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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

#51

Post by tgv »

Ekaros wrote: post #38
sorokod wrote: post #20

A bit surprising that no one seriously attempted to find a counter example before.

I don't think it is unexpected if the counter examples are really really big. It haven't been so long we have gotten lot of processing power and memory. If the counter examples only appear when you are in territory that is unfeasible by hand it is not so surprising.

OTOH, people have been using computer-assisted "configuration generation" for more than 50 years now. The 4-color problem was done on hardware that was already obsolete 30 years ago. I also would have expected mathematicians to take their hypotheses a bit more seriously. Especially since this does toppled at a rather modest problem size.

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

#52

Post by alasdair_ »

agensaequivocum wrote: post #36
Jeff_Brown wrote: post #31

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

All knowledge ultimately relies on some self-evident first principals which are not demonstrable.

ALL knowledge does? Can you prove that? :)

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

#53

Post by kongolongo »

Spivak wrote: post #50
quchen wrote: post #19

Earlier quoted context omitted.

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.

Here's a hypothesis: no positive number is evenly divisible by 3 trillion one. True up to 3 trillion then false at 3 trillion 1.

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

#54

Post by constantcrying »

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?

It is probably mostly a reporting bias. Almost all new mathematical discoveries don't have simple pictures to draw and are about highly abstract concepts which are difficult to write about for a general audience.

Analysis is an extremely active field, PDEs have almost endless amounts of open research questions. Usually they are not very flashy and can be very non-geometrical.

Algebra is extremely hard to write about for a general audience. Trying to communicate any result which does not have a simple visual interpretation seems like a nightmare.

Numerics are often things where advances are hard to relay to an audience, as it usually is about incremental improvements instead of breakthroughs.

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

#55

Post by constantcrying »

vkazanov wrote: post #24
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?

Well, algebra and calculus have been around for a while. And then, Calculus took about 3 years for me to begin to scratch the surface... and realise I will never need it as numeric methods took over completely with the advent of cheap compute.

Nothing is more important for numerics than a veey good grasp of analysis.

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

#56

Post by constantcrying »

klyrs wrote: post #27
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?

Algebra and Calculus are more or less "solved." Unless by algebra, you mean abstract algebra... but the open questions there tend to be quite esoteric. That said, we did recently see a novel approach to solving quadratic equations ( https://www.sciencealert.com/math-genius-finally-discovers-e... ). I'd say that HN posts a lot of quanta articles, and quanta has a "bias" towards results that can be explained to a semi-…

>Algebra and Calculus are more or less "solved." Unless by algebra, you mean abstract algebra

Or unless by calculus he means analysis, which is really active. Especially things like PDEs.

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

#57

Post by constantcrying »

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?

They were undergrads. Nobody expects undergrads to solve decade old research problems in a few months.

These seminars are about deep dives into particular mathematical questions. Maybe including some recent "doable" unsolved problems.

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

#58

Post by Extigy »

Spivak wrote: post #50
quchen wrote: post #19

Earlier quoted context omitted.

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.

It can happen, the Pólya conjecture is the usual example which holds until n = 906150257.

Another fun one I just found is the statement “n^17 + 9 and (n + 1)^17 + 9 are relatively prime”. The first counterexample is at n=8424432925592889329288197322308900672459420460792433.

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

#59

Post by zmgsabst »

lcnPylGDnU4H9OF wrote: post #49
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.

We call it LIGO these days.

Which is to say, many of these ideas turn out to work after some modifications.

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

#60

Post by Sxubas »

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?

From the article:

> Stange added that none of this would have happened without the low-stakes summer project. “Serendipity and an attitude of playful exploration both have such a huge role in discovery,” she said.

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