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Students’ insight proves that the local-global conjecture doesn’t hold

quantamagazine.org

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Re: Students’ insight proves that the local-global conjecture doesn’t hold

#41

Post by jagged-chisel »

gowld wrote: post #25

What would happen if the conjecture happened to be true, and they did their data collection/computation, and got a bunch of data (called "0%" in mathematics) consistent with the conjecture? Would their summer research be an utter failure? Would they be unimpressive mathematicatians? How much of math success is being lucky enough to stumble upon a tractable problem?

If their approach is reproducible and correct, they have successfully disproved the conjecture. If errors are found, corrections will be made.

They had started attempting to prove the conjecture. This result could be considered a failure of that proof. Showing they’re capable of performing the research makes them good mathematicians.

I believe the answer to your last question is “lots.”

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#43

Post by svat »

Aardwolf wrote: post #6

A lot of recent math discoveries seem to be geometrical: circle packing, infinitely tiling patterns, neighbor coloring theorem, ... I don't seem to see as much discoveries in other areas like number theory, calculus, algebra etc... Is this a bias of what gets covered on HN, or are this type of geometrical problems the currently most active field of mathematics?

> other areas like number theory…

In case you didn't notice: the article is about a discovery in number theory. The primary mathematician in this story (search the article for "Stange is a number theorist…"), the ones referenced and quoted ("Elena Fuchs, a number theorist", James Rickards, Peter Sarnak, Alex Kontorovich, Jeffrey C. Lagarias, …) are all number theorists, and the paper itself (https://arxiv.org/abs/2307.02749) was posted under math.NT.

Apollonian circles are geometry, but the conjecture is about the integers that show up as the curvatures of packings, and specifically about the "certain numerical buckets" they happen to fall into. Of course mathematics is ultimately a connected whole; e.g. Jean Bourgain mentioned in the article would not be considered primarily a number theorist.

[And of course there's a bias in what gets covered: researchers work in all areas and it's far from true that "this type of geometrical problems the currently most active field of mathematics", but the ones that can be turned into a good story (and geometry is easier to explain / show) are more likely to get picked up by media like Quanta; and some of them are more likely to be posted to HN and to be upvoted. And some of them are likely to be interpreted as about geometry anyway!]

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#44

Post by alanbernstein »

Jeff_Brown wrote: post #31

Does mathematics contain a lot of theorems that rely on unproven assumptions?

Probably, but we call them conjectures to indicate that we know that, and only call them theorems when we have tricked ourselves into believing otherwise

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#45

Post by chongli »

Jeff_Brown wrote: post #31

Does mathematics contain a lot of theorems that rely on unproven assumptions?

A theorem is just a statement which has been proved. Proofs are logical deductions that begin with a set of assumptions (called the hypothesis of the theorem) and follow a sequence of valid steps to reach a result (called the conclusion of the theorem).

Without any assumptions at all, you have nowhere to go. There’s nothing you can conclude if you begin by assuming nothing.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#46

Post by svat »

TheRealPomax wrote: post #40

tl;dr: the conjecture is that if you start with three touching circles, each of which has an integer curvature (i.e. 1/r is a whole number) and you then draw a circumscribing circle around those, then starting filling in the gaps between the circles with ever smaller circles, every one of those smaller circle will also have an integer curvature. Turns out, they don't, but no one actually sat down to do the grunt work…

That the curvatures are integers is proved (noticed by Soddy in 1936 as "a fairly straightforward consequence of Descartes’ equation", as the article mentions), and a lot more is also proved about the integers' distribution; the conjecture specifically is about whether every sufficiently large “admissible (passing local obstructions) integer is the curvature of some circle in the gasket” — see the second page of https://arxiv.org/abs/1205.4416, or indeed the second page of the paper the article is about (https://arxiv.org/abs/2307.02749), which puts it even more concretely:

> Conjecture 1.1 ([GLM+03, FS11]). Let A be a primitive Apollonian circle packing containing curvatures equivalent to r (mod 24). The set of positive integers x ≡ r (mod 24) not occurring in A is finite.

which they disprove with:

> Theorem 1.3. There exist infinitely many primitive Apollonian circle packings for which the number of missing curvatures up to N is Ω(√N). In particular, the local-global conjecture is false for these packings.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#47

Post by anon291 »

Jeff_Brown wrote: post #31

Does mathematics contain a lot of theorems that rely on unproven assumptions?

Godel's incompleteness theorems say that all mathematics that is complicated enough to encode basic arithmetic must rely on unproven assumptions. Unproven assumptions are the basis of all mathematics. For example, even numbers rely on very unproven assumptions. We assume, for example, that there is always a number following another number, but it is not at all obvious what that means.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#48

Post by Someone »

amelius wrote: post #13
phoe-krk wrote: post #2

Assume that something is true, write code to generate cases, run it, plot the generated data, expect that the plots show that everything matches the assertion, notice it doesn't, disprove the thing previously considered true. Nice and elegant.

Now try that on this problem: https://en.wikipedia.org/wiki/Riemann_hypothesis

Mathematicians have been working on that for over 160 years. See https://en.wikipedia.org/wiki/Riemann_hypothesis#Numerical_c....

Fun quote (at least for me) from that page (https://en.wikipedia.org/wiki/Riemann_hypothesis#Littlewood'...:

“It has been computed that π(x)

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#49

Post by lcnPylGDnU4H9OF »

phoe-krk wrote: post #2

Assume that something is true, write code to generate cases, run it, plot the generated data, expect that the plots show that everything matches the assertion, notice it doesn't, disprove the thing previously considered true. Nice and elegant.

I'm reminded of a famous experiment in physics history: https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_exper....

Similarly, the experiment was done with the intention of proving something that is widely believed to be true (I mean, obviously light travels through something) only to undeniably disprove it.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#50

Post by Spivak »

quchen wrote: post #19
ducttapecrown wrote: post #16

Earlier quoted context omitted.

This one is true though.

Unless it's not. I'm sure lots of mathematicians have open bets about that. I still remember my supervisor hoping for no Higgs "because then physics will be boring for the foreseeable future".

I think anyone that's betting on it being false is nuts at this point. It would be astounding if it held for the first three trillion then broke down.

Generally patterns like this get more regular as the numbers get bigger not less.

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