"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.
> 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.
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.
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?
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?
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.
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?
After 20 years, math couple solves major group theory problem
(quantamagazine.org)437 points by isaacfrond 20 February 2025 | 121 comments
Comments
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.
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?
[1] https://en.wikipedia.org/wiki/Abstract_interpretation
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.
https://en.m.wikipedia.org/wiki/McKay_conjecture