I’ve become very interested in the classification of finite simple groups. It would be super cool for you to explore this area more! For example, the monster group and it’s derivatives are really well explored typically. But the pariahs have basically no videos about their properties or even discovery. Would be cool to explore those ones.
I wonder if the classification of finite simple groups have an associated moduli space. Moduli spaces are solutions to geometric classification problems. A group is geometric by its group representation. One example is the dihedral group D_4 being symmetries of a square. In that way there would be a universal object (ie base space, classifying space, topos, or scheme, etc.) that would describe all finite simple groups. Maybe this would be of Lie type as most finite simple groups are in some way Lie?
Hmm, interesting question. I have never seen anything like that, but I somewhat can imagine a space which is mostly regular (Lie-type) and then has some weird isolated points (sporadic) 🤔
@@VisualMath Assuming every finite simple group have an associated a Lie algebra g of some algebraic group G, it would be possible to consider each group scheme of that algebraic group G. This Lie algebra may be semisimple which have their classification. Having a (group) scheme is the best case since the functor of points of that scheme describes a fine moduli space. For the infinite families of simple groups, I think the object that completely classifies each infinite family in this way would be an extra step needed before combining them in some way.
Hmm, how does that work in practice, say for the sporadic groups? The monster for example has an associated Lie algebra, but that one is insane and I cannot actually do anything with it.
@@VisualMath It’s using algebraic geometry which I struggle with. At least one approach is to find a scheme to describe all objects being classified. There is a way of “gluing” schemes together to form a larger schemes. It also possible to consider the group itself instead of its Lie algebra. Finite groups are algebraic groups. Every finite algebraic group is reductive. Using geometric invariant theory a moduli space can be constructed from a reductive algebraic group. This geometric invariant theory applies to all finite groups. On their own each finite simple group has an associated moduli spaces which classifies something.
@@Jaylooker Right. So the crucial question is "What do these module spaces (coming from the reductive groups) actually encode?". I have no idea; let me know if you have thoughts on this 🤔
Great video! Thank you. The visualization of J2 was especially helpful. Please make more videos and make them longer and explain more. I want to make some visualizations of the Monster group, but after seeing your video with J2, I see that the Monster must be even more complex, probably would just turn into a circle with dense lines in the middle? I've watched lots of videos about the Monster group, and yours is the only one to talk to me, not too dense, not to abstract and hand wavy. More videos please.
Thanks for all the feedback! A video about the monster itself sounds like fun; I will consider doing that. A graph for the monster is indeed large - it has 196883 vertices - and if you illustrate it looks like a mess 🤔But that shouldn't stop us doing it 😀
26
Well, I got the order of magnitude correct 😅
I’ve become very interested in the classification of finite simple groups. It would be super cool for you to explore this area more!
For example, the monster group and it’s derivatives are really well explored typically. But the pariahs have basically no videos about their properties or even discovery. Would be cool to explore those ones.
Large is always good 😄 so the monster is clearly overrepresented
But I like the idea, and will have a look into this. Thanks! ☺
I wonder if the classification of finite simple groups have an associated moduli space. Moduli spaces are solutions to geometric classification problems. A group is geometric by its group representation. One example is the dihedral group D_4 being symmetries of a square.
In that way there would be a universal object (ie base space, classifying space, topos, or scheme, etc.) that would describe all finite simple groups. Maybe this would be of Lie type as most finite simple groups are in some way Lie?
Hmm, interesting question.
I have never seen anything like that, but I somewhat can imagine a space which is mostly regular (Lie-type) and then has some weird isolated points (sporadic) 🤔
@@VisualMath Assuming every finite simple group have an associated a Lie algebra g of some algebraic group G, it would be possible to consider each group scheme of that algebraic group G. This Lie algebra may be semisimple which have their classification. Having a (group) scheme is the best case since the functor of points of that scheme describes a fine moduli space. For the infinite families of simple groups, I think the object that completely classifies each infinite family in this way would be an extra step needed before combining them in some way.
Hmm, how does that work in practice, say for the sporadic groups? The monster for example has an associated Lie algebra, but that one is insane and I cannot actually do anything with it.
@@VisualMath It’s using algebraic geometry which I struggle with. At least one approach is to find a scheme to describe all objects being classified. There is a way of “gluing” schemes together to form a larger schemes.
It also possible to consider the group itself instead of its Lie algebra. Finite groups are algebraic groups. Every finite algebraic group is reductive. Using geometric invariant theory a moduli space can be constructed from a reductive algebraic group. This geometric invariant theory applies to all finite groups. On their own each finite simple group has an associated moduli spaces which classifies something.
@@Jaylooker Right. So the crucial question is "What do these module spaces (coming from the reductive groups) actually encode?". I have no idea; let me know if you have thoughts on this 🤔
Great video! Thank you. The visualization of J2 was especially helpful. Please make more videos and make them longer and explain more. I want to make some visualizations of the Monster group, but after seeing your video with J2, I see that the Monster must be even more complex, probably would just turn into a circle with dense lines in the middle? I've watched lots of videos about the Monster group, and yours is the only one to talk to me, not too dense, not to abstract and hand wavy. More videos please.
also, consider a headset with microphone, it was a little hard for me to follow your speech sometimes with your very mild and pleasant accent
Thanks for all the feedback!
A video about the monster itself sounds like fun; I will consider doing that. A graph for the monster is indeed large - it has 196883 vertices - and if you illustrate it looks like a mess 🤔But that shouldn't stop us doing it 😀