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@positivelyprime
The most ~~*beautiful*~~ physics and maths ~~*equations*~~

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I will forever maintain that writing e^πi + 1 = 0 is the mathematical equivalent of putting ketchup on wagyu beef.
e^πi = -1 is beautiful and geometric and nice why would you obscure the actual meaning by rewriting it as e^πi + 1 = 0 just to forcefully cram some extra numbers you like in there!! you're hurting the equation please stop!!!
Does anyone have any recommendations for books which deal with the general theory of the representations of compact Lie groups and their Lie algebras?
Before recommending anything I want to say that to properly understand the general theory for semisimple Lie algebras, it is very useful to treat the sl_2 = SU(2) case (and sl_3 = SU(3) case really) in detail, since sl_2-triples are a very important tool in this case. As such many books include these as specific examples, sometimes entire chapters.
Hall's Lie groups, Lie algebras and representations is good as an elementary introduction. However, it focuses (understandably) on matrix Lie groups which means that the theory is presented in a slightly less general way ar first, although towards the end of the book this is rectified I think. The book does treat sl2 and sl3 quite nicely iirc. Would recommend for a first meeting, but not if you know more.
Duistermaat and Kolk Lie groups is very geometric, too geometric for me, but depends on your background and purpose. It definitely talks about representation theory, but not always in a clear way if you do not already know the concepts. I distinctly remember associated vector bundles being called something different which confused me at first, skill issue though.
Fulton and Harris' Representation theory is a classic, and very good, book on representation theory. Especially if you care about the Lie algebra side of things more it is great (together with Humphrey's, although I havent studied that book much). However, since it deals with quite a lot of general theory, it can be hard to distinguish the essential features of the theory of cpt Lie groups. For example, I don't think the Peter-Weyl theorem is even mentioned? Even though for my applications that is one of the main reasons to care about representations.
Bump's Lie groups is also a good book, but it assumes basic knowledge of representation theory of cpt Lie groups in all but name. This can be supplemented by Hall though. For example, I think Peter-Weyl is proven around page 20 or so lol. So, better if you already know a bit imo.
This post tuned out longer than expected but I hope it is useful!
Thank you for the detailed response!
To give some more context, I have done some representation theory of Lie groups and Lie algebras before and it did mostly stick to the representation theory if sl₂ℂ but we did do a bit about sl₃ℂ (though I intend to refresh that stuff cause it's been over a year now since I looked at this stuff).
Also I'm approaching this from the view of equivariant homotopy theory. Here, knowledge of the closed subgroups of the (compact) Lie group is important to things (there's actually a post I'm working on where I explain one such instance of this actually). So if there's some sort of general theory about understanding what subgroups can show up in a compact Lie group would be helpful to know (if there even is such a theory?).
Also the reason I'm interested in representations is again they play an important role in the structure of stable equivariant homotopy theory. So having a general framework for understanding the representation theory of compact Lie groups would be really helpful, cause we didn't really get to that point in my undegrad course. (Explanation for why they're important will be under the cut for anyone interested)
The point of this is to make clearer my reasons for wanting to learn more about Lie groups in the hopes that it might make the suggestions more narrow if you have the time?
That sounds interesting! I am not familiar with genuine homotopy theory, let alone G-equivariant homotopy theory, so I cannot really help you much in that regard.
Regarding subgroups: perhaps you can look at the Frobenius reciprocity theorem if you do not already know about this. For H a closed subgroup of cpt (connected) G this establishes an adjunction between the restriction functor and induced representation functor. For finite groups this is even a two way adjunction (i.e. one is both the left and right adjoint of the other) but for cpt Lie groups this is not the case anymore iirc, so be careful there. In the context of Lie groups, induced representations can be seen as smooth sections of a vector bundle over G/H (if I'm not mistaken) so this also has a more geometric meaning. I do not know if this can be used to classify or characterise all the closed subgroups in some meaningful way, but it is a big result for closed subgroups of Lie groups.
Alternatively, and much weaker, there is the maximal torus subgroup. This helps the study of compact (connected) Lie groups and is related the the maximal abelian subalgebra of the associated Lie algebra. There are some results that every g in G is conjugate to some element in the torus, for example. So this probably does not help to study all the closed subgroups, but at least it is a very important example of a closed subgroup.
Does anyone have any recommendations for books which deal with the general theory of the representations of compact Lie groups and their Lie algebras?
Before recommending anything I want to say that to properly understand the general theory for semisimple Lie algebras, it is very useful to treat the sl_2 = SU(2) case (and sl_3 = SU(3) case really) in detail, since sl_2-triples are a very important tool in this case. As such many books include these as specific examples, sometimes entire chapters.
Hall's Lie groups, Lie algebras and representations is good as an elementary introduction. However, it focuses (understandably) on matrix Lie groups which means that the theory is presented in a slightly less general way ar first, although towards the end of the book this is rectified I think. The book does treat sl2 and sl3 quite nicely iirc. Would recommend for a first meeting, but not if you know more.
Duistermaat and Kolk Lie groups is very geometric, too geometric for me, but depends on your background and purpose. It definitely talks about representation theory, but not always in a clear way if you do not already know the concepts. I distinctly remember associated vector bundles being called something different which confused me at first, skill issue though.
Fulton and Harris' Representation theory is a classic, and very good, book on representation theory. Especially if you care about the Lie algebra side of things more it is great (together with Humphrey's, although I havent studied that book much). However, since it deals with quite a lot of general theory, it can be hard to distinguish the essential features of the theory of cpt Lie groups. For example, I don't think the Peter-Weyl theorem is even mentioned? Even though for my applications that is one of the main reasons to care about representations.
Bump's Lie groups is also a good book, but it assumes basic knowledge of representation theory of cpt Lie groups in all but name. This can be supplemented by Hall though. For example, I think Peter-Weyl is proven around page 20 or so lol. So, better if you already know a bit imo.
This post tuned out longer than expected but I hope it is useful!
This Hank Green situation is kind of sad. Aside from the outdated beliefs about AI, it's clear that most of the aggrieved population have never done research. Like research research.
Everyone seems to think you can just... Google stuff? And it comes up? bruh the moment you get anywhere near the frontier of knowledge, accessibility craters. Hard.
A real life example: during my PhD, my whole (subsubsub-)field existed basically entirely because a certain sensible conjecture turns out to be false, so we need a weird subtle painful approach rather than just using the obviously true fact.
How do we know it's not true? A counterexample was given in 1913 in a paper written in french, which no one actually cites, has no existing english translation, is nigh impossible to find a pdf of (no one could tell me its name - only its author's name), and is over 100 pages long, seems to be (I think) concerned largely with only tangentially related stuff, and is impenetrably dense due to lacking many modern concepts and notations. No one I'd ever met had ever read it, let alone understood it. And no other published proof exists - that I know of. It is not a simple proof. I wouldn't be able to prove it myself.
One day, my supervisor and I were emailing a colleague, and I offhandedly mentioned that it was the "only known counterexample". The colleague responded - oh! That's not true. Here's a handful more.
He did not supply any proofs. My supervisor went and figured out and wrote down a proof for one, and satisfied himself. He did not publish it, and it is basically illegible. He tried to explain it to me, but it was unintelligible. I could not make heads or tails of it. I still don't know, myself, for absolute certain, that a counter-example even exists. I had all the access in the world to journals and leading researchers in the area and I still couldn't get access to a remotely legible explanation of the foundational fact that spawned my field.
Research is THICK with folklore, and often the only way to learn something vital is to be wrong in the right way in front of the right person at the right time.
Y'know what tool is capable of cutting through a huge proportion of that reliance on luck and the hundreds of hours that go into just trying to figure out what's known and where to find it (even for established researchers familiar with their field)? yuh. LLMs. and nowadays they're basically reliable. They check online. They check themselves. And they've read the entire literature and picked up on much of the unwritten folklore whose presence in the literature is only the shadow it casts.
You'd have to be insane not to be using them at this point. Even just to ask "is it known whether or not blah?". Genuinely an invaluable tool. You don't have to offload your cognitive. You don't have to sell your soul or publish slop. You can just... benefit. And people are coming thick and fast at Hank Green - who has contributed so, so much to humanity - for using LLMs. For research. For his videos.
Sad!
Perhaps this is slightly contained within math research and not as relevant in other sciences/humanities, but I'm slowly starting to slightly agree. At this point I think it is basically a given that LLMs will become very valuable in the near future for research, but it is perhaps easy to use it 'wrongly'. The reasons why a researcher would use an LLM (scouting the field for sources and folklore, concept testing for small but tedious calculations, perhaps proof checking?) are different from the interests of your average student (i.e. generating proofs for homework assignments), which I think muddles the discussion quite a bit.
However, I will add that in my experience ChatGPT is still quite prone to hallucinations. This makes it more of an unreliable tool to me with a high ceiling but also the potential to waste half an hour trying to convince you something works before admitting defeat.

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This Hank Green situation is kind of sad. Aside from the outdated beliefs about AI, it's clear that most of the aggrieved population have never done research. Like research research.
Everyone seems to think you can just... Google stuff? And it comes up? bruh the moment you get anywhere near the frontier of knowledge, accessibility craters. Hard.
A real life example: during my PhD, my whole (subsubsub-)field existed basically entirely because a certain sensible conjecture turns out to be false, so we need a weird subtle painful approach rather than just using the obviously true fact.
How do we know it's not true? A counterexample was given in 1913 in a paper written in french, which no one actually cites, has no existing english translation, is nigh impossible to find a pdf of (no one could tell me its name - only its author's name), and is over 100 pages long, seems to be (I think) concerned largely with only tangentially related stuff, and is impenetrably dense due to lacking many modern concepts and notations. No one I'd ever met had ever read it, let alone understood it. And no other published proof exists - that I know of. It is not a simple proof. I wouldn't be able to prove it myself.
One day, my supervisor and I were emailing a colleague, and I offhandedly mentioned that it was the "only known counterexample". The colleague responded - oh! That's not true. Here's a handful more.
He did not supply any proofs. My supervisor went and figured out and wrote down a proof for one, and satisfied himself. He did not publish it, and it is basically illegible. He tried to explain it to me, but it was unintelligible. I could not make heads or tails of it. I still don't know, myself, for absolute certain, that a counter-example even exists. I had all the access in the world to journals and leading researchers in the area and I still couldn't get access to a remotely legible explanation of the foundational fact that spawned my field.
Research is THICK with folklore, and often the only way to learn something vital is to be wrong in the right way in front of the right person at the right time.
Y'know what tool is capable of cutting through a huge proportion of that reliance on luck and the hundreds of hours that go into just trying to figure out what's known and where to find it (even for established researchers familiar with their field)? yuh. LLMs. and nowadays they're basically reliable. They check online. They check themselves. And they've read the entire literature and picked up on much of the unwritten folklore whose presence in the literature is only the shadow it casts.
You'd have to be insane not to be using them at this point. Even just to ask "is it known whether or not blah?". Genuinely an invaluable tool. You don't have to offload your cognitive. You don't have to sell your soul or publish slop. You can just... benefit. And people are coming thick and fast at Hank Green - who has contributed so, so much to humanity - for using LLMs. For research. For his videos.
Sad!
I am always conflicted about the Fields medal being called the "Nobel prize" of mathematics. They are more different than alike! The Fields medal is given only every 4 years to 4 young researchers (younger than 40), which is a completely different demographic than the actual Nobel prizes. The Fields medal is of course very prestigious still, but is not meant to be a "best math done this last decade" type of prize.
On the other hand, this does get the public more invested in mathematics! And it is wonderful that women are getting these prizes too (for being really good mathematicians!), forming role models for hopefully more gender diversity in math in the future!! All good things!! But the public doesn't actually understand the scope of the prize!!!
If you're ever struggling with math, Fields Medalist Yu Deng recommends yuri.
Im gonna pavlov myself into getting hard over operator theory
i think every textbook should have this
From Information Theory, Inference, and Learning Algorithms (McKay)

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remember: worst case scenario you die, and hell is real, and you go there
let f(x) be a polynomial with non-negative integer coefficients such that f(10) is prime. If all coefficients are ≤49598666989151226098104244512918, then f(x) is irreducible over Z[x]. Moreover, they proved that this bound is also sharp. In other words, coefficients larger than 49598666989151226098104244512918 do not guarantee irreducibility.
numbers are so fucking stupid man
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Anya is live and ready to show you everything. Watch her strip, dance, and perform exclusive shows just for you. Interact in real-time and make your fantasies come true.
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I made a Möbius transformation visualizer together with Taketo Sano in a hackathon(http://hackday.jp/), and we got GOLD prize!
Parabolic Möbius transformation has a fixed point, and other Möbius transformations have two fixed points. (cf.http://hyrodium.tumblr.com/post/138314686744)
Bring two fixed points of a Möbius transformation closer to a point, then the transformation changes into a parabolic Möbius transformation.
Clifford the big red algebra