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After 20 years, math couple solves major group theory problem
- thih9 2y agoThis is about McKay conjecture https://en.m.wikipedia.org/wiki/McKay_conjecture https://en.m.wikipedia.org/wiki/McKay_conjecture
- alberto_ol 2y agoNon mobile link https://en.wikipedia.org/wiki/McKay_conjecture https://en.wikipedia.org/wiki/McKay_conjecture
- deleted 2y ago[deleted]
- ChrisMarshallNY 2y agoDamn. 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.
- nthingtohide 2y agoI 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.
- grandempire 2y agoIt’s always less impressive to know how it works and fewer people care.
- nthingtohide 2y agoHere's an example of what I meant Lecture by Douglas Hofstadter: Albert Einstein on Light; Light on Albert Einstein https://youtu.be/ePA1zq56J1I?si=7WuiRa3GNSmMXArZ https://youtu.be/ePA1zq56J1I?si=7WuiRa3GNSmMXArZ
- deleted 2y ago[deleted]
- lo_fye 2y agoWay to go, Math!
- socceroos 2y agoWhat a mathive achievement!
- yamrzou 2y ago“While working together on the McKay conjecture, Britta Späth and Marc Cabanes fell in love and started a family.”
- Out_of_Characte 2y agoThey found their succession
- anthk 2y agoIntersection set with genes from set A and B.
- inglor_cz 2y agoWith the kids being homeomorphic to the parents.
- echoangle 2y agoWouldn'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.
- anthk 2y agoTrue; that would be dangerously close to incest. Subset of a union, then.
- veidr 2y agoFFS you nerds, get a room (and then form a working group to devise an algebraically provable disincentive to incestuous calculation)
- Out_of_Characte 2y agoWay ahead of you, pal. I am my own grandpa. https://youtu.be/eYlJH81dSiw?si=NNPDjaWV-OzrP2JU https://youtu.be/eYlJH81dSiw?si=NNPDjaWV-OzrP2JU
- charlangas 2y agoHow 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?
- getnormality 2y agoIn 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.
- btilly 2y agoYou'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.
- isotypic 2y ago> 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?
- btilly 2y agoHuh. 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-second-and-third-generation-proofs-of-the-classification-of-fin 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.
- Jun8 2y agoHah, 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?
- Chinjut 2y agoDon'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.
- Imustaskforhelp 2y agoDon'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.
- jandrese 2y agoDefinitely 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.
- grandempire 2y agoMany 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.
- __MatrixMan__ 2y agoI'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.
- w0de0 2y agoWith the right photo, this could be an Onion headline poking fun at lonely math nerds.
- deleted 2y ago[deleted]
- PhillyPhuture 2y 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.
- Der_Einzige 2y agoUnfortunately 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.
- egoism 2y agore: Your Comments of Hegel elsewhere. your name is reference to Stirner is it not? Hegel's realm of ideality is just that, a realm of ideality. Hegel's totality as well, a spiritual abstraction upon the world, but a good one to deny the Abrahamics and other mental rots of the west. The irony of totality is also, that it resolves back to the self, the process of becoming, the unique. Dialectical analysis is a never ending, self-reflected, and interpenetrated spiral toward an asymptotic singularity.... but that is exactly what being is. Some X posited/constructed condition is always appearing, but X is not only X, the synthesized revelation of these arise in a new more real truth, along with a final truth that this process leads back, to some condition is always appearing.
- mkzetta 2y agoHow can you claim to know where it ends? This itself sounds like mental rot.
- veidr 2y agoIsn'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...)
- isaacfrond 2y agoSo 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.
- antonf 2y agoDid you miss definition of G, or I didn’t have enough caffeine yet?
- throwpoaster 2y agoI assume from context it's the original group, but I'm only a cup deep.
- looofooo0 2y agoCorrect
- jmount 2y ago> 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.
- MrMcCall 2y agoAll 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
- xenago 2y agoWhat 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
- PunchTunnel 2y agoWhat browser are you working in? I'm not having any trouble in Brave, Chrome, or Opera.
- xenago 2y agoEdge and Firefox both replicate the issue. It is on all browsers
- Imustaskforhelp 2y agoon zen , it works
- forgotpwd16 2y agoHeh, have read Quantum's articles before but this is the first time I noticed it when selected "McKay conjecture" and couldn't right-click to search. Can copy with C-c though. (Or can use an extension/userscript/bookmarklet to allow right-click.)
- justinpombrio 2y agoThe paper: https://arxiv.org/abs/2410.20392 https://arxiv.org/abs/2410.20392
- rurban 2y agoTheir proof: https://arxiv.org/abs/2410.20392 https://arxiv.org/abs/2410.20392 (2024)
- wglb 2y agoThis 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.
- racl101 2y agoThe couple that maths together stays together.
- JayStavis 2y agoI 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?
- fazkan 2y agoThis 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. [1] https://en.wikipedia.org/wiki/Abstract_interpretation https://en.wikipedia.org/wiki/Abstract_interpretation
- markisus 2y agoIt’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.
- jlo2365 2y agoI 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.