After 20 years, math couple solves major group theory problem
quantamagazine.org437 points by isaacfrond a day ago
437 points by isaacfrond a day ago
"There was a risk that such a single-minded pursuit of so difficult a problem could hurt her academic career, but Späth dedicated all her time to it anyway."
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")
> Breakthroughs, BY DEFINITION, come from people going against the grain. Breakthroughs are paradigm shifts.
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.
Sounds like an argument over semantics and the meaning of the word "breakthrough".
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.
Yup, it's semantics, because the comment I answered stresses "by definition." My point is partly that that isn't the definition.
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."
Yea but this is HN where everyone is a disruptor and doesn’t play by the rules
> Often with reasons about fear of failure.
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.
Paradigm shifts require an accumulation of mundane experiments that present contradictions in a model. The renegade hacker isn't enough.
> Breakthroughs, BY DEFINITION, come from people going against the grain.
They are what Gladwell calls, in "David and Goliath", being unreasonable in the face of so-called "prevailing wisdom".
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 (there might still be some judgment, but I can ignore that).
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
I bet there’s a good number of us. How do we make this a reality? :)
I bet there is too! But I think it is also hard to form this community. Given the increased friction from the general community I think it might be time to put serious effort in determining how to make this a reality.
I don't have a great idea for it tbh. I'll pitch a bad one at least to put something out there. Maybe we can convince some billionaire to perform a post scarcity experiment? To buy large chunk of land, gather people who have both high passion and expertise in domains, and let them run wild. Less like Star Trek and more like Eureka[0]. I don't think it is a realistic expectation to get billionaire funding, but I think the idea of looking at something like Eureka (or this category of groups you see in many Sci-Fi stories) is worth drawing inspiration from: effectively post scarce (possible on small scale?), high levels of freedom (very doable), high levels of creativity and expressiveness (this is the experiment, to see if it is decreased/maintained/increased).
I get where you're coming from, but this is just Prsopera, or any of the other failed libertarian citiy experiments.
I'm a Swedish game developer and I feel exactly the same way. I have my dream games I work on every now and then making very little slow progress. My wildest dream would be just being able to dedicate myself to it full time. But, there are bills to pay.
Given what universities charge, they should more than be able to cover comfortable salaries for all researchers so they never need to worry about going broke. Tenure is a very useful tool!
What are Harvard's coffers up to these days? Over 2bn? IIRC that's free and clear.
I worked at a pro audio company where one guy spent 5 years on a power supply. It succeeded, and I always appreciated the management for supporting him.
Do recall the specific problem he was trying to solve?
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.
Haha the blue led story is literally people going against the grain, a great example. Worked on it after being ordered not to. The original owner of the company believed in the inventor, too, which probably helps.
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.
And you can thank this guy for the LEDs that made it possible for you to even read about it on a screen https://youtu.be/AF8d72mA41M
There are tens of millions of people doing a repetitive work every day, instead of being entrepreneurs. Just let them be, not everybody needs to dedicate their existence to maximizing their career opportunities, at any level.
> Just let them be
Just make sure "let them be" applies in both directions
Unfortunately the average persons hatred of autistic or nerdy people implies that many believe the world would be a better place if “obsessive types” didn’t exist.
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.
"the average persons hatred of autistic or nerdy people"
This is a wildly inaccurate picture of the average person. I don't even think this is true of 10% of people.
Probably enough for 90% of such folks to be tormented quite effectively during childhood.
This is a great point. If a small percentage of the population is openly and aggressively negative towards another part of the population, it makes sense that they would both artificially appear as if they are a larger part of the population, to each other. I think it goes both ways.
I _highly_ doubt 90% of nerdy people were tormented during childhood. Have you spent time in/around academia? These people are massive nerds and horrendous childhood torment is pretty rare.
It was 100% in my personal experience.
I’m not saying it doesn’t happen. I’m not even saying it doesn’t happen at all higher rate than the general population.
But 90% is just too much to be feasible.
Well all good stats are made up.
My point was more that its doesn't take a large chunk of a population to mistreat others to make mistreatment almost inescapable.
For child sexual abuse, which is studied more than "anti-nerd bullying" offenders are just a few percent of the population, if that (I can't actually find good sources for population incidence of being an abuser) but the offended against are 1/6 to 1/3 of the population.
Another interesting point is that some nerdy folks might not even notice the bullying that much - I once had a friend from high school say that they always felt bad for how much bullying I experienced, and I said "what? what do you mean?" I felt that many of the folks in high school were terribly evil people, but I didn't get that upset by the bullying itself. Just determined to move as soon as I could. Not like as a strategy to over come the bullying, just to get among people I can respect.
Our son was specifically chosen to be in his 4th Grade class because it spent part of the day hosting the spectrum kids within their "regular" class. He was chosen for that honor because of his kindness.
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 don't think the average person hate autistic or nerdy people.
Yes, they do. They wouldn't say it, but look at how autistic people get treated by their peers, teachers, and bosses.
How do you know their peers, teachers, and bosses aren’t autistic too? Autistic people can’t be peers, teachers, and bosses? Autistic people can be assholes too. Maybe that’s what it really is: some people are just assholes.
Obviously everything is a spectrum but I agree. If anything, "a touch of the 'tism" is the new "I have OCD" because you like a clean desk, something people are fine with saying without regards to it being true.
They absolutely don't.
The average person is probably uncomfortable around autistic people because they don't know how to deal with them and, when you see someone like that, it's usually best to avoid interacting with them. Not because autistic people are dangerous but people acting out of the ordinary sometimes are.
The lumping in of nerdy with autistic is ridiculous too. The average person just doesn't care about your interests unless they are in common. Nerdy typically just means having a niche interest or hobby.
Your parent reeks of a persecution complex.
I guess you never studied in a school and therefore not aware of bullying and its stereotypical targets.
It's shocking that you can just bury your head in the sand about the colloquial usage of "autist" (often happening at the same time as "Incel") replacing "retard" in the lexicon of the average zoomer.
It's not a persecution complex to identify the ways that hate gets used, deployed, and repackaged for a new generation. Putting your head in the sand about stuff like this is why Trump winning the election was SO shocking to some and extremely obvious to others.
It's shocking that you seem to think that when I said, "the average person" you thought I meant everyone.
The average zoomer isn't your average person. People are sometimes cruel. The average person is not. The average person didn't vote for Trump.
I also took most issue with grouping a autism with "nerdy". Being a nerd has never been easier.
> It's not a persecution complex to identify the ways that hate gets used, deployed, and repackaged for a new generation.
Okay, but that's not what happened. Cool strawman though.
Probably the most powerful man in the world right now openly self-identifies as autistic. Obviously there are very many autistic people who get treated very badly, but I don't think it's reasonable to say that the average person "hates" autistic people.
that guy is arguably causing branding problems for less powerful autistic people
Yeah I wouldn't want to be an autistic person that doesn't suck right now.
I imagine it's a lot like my experience of having ADHD and being trapped between wanting leeway and support but also wanting to be held accountable and considered capable of improving and watching the Tiktokification of my disorder force me to argue that actually people with ADHD can do things and no having it doesn't fully excuse you from ever meeting commitments or doing the fucking dishes, especially if you're rejecting all and any treatment or strategy.
If anything autistic/nerdy people are lionized these days with tons of people larping as them online, claiming they are autistic because they sometimes feel awkward at a social event.
I think the claim of being autistic is lionized but actually having detrimental symptoms of autism is still very stigmatized.
Isn't it basically the same? Nazi Germany in 1934 was relatively advanced, too.
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...)
> When the couple announced their result, their colleagues were in awe. “I wanted there to be parades,” said Persi Diaconis (opens a new tab) of Stanford University. “Years of hard, hard, hard work, and she did it, they did it.”
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.
All our acts are butterfly wing beats that influence those with whom we interact, for either good or ill. And those waves resonate back within our being as happiness or its opposite, depending on our intentions and actions.
"Positive vibration, yeah." --Bob Marley
So what the McKay conjecture is saying is this.
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.
It’s interesting that the conjecture was proven via case by case analysis, with each case demanding different techniques. It’s almost a coincidence that all finite groups have this property, since each group has the property because of a different “reason”.
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.
I don't think that is that unusual in group theory proofs to be honest. You often break things down into related things and then prove for each collection of related things. And some of those proofs might be straightforward and some might be open problems for years requiring much more advanced techniques.
Hah, serendipity: I was reading the Groups part of the Infinite Napkin after it was posted on HN recently. I understand the definitions, etc. but still haven’t grasped the central importance of groups.
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?
Don't listen to ChatGPT. There are 44 non-abelian and 6 abelian groups of order 72. I wouldn't say these particular numbers are terribly important, but they are correct.
Definitely don't use ChatGPT to try to learn about something you have no knowledge of. It's impossible to separate the bullshit from the truth if you don't already have some foundation in the field, and chatbots often try to sneak some bullshit in there.
Don't listen to ChatGPT.
As someone "junior programmer" , I think I should just keep this information . Many times I have tried to not use chatgpt. but its just so lucrative.
I really need better control of myself. Going to block chatgpt at a dns level.
Groups are important because they are the algebraic way to describe symmetry: if you have some operation that leaves a thing invariant (e.g. rotating an equilateral triangle so that a vertex lands on where another started), then the operation is invertible and the inverse leaves the thing invariant. You can compose such operations and the composition will still be invariant. The identity function always leaves everything invariant. So your symmetry operations form a group.
Slightly trickier is that every group is a set of symmetry operations for something. So groups exactly capture the idea of symmetry. To a mathematician, "group" and "symmetries" are synonymous.
Finite groups can be interesting as the symmetries of e.g. molecules (e.g. rotating atoms around onto each other), which can tell you something about molecular structure, energy levels, spectra, bonding potential, etc. Infinite groups appear in physics (e.g. the laws of physics are the same when you rotate or translate your coordinates by arbitrary amounts). Symmetry also comes up as a way to study other mathematical objects, and mathematicians might just want to know what all possible groups look like.
The presence of ann invariant implies invertibility! That doesn’t seem true.
I've only taken three (undergrad) classes involving groups so I'm far from an expert but my feeling is that their underlying structure is a bit like prime numbers.
Nobody bothers explaining why the primes are spaced like they are, rather people explain other phenomena by pointing out that certain things about it are prime or not.
For instance, there's this thing about having a nice neat formula for factoring second degree polynomials (the quadratic formula). One also exists for cubics and quartics (though they don't usually have you memorize these) but none exists for quintics. It took mathematicians a while to prove that such a thing doesn't exist (how to prove a negative?) but they managed it by arguing that all such things have an underlying finite group smaller than a certain size and look, we've listed them here, and there's no such group corresponding to a quintic formula, therefore there is no quintic formula.
So they're useful as a sort of primordial complexity that can be referenced without extensive explanation since properties about the small ones can be checked by hand. And as it turns out, quite a lot of things form groups if you bother you look at them that way.
> Nobody bothers explaining why the primes are spaced like they are
I take it that means you also haven't taken (m)any number theory classes then ;-). Because people wildly care about that.
This is in a sense the background of another well-known conjecture, the Riemann conjecture...
Fair point, and correct (just one class). Myself, I'm fine with a field being important because well it's a mystery of the universe and well we just couldn't not scratch the itch. But I know that not all askers want answers like that, so I was attempting to dust off the "it's also useful" neurons :)
Many mathematical structures are groups so understanding them helps you get insight into the concrete situation you are trying to solve. For example is the problem I am seeing abelian? If so then it must look like X or Y.
This reminds me of the husband-wife duo of Patrick and Radhia Cousot, who together created Abstract Interpretation [1]. Useful technique, learned about it in my formal verification class.
Damn. That's some dedication. I really like the personal story, therein. You don't always see that, in STEM stuff.
I hope that their relationship deals well with the new reality, now that their principal goal has been achieved.
I think experts in all kinds of fields should write more about their thinking process. Just showing final result often feels like it was easy for them and discourages other from even attempting to understand and contribute. Especially in mathematics, the ideas are simple (they have to be because no one would be summing infinite series in their head like Ramanujan) but they are hidden behind lot of symbolic jargons.
It’s always less impressive to know how it works and fewer people care.
Here's an example of what I meant
Lecture by Douglas Hofstadter: Albert Einstein on Light; Light on Albert Einstein
I started "Prime Target" on Apple TV last night and I knew the premise of this story sounded familiar! The protagonist is obsessed over a prime number problem.
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?
The couple that maths together stays together.
This is a terrific article. It led me to a couple of hours tracking articles about related efforts, not the least of which was John Conway's work.
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.
“While working together on the McKay conjecture, Britta Späth and Marc Cabanes fell in love and started a family.”
They found their succession
Intersection set with genes from set A and B.
Wouldn't it be more like a subset of the union of A and B? Intersection set sounds like the child only has the genes that both parents share.
Way to go, Math!
What a mathive achievement!
This is about McKay conjecture
How do these kinds of advancements in math happen? Is it a momentary spark of insight after thinking deeply about the problem for 20 years? Or is it more like brute forcing your way to a solution by trying everything?
My experience from proving a moderately complicated result in my PhD was that it's neither. There wasn't enough time to brute force by trying many complete solutions, but it also wasn't a single flash of insight. It was more a case of following a path towards the solution based on intuition and then trying a few different approaches when getting stuck to keep making progress. Sometimes that involves backtracking when you realize you took a wrong path.
Yeah agreed - there are actually many, smaller flashes of insight, but most of them don't lead to anything. I once joked that you could probably compress all the time I was actually going in the right direction in my PhD down to about a month or two. That's a bit glib, often seeing why an approach fails gives you a much better idea of what a proof 'has to look like' or 'has to be able to overcome'. But many months of my PhD were working on complete dead-ends, and I certainly had a few very dark days because of that. Research math takes a lot of perseverance.
In this case, a ton of progress had already been made. The conjecture had been proved in some cases, reduced to a simpler problem in others. This couple went the last mile of solving the simpler problem in some particularly thorny cases.
You're really standing on the shoulders of giants when you rely on the classification of finite simple groups.
You're really standing on the shoulders of giants when you rely on the classification of finite simple groups.
Giants whose work has (dirty little secret) never truly been verified. The proof totals about 10,000 pages. At the end of the effort to prove it there were lots of very long papers, with a shrinking pool of experts reviewing them. There have been efforts to reprove it with a more easily verified proof, but they've gone nowhere.
Hopefully, the growing ease of formalization will lead to a verification some day. But even optimistically that is still a few years out.
> There have been efforts to reprove it with a more easily verified proof, but they've gone nowhere.
My understanding was that the so called "second generation proof" of the classification of finite simple groups led by Gorenstein, Lyons, Solomon has been progressing slowly but steadily, and only the quasithin case had a significant (but now fixed) hole. Are there other significant gaps that aren't as well known?
Huh. Looking at it, they have made a lot more progress than I was aware of. I will correct my opinion. https://mathoverflow.net/questions/114943/where-are-the-seco... was very informative on this.
Still they have a couple more books of proof left, and I have to wonder how carefully it will be reviewed. This will still be a massive improvement, but I'd be a lot happier if the entire proof could be formalized.
Plus there is still a possibility that there proves to be another significant hole.
If any theorem needs to be formalized, this is the one. No other theorem is this big, this hard to prove, and this important to get right.
At the moment we seem to have exactly one guy equipped to do this, and he'll be busy with something else for the next few years: https://www.reddit.com/r/math/comments/176vtju/kevin_buzzard...
The structure of DNA was seen in a vision in James Watson's dream. Some say it's subconscious problem solving and I think most down to earth people agree with that, but some less down to earth people will absolutely attribute it to god (I'm in the latter). If we were to entertain a silly proposition, something in the universe could just move our story along, all of a sudden. These paradigm shifts just seem to appear.
> The structure of DNA was seen in a vision in James Watson's dream
I believe this is apocryphal. Watson likely said this because he stole Rosalind Franklin's research.
I disagree. What he stole was the pictures she took. He really did come up with the structure.
Mind you he was helped by a nice coincidence. Franklin knew all 230 space symmetry groups, and so had to sort through them. Watson only really knew one - his PhD thesis was on a protein with the same group as DNA.
I can only think of that example because it’s the only one I know of where the scientist admitted to something divine.
This is probably something I need to research more, because the questions scientists are dealing with are too deep to ignore the one big question, and I wonder who struggled with it.
Not divine specifically but I like the Wikipedia List of Dreams: https://en.wikipedia.org/wiki/List_of_works_based_on_dreams#...
Subconscious problem solving is definitely a thing as far as I'm concerned.
It's happened several times that I struggled with a bug for hours, then suddenly came up with a key insight during the commute home (or while taking a shower or whatnot), while not actively thinking about the problem. I can't explain this any other way than some kind of subconscious "brainstorming" taking place.
As a side note, this doesn't mean the hours of conciously struggling with the problem were a waste. I bet that this period of focus on the problem is what allows for the later insights to happen. Whether it's data gathering that alows the insights to happen, or even just giving importance to the problem by focusing on it. Most likely it's both.
Subconscious synthesis of seemingly unrelated strands of thought is the basis of assessing an event's meaning, for me. A day or two will pass after the event, and I notice the meaning evolves even without new information and without conscious thinking about the event.
For me, the subconscious is the wellspring of sudden insights.
an alternate hypothesis is that all thinking is unconscious with consciousness being the sum of many unconscious processes. So, while it's true as you say "that some kind of subconscious "brainstorming" taking place", that would just true all the time, and you only notice it when your conscious thoughts are of something else.
(n.b. historically speaking in the literature, "subconscious" was the word used to describe Jung's "woo-woo" ideas about a collective subconscious shared across populations, and unconscious was the word used for ideas about a single brain a la Freud which is what we are talking about here)
With the right photo, this could be an Onion headline poking fun at lonely math nerds.
What an awful webpage. They hijack the click and right click actions. Can't triple click to select a paragraph. Can't drag selected text. Ugh
What browser are you working in? I'm not having any trouble in Brave, Chrome, or Opera.