After 20 years, math couple solves major group theory problem
https://www.quantamagazine.org/after-20-years-math-couple-solves-major-group-theory-problem-20250219/I feel like this sentence is in every article for a reason. Thank goodness there are such obsessive people and here's a toast to those counter-factuals that never get mentioned.
> I feel like this sentence is in every article for a reason.
I think we do a lot of disservice by dismissing the role of the dark horses. They are necessary. Like you suggest, there are many that fail, probably most do. But considering the impact, even just a small percentage succeeding warrants significant encouragement. Yet we often act in reverse, we discourage going against the grain. Often with reasons about fear of failure. In research, most things fail. But the only real failure is the ones you don't learn from (currently it is very hard to publish negative results. Resulting it not even being attempted. The system encourages "safe" research, which by its nature, can only be incremental. Fine, we want this, but it's ironic considering how many works get rejected due to "lack of novelty")
This is wrong. It's not inherent in the meaning of the word "breakthrough" that a breakthrough can occur only when someone has gone against the grain, and there are countless breakthroughs that have not gone against the grain. See: the four-minute mile; the Manhattan Project; the sequencing of the human genome; the decipherment of Linear B; research into protein folding. These breakthroughs have largely been the result of being first to find the solution to the problem or cross the theshold. That's it. That doesn't mean the people who managed to do that were working against the grain.
> Yet we often act in reverse, we discourage going against the grain. Often with reasons about fear of failure.
I don't know which "we" you're referring to, but just about everybody would agree with the statement that it's good to think creatively, experiment, and pursue either new lines of inquiry or old lines in new ways, so, again, your claim seems clearly wrong.
If you're discussing just scientific research, though, sure, there are plenty of incentives that encourage labs and PIs to make the safe choice rather than the bold or innovative choice.
Running the 4 minute mile, climbing everest - those are achievements rather than breakthroughs.
I'd also class the atomic bomb as an achievement - it was the expected/desired result of a massive investment program - though no doubt there were many breakthroughs required in order to achieve that result.
Even if we decide that breakthroughs require some kind of discontinuity, break, or, as the comment said, "paradigm shift," such discontinuity isn't necessarily "against the grain," as this would imply some kind of resistance to or rejection of "the grain."
If that were it, I would agree.
But I don't agree. I think people who discourage going against the grain are more fearful of the loss of economic input. It's unproductive to do something you know will fail; it's very expensive to encourage that failure.
They are what Gladwell calls, in "David and Goliath", being unreasonable in the face of so-called "prevailing wisdom".
Personally I think governments should fund more moonshot solo or small team efforts because high risk / high reward pays off when you reduce the variance by spreading it out over so many people. But it looks like we’re going headstrong the other direction in terms of funding in the U.S. right now, so I’m not optimistic.
> I want financial independence for the sole reason that I can work on interesting problems like this without any outside nagging or funding issues from anyone else
It's amazing to me how much thought and work has gone into the seemingly trivial things we encounter on a daily basis.
I think of this every time I see a blue LED. Or a rice cooker!! So easy to take for granted.
Is the nifty part about the rice cooker the temperature cutoff at 105C? Induction? That my cats turn on my zojirushi three times a day and open the lid and it doesn't harm it because it knows there's nothing in the pot?
One of these days I need to track down who actually got the patent for using IR LED in a ring around a camera lens to see in the dark.
If it was at&t I am gunna be pissed.
> Just let them be
Just make sure "let them be" applies in both directions
Hans Asperger could only save his Austic children from nazi death camps by convincing the nazis that they had value to produce rockets and bombs.
It’s quite remarkable that the USA is so advanced given how deep and ruthless our anti-intellectualism goes.
This is a wildly inaccurate picture of the average person. I don't even think this is true of 10% of people.
He has been taught to love others since he was born, and the Path of Love has borne fruit for all those around all four of us.
All the people who say it can't be done have never tried consciously evolving with Divine help.
{Complete lyrics} --Sinead O'Connor in Massive Attack's "What Your Soul Sings"
I think the difference is^W was that USA celebrated it in a Homer Simpson kind of "Ha ha! NERDS!" way, while meth-Hitler was like "let's sterilize them but try to extract math from them to... (whatever batshit goal)"
Anti-intellectualism seems to be a thing when the intellectual/moderately-competent people have already brought success. (Until then it's more like anti-witchcraft, or whatever...)
That sort of positive support was one of the elements I really liked in working on combinatorial problems. People like Persi Diaconis and D.J.A. Welsh were so nice it makes the whole field seem more inviting.
Suppose I'm interested in representing a Group as matrices over the complex numbers. There are usually many ways of doing this. Each one of them has a so-called character, which is like fingerprint of such a representation.
Along another line, it has been known that all groups contain large subgroup having an order which is a power of a prime--call it P. This group in turn has a normalizer in which P is normal--call it N(P).
The surprising thing is that the number of characters of G and of N(P)--which is is only a small part of G--is equal.
*technical note in both cases we exclude representation the degree of which is a multiple of p.
But the article says that mathematicians are now searching for a deeper “structural reason” why the conjecture holds. Now that the result is known to be true, it’s giving more mathematicians the permission to attack it seriously.
For example, article says there are 50 groups of order 72 (chatGPT says there are 50 non-Abelian, 5 Abelian), this seems to be an important insight but into what?
I hope that their relationship deals well with the new reality, now that their principal goal has been achieved.
Unrelatedly, I'd be curious what this couple thinks about using AI tools in formal math problems. Did they use any AI tools in the past 2 years while working on this problem?
Mind you, my math is enough for BSEE. I do have a copy of one of my university professor's go-to work books: The Algebraic Eigenvalue Problem and consult it occasionally and briefly.