234: Chapter 109 Paper Completion_2 234: Chapter 109 Paper Completion_2 This ti point is actually unrelated to when Elder Yuan is available, it’s simply because Qiao Yu thinks that if his direction is correct, by next Wednesday, he should more or less have the results ready.
At that ti, whether he looks over the paper for his senior first, or really takes it to Huaqing to have his grandmaster review it, it will be more convenient.
Before confirming this direction is correct, he has no mood to deal with other things.
Chen Zhuoyang said excitedly: “Next Wednesday?
No problem!
By the way, I don’t really expect Elder Yuan to help revise the paper, as long as you let Elder Yuan see where the problems are and give specific comnts, that’s enough.”
Qiao Yu nodded, affirmatively said: “OK!
Just rember to give the paper before Wednesday.”
“Thank you so much, little junior brother, I’ll go back to work on the paper first.”
“No worries, you go ahead and busy yourself!”
“Sigh…” Watching Chen Zhuoyang leave, Qiao Yu sighed, suddenly realizing that he was having more and more things to do.
Study, read books, coax his ntor and his ntor’s ntor, do projects, write papers, participate in selections, then go to IMO for a dal, incidentally surpass his peers…
Now he also has to worry about his senior’s doctoral thesis.
How many things does he have to do alone?
This is probably the legend of ‘the capable do more work’!
Hmm, the task of revitalizing the Huaxia mathematics community, it seems it must be he who bears it!
After all, he is already sixteen, no longer that fifteen-year-old kid!
Thinking of this, after being filled, Qiao Yu was rejuvenated again, sitting upright in front of the computer, work, work…
For revitalizing Huaxia mathematics, and to show so Qiao Family colors to his ntor, grandmaster, and senior, he must sohow work out the Qiao Family Upper Bound Theorem!
…
Mathematical research often has a very interesting characteristic, which countless mathematicians have encountered, that is during the research process, it can get stuck at so step, or on so issue, unable to make any progress for a long ti.
Yes, truly stuck there.
Sotis with an epiphany, this hurdle is overco, only to feel suddenly enlightened, the road ahead is broad and smooth.
But unfortunately, for the vast majority of mathematicians in this world, encountering this hurdle might last a lifeti, so the topic cos to an unsuccessful end, the past work and materials sealed there, fantasizing that soday, they might suddenly have an epiphany, allowing these studies to see the light of day in the future soday, but more likely, there will be no future.
Qiao Yu is actually the sa, except that his talent is a little higher than countless ordinary mathematicians.
When he, with the hint from Qiao Xi, realized that finding parater commonality, to him this problem didn’t seem to be a problem anymore.
All the reasoning and proof processes before were already finished, finding commonality can simplify, the commonality is hidden in the not-so-obvious connections behind those paraters, as long as the work is detailed enough, Qiao Yu felt this is definitely the right direction!
And indeed it was so.
In three days, Qiao Yu hardly left his room except to eat, not even reading books, throwing himself wholeheartedly into this work, and then he truly discovered the existence of commonality.
The higher the level of modular forms, the more complex the curve, so k ~ curve complexity.
The pri p controls the local geotric behavior of the curve in the -adic number field, different pris correspond to different geotric constraints, the pri p is also related to curve complexity, so p ~ local geotric complexity.
The parater q in quantum cohomology reflects the influence of quantum geotric objects on the global complexity of the curve, this is a further quantification of the geotric complexity of the curve, so q ~ global geotric complexity.
In other words, although different geotric paraters co from different sources, they all reflect the complexity of the curve from different perspectives.
What is this?
This is the definition condition of parater unification.
So on Friday night, Qiao Yu designed a unified geotric constraint parater θ, and proposed a second hypothesis: the geotric constraint parater θ is so weighted combination of the modular forms level, -adic number field pri, and quantum cohomology paraters, they together reflect the global complexity of the curve.
Based on this hypothesis, it is clear to get a basic structure: θ = f(g, k, p, q).
Of course, at this step, it is clearly not enough.
Because the weight of each parater is different, to make the structure mathematically reasonable, a combination thod that can perfectly reflect the weights of each parater is needed.
Next is the work of calculation and verification, complex, but not difficult.
But in one night, he concluded that the growth of k grows logarithmically with the deficiency g, so: k ~ g log(g); the complexity of local geotry increases exponentially with the increase of deficiency, so p ~ e^g/2; in quantum cohomology, the relationship between the parater q and the deficiency g directly calculated an approximate value: q ~ g^3/2.
The formula naturally appeared: θ = f(g, k, p, q) = g ⋅ log(k) g^2 ⋅ log(p) g ⋅ q
After directly substituting the expressions of the three paraters, it shows: θ = g ⋅ log(g log(g)) g^2 ⋅ log(e^g/2) g ⋅ g^3/2
At this step, only the deficiency g remains as an important parater.
Next is the simplest simplification work: θ = g ⋅ (log(g) log(log(g))) g^3/2 g^5/2
After three days of working day and night in front of the computer, on February 21, 2025, Friday at 11:37 pm, Qiao Yu finally typed the final formula for the estimation of rational points on curves on his computer: N(X) ≤ C(θ) = θ^g
θ is the geotric constraint parater he designed, g is the deficiency.
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