"... and then there's one other group called the quarternion group which is a counterexample to almost anything". I just love that remark. True for basically all finite subgroups of SU(2)
I like how the Royal navy was interested in cannonball packings because of space reasons, but cannonballs are still the main metaphor for adding squares.
these groups are the most bizarre objects i have ever heard of, having heard of them i have not since been able to stop thinking about them. Why are there not more people talking about this?
That they exist is amazing. But once they are there, the fact that there are no more of them and that we can know that seems absolutely mesmerizing to me.
Draw out stuff like A=(1, 2, 3, …, 10, 11) and B=((3, 7, 11, 8)(4, 10, 5, 6)) and apply A and/or B to a list and draw out what states (lists) you end up with as a graph or digraph, where edges mean applying A or B takes you to the resulting list. Different sets of permutations (other than our choice and A and B and for there to be exactly 2 of them) result in some nice somewhat uniform looking graphs, but they're strange. They contain lots of partial cubes, octahedrons, icosahedrons, truncated cubes and the such, with a few branching parts and a few huge cycles. They're hellishly hard to calculate and draw, much of the problem is that you start off not knowing what you're drawing.
Thank you so much for sharing your knowledge with all of us, what a gift. I started watching your Galois theory lectures and only after recently watching 3b1b's video on the monster group did I make the connection that you were THE Richard Borcherds! 🤯
Spent several hours trying to learn about this today, I think the most fascinating thing I've heard so far is how string theory is useful in mathematics and related to this.
The book(s) by Aschbacher and Smith that takes care of a case that had been overlooked runs 1300 pages. I asked Aschbacher why these proofs are so long; he said because they are hard. But it must also be that there is an immense amount of calculation of cases involved.
16:30 shows a slide claiming that the smallest non-cyclic group is the icosahedral group of order 60 but I think the smallest non-cyclic group is actually the dihedral group of order 6?
I actually think a potatoe is much prettier than a crystal. Something highly symmetrical is almost always the same whatever way you rotate/reflect/transform it. Somewhat boring. Contrast a potatoe: always different and unpredictable however you put it. So it probably depends on your psychological disposition whether you esthetically prefer symmetry over asymmetry.
Where did the idea of looking at the centralizer of involutions to study finite simple groups come from? Like what does it tell you over other things like representations and the like? And what drew Brauer to it?
There are non-living metal-phosphorus compounds that have an icosahedral structure. It could be the seed on which the radiolarian grows it's structure. In fact many non-living metallic compounds resemble living structures. Who knows, it might be the framework life has evolved to use, to grow itself on.
Maybe a uniform shape like that requires only a very short, simple, and easy to copy molecular description. Maybe it was advantageous to be covered in folds/ridges and the physics on that scale make a more chaotic scrunched up pattern harder to achieve.
"abstract polytopes" have a single "-1" dimensional feature that is touching all the points (zero dimensional features) in the space, not sure if that helps though, unless you say that there's just one "-1" dimensional vector and it points everywhere
The evolution of structurally advantageous angles would probably favor the formation of the most evolved platonic solid. Maybe the same reason why viruses take the same form, even though they aren't perfect icosehedron. Very interesting to consider.
Actually Rotman's book covers group theory only while this book covers extra subjects. None of those cover representation theory that you might find in e.g. the book of Fulton and Harris "Representation theory: a first course", and also the two books by A. Knapp "Basic algebra" and "advanced algebra"
