, mathematicians were still very interested in his views.
After all, Lin Ran was the creator of the Göttingen myth, considered able to match seven years of other mathematicians’ thinking in seven days.
Mathematicians felt that even casual chat-inspired ideas could give everyone new enlightenment.
Of course, Lin Ran did not disappoint their expectations; he told Fox in advance that this year he would lecture on his latest achievement at the talk: the proof of the Mordell conjecture.
Mordell conjecture: on algebraic number fields, curves with genus greater than 1 have only finitely many rational points.
Okay, that’s too complex; just what genus is, for those without professional training, is like a heavenly script.
Simply put, it is about “points” on “curves”.
Imagine a curve drawn with a mathematical equation, like a circle (x + y = 1) or more complex shapes.
These curves can be “simple”, like a circle, with no holes.
Or “complex”, like a doughnut or shapes with more holes.
Mathematically, “genus” measures complexity: genus 0 or 1 is simple, greater than 1 is complex.
The core of the conjecture: if you use rational numbers, like integers or fractions, as coordinates, on these complex curves with genus greater than 1, the findable points are finite, not infinite.
For example, a simple curve like an elliptic curve may have infinitely many rational points, but complex curves do not; there is always an upper limit.
Why important?
It connects algebra, geometry, and number theory, helping mathematicians understand the deep laws of numbers and shapes, like proving “infinite points won’t run wild”.
Everyone can think of it as: in the mathematical world, some “maps” have finite “waypoints”, not endless.
This year’s New York Mathematicians Conference was held in the auditorium of New York University Courant Mathematics Research Institute, with buzzing anticipation louder than a beehive.
Ever since Fox released the news, all notable mathematicians in America gathered together.
Even if not researching this field, everyone prepared thoroughly in advance, studying the Mordell conjecture and related papers to avoid not understanding Lin Ran’s academic lecture.
There was a subtle saying in the mathematics community that if Lin Ran continued working at the White House, sooner or later the New York Mathematicians Conference would surpass the quadrennial International Congress of Mathematicians in importance.
Lin Ran walked to the podium from the first row; on stage, besides the microphone and pre-prepared blackboard, there was nothing else.
He tapped the microphone to ensure the voice was clear enough:
“Fellow colleagues, I have always been a Mathematics Department professor at Columbia University, but I may spend more time exchanging with you all than with Columbia University classmates, which makes me somewhat ashamed. I hope to leave the White House soon and return to academia to exchange with more colleagues in the mathematics community.”
Lin Ran’s opening remarks caused an uproar below; this was the first time Lin Ran expressed weariness of Washington and a desire to return to academia.
So after he finished, Fox below immediately said loudly: “Professor, Columbia welcomes you. I believe if the president knows this news, he will be so happy he can’t sleep.”
Princeton’s mathematics professors looked unhappy; they felt Princeton’s mathematical holy land status was at risk.
“Haha.” Lin Ran did not answer directly and continued: “Today, I will mainly talk about the proof of the Mordell conjecture, and I will also show multiple paths to the final destination.”
The audience leaned forward, whispering among themselves.
Proving the Mordell conjecture is already impressive, and you want to use multiple methods.
“Worthy of the professor.”
“This is the professor’s style; he always achieves what the outside world thinks impossible.”
“Worth my special flight from Toronto.”
Lin Ran wrote the number “3” on the blackboard.
“The fusion paths I used all embody the deep interactions of number theory, algebraic geometry, and height functions. I hope everyone can gain some inspiration for the future unification of mathematics from them.”
Lin Ran looked at the focused expressions of the audience below and continued: “First, consider a path based on the Shafarevich conjecture. Although it is not fully proven itself, assume we can prove the finiteness theorem for Abelian varieties.
Through Paschen’s technique, we reduce the curve problem to finite covers over number fields, thereby proving the finiteness of rational points.
Here, algebraic geometry provides the foundation: using finite flat group schemes and p-divisible groups, transform geometric objects into finite structures, avoiding the tricky arithmetic Riemann-Roch theorem.”
He paused, scanning the room; Lin Ran already sensed most mathematicians were starting to struggle with understanding.
“Second, I introduce the application of the Tate conjecture: through the finiteness of étale cohomology, compare the cohomology of the Jacobian variety with height functions.
Imagine Siegel modular varieties as a bridge, comparing metrics and naive heights to bound the height of points; beyond it, no more rational points, without violating the analytic properties of L-functions.
This embodies the fusion of number theory’s Galois representations and geometry’s moduli spaces.”
Andrew Weil raised his hand and asked: “Professor, how does this fusion avoid infinite descent? Isn’t it circular reasoning?”
Lin Ran smiled: “Good question, Andrew.
At this point, we borrow ideas from Diophantine approximation, like Harvey did, using height inequalities and Vojta’s techniques to verify low-genus cases.
This is not isolated; it is a combination of multiple methods: number theory’s L-functions plus geometry’s scheme theory, plus computational sieve methods. This embodies the interdisciplinary spirit shown by Grothendieck in Algebraic Geometry, yet not limited to EGA.”
Andrew still felt there was an issue: “But is your height bound effectively computable? After all, the core of the Mordell conjecture is finiteness, not the specific number.”
“Of course,” Lin Ran replied without thinking, his mind flashing to the derivation games he once did in spare time: “By sharpening Bombieri’s refinement, I shrink the bound to a log(h) factor, making it applicable to actual verification.”
“Alright, that was the overall framework; next is the specific technical level.
First, introduce the fusion of Abelian varieties and height functions. Everyone recall, Abelian varieties are higher-dimensional generalizations of elliptic curves, smooth, proper, connected algebraic group schemes.
We start from the Jacobian variety Jac(C) of the curve C, a g-dimensional Abelian variety capturing the curve’s points and divisor information. 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 Abelian variety A.
Specifically, for A over number field K”
“. Through the Zarhin trick, we turn (A × A^∨)^4 into principally polarized varieties, reducing to polarization degree 1, which is the cornerstone of the entire proof.
Next, prove Finiteness II: for fixed A, the set of varieties isomorphic to A is finite. This involves p-divisible groups and p-adic Hodge theory, computing height changes under isogenies, ensuring the height set is finite, thus deriving isogeny finiteness of Abelian varieties.”
After explaining Faltings’s proof method, Lin Ran also gave two other paths: one based on Diophantine approximation proof ideas, the other starting from the p-adic Hodge theorem proof.
The latter two paths had no specific technical details, meaning whoever figures them out can publish papers.
It was practically publicly giving away benefits.
After the full half-day academic report, Lin Ran said: “Colleagues, as we trace the proof journey of the Mordell conjecture, from Arakelov geometry to Tate conjecture’s Galois representations, to Shafarevich’s finiteness and Paschen techniques, what we see is not just the conquest of a theorem, but a great fusion in the field of mathematics.
Scheme theory in algebraic geometry is the cornerstone, L-functions and representation theory support number theory, and arithmetic metrics of height functions bridge the gap between infinite and finite.
This is not an isolated victory, but the intersection of different subfields: geometry’s elegance, number theory’s profundity, analysis’s rigorousness, together achieving the solution of the Mordell conjecture.
Grothendieck’s Algebraic Geometry is excellent; he told me countless mathematicians are contributing to the grand unification of mathematics from their own angles.
What I did here today is just throwing out a brick to get a jade gem.”
Lin Ran briefly introduced the Chinese allusion of throwing out a brick to get a jade gem.
Then continued: “I hope everyone, based on the concept of mathematical fusion, can solve more and better problems, making a contribution to mathematical unification.
The moon landing requires the joint effort of tens of thousands of engineers; similarly, I believe the grand unification of mathematics is absolutely not something one or a few mathematicians can achieve.
Here, I encourage everyone together.”
