alt Hacker NewsAfter 20 years, math couple solves major group theory problem
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> 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.
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.
The paper: https://arxiv.org/abs/2410.20392
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?
Their proof: https://arxiv.org/abs/2410.20392 (2024)
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.
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.
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.”
Way to go, Math!
What a mathive achievement!
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?
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
"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.