"... 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)
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.
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.
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! 🤯
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?
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.
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.
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.
"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 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.
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.
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"
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.
![NOSFERATU - Official Trailer [HD] - Only In Theaters December 25](http://i.ytimg.com/vi/nulvWqYUM8k/mqdefault.jpg)
You, sir, are what makes RUclips great.
"... 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)
Does it tend to 203125
It is like listening to mathematical zen master... Thanks for enriching the world with your efforts
Absolutely loved this, reminded me why I find math meaningful and gave me a new reading list
Thank you for this amazing lecture
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?
It amazes me that the Mandelbrot set is used for so many things. Even the spots on a leopard.
I’m the same. I got sucked in by the Monster Group and E8. It’s absolutely fascinating.
@@gx2music best of fun with that
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.
It looks like there is an error on the slide at 52:04. The forth coefficient of the j-function should probably be 21493760 not 21493769.
You are correct: this is indeed a typo.
Very comprehensible talk :)
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.
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.
if the Mathieu groups "live inside" the Conway group, how can the Conway group be simple ?
A simple group can have subgroups (besides itself and the trivial group) as long as they aren't normal subgroups.
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! 🤯
go off king
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?
I'm not sure, but Brauer's paper on it is "On groups of even order" doi.org/10.2307/1970080
Wow. Such a great lecturer. I’ve always preferred analysis over algebra but his videos are on the way to change that
Update: definitely prefer algebra now. He’s converted me 😂
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.
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.
Thank you so much for this talk.
Brilliant, thank you
Awesome!
ye
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.
"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 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.
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.
40:00 The number of cannonballs are correct for a square pyramidal pile, but this is clearly a tetrahedral pile! :)
The symmetry group of the tennis ball is Z2 x Z2, right?
Thankyou
Very enlightening and interesting 🤔 👍
So glad I found you channel.
nicely described :)
I've been pronouncing Mathieu all wrong: Math eye U.
Mathieu is French, so his name should be pronounced a-la French: Matt-yeu
53:11 "Actual physics" :-)
Excellent presentation Sir .Plz suggest a proper.book of Group theory .DrRahul Rohtak Haryana India
It strongly depends on what level you are asking for. Rotman's book "An introduction to the theory of groups" is a good starting point.
@@rgicquaud Sir Is the book by JAGallian proper ? Iam asking for an advanced book on Group theory. DrRahul Rohtak Haryana India
@@dr.rahulgupta7573 I don't know the book you mention. I am not specialized in group theory. Rotman's book is roughtly a graduate book on group theory
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"
@@rgicquaud Thanks Sir for your kind information.
Video anniversary!
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.
the stone in 03:14 looks a bit suspicious