I look forward to seeing your videos as much as I look forward to videos from 3Blue1Brown and StandUpMaths. Excellence work! Please keep making awesome maths content! (If you ever felt up to making videos about homotopic type theory, category theory, or geometric algebra, I would welcome them wholeheartedly, too. 😁 )
Thank you for the nice comments. You're in luck: A series on geometric algebra is planned for next year. For category theory, I would need a lot of assistance because I don't understand it very well. And homotopic type theory? No idea what that is. Is it related to type theory as in computer science?
I love how it's obviously MS paint from 6:00 onwards haha. But seriously, it's great that you provide so many visuals along with your detailed voice narration, even if the visuals are not super fancy. They're clear enough to explain the point but probably simple enough it allows you to produce your videos fast enough.
That's exactly right. Some youtube channels have amazing graphics, but they publish only every 2 months. I want to go faster than that, so I can't always make the visuals super fancy.
This has been fantastic series, as a particle physicist, I dream you extended this to group decomposition, Young Tableaux and weight diagrams... Those things scare me
Yeah, they scare me too 😆 Maybe you can get some useful information from this playlist by John Baez: ruclips.net/p/PLuAO-1XXEh0a4UCA-iOqPilVmiqyXTkdJ The first few videos talk about Young tableaux and their role in group classification. (Careful: The videos in the playlist may be in reverse order). I hope this helps!
20:10 I wonder how this relates to the generalized Stokes' theorem: which written in LaTeX is ```(\int_{\partial \Omega} \omega = \int_{\Omega} d\omega)```. It essentially says that in Calculus on smooth manifolds, the boundary is the opposite of the interior, and if you know one, you can figure out the other.
Good question! A homomorphism is a mapping between 2 monoids or groups. It satisfies an important property as explained in the video. An isomorphism is a homomorphism that goes both ways, so it's a bijection (which means it maps 1-on-1 in both directions). The word "morphism" is just a term that covers both homomorphisms and isomorphisms. This term is not often used in group theory, I stole it from category theory. I hope that clarifies things!
I found this series incredibly easy to follow, so thank you, but I always find it difficult to understand the "why" of group theory. With e.g. calculus or linear algebra, I can really intuit why it's useful from its applications - but with group theory I'm not sure what its applications are. Were the pioneers of group theory trying to solve a certain class of problems that demanded the theory? Or is this form of abstraction more about trying to understand the fundamentals of something? I'm not trying to impose an application of the theory if its not merited, but I always find it hard to motivate myself to learn more when I can't see them clearly. Thanks again!
Thank you for this excellent question! Group theory has many applications, and most of them are centered around symmetries. For example, a simple physical or chemical system like a molecule can exhibit a number of symmetries. Think of a hexagonal benzene molecule. If you rotate it over the correct angle, it stays exactly the same. These symmetries can be used to simplify equations and models, greatly reducing their complexity. But here is an even deeper application: the symmetries of the universe dictate certain properties of the laws of nature. See the series on linear algebra, and specifically video number 6, for some very good examples.
Sorry for the bother but, would it be feasible for you to provide proper subtitling? Asking since the autogenerated ones often fail to transcribe what was said properly, or just drops entire sentences, and consequently I'm forced to try to reconstruct what might have been said based on what the visuals are and what remains of the text, if any of it does.
I appreciate your candor, and I have empathy for your situation. The problem is that creating explicit subtitles is a lot of extra work, and we just don't have the resources to do that at the moment. We're only a small team and we already have our hands full with the research, writing the scripts, producing the animations, recording the audio, etc. Maybe we could figure out a way to outsource the production of the subtitles, but it will probably not happen anytime soon.
@@paradox9551 Click on video settings gear -> Subtitles -> Add translation. For some people on some videos it might be absent and I don't know the logic behind it. So if you are lucky enough you could provide translation and if author of video approves it it will be displayed to the rest of the viewers.
@@paradox9551 I have looked into the process of subtitling in a bit more detail, but I still think it would take too much work on our side, even if someone else provides the input text. Still, I really appreciate your offer. Thank you. We may get back to this in the future when we have more "room" in our process.
@@AllAnglesMath you can actually just take the autogenerated subtitles as a base and then fix any errors. this takes like ~an hour of work so it's not much effort
I noticed: you said matrix multiplication isn't commutative, then later at the end of the video you said linear transformations and matrices act like homomorphisms for vector spaces. Does this imply that homomorphisms are not commutative, in the case of monoids and groups?
That's a good question. First of all, the only similarity between matrices and morphisms is that they both preserve some kind of structure. But the structure of a monoid is very different from the structure of a vector space. So you shouldn't think that matrices "are" homomorphisms in the formal sense. It's only an analogy. Commutativity only applies to binary operations such as addition or multiplication (of numbers, matrices, or other objects). The operation must be binary, which means that it must take 2 inputs. It is commutative if you can swap those 2 inputs around. A homomorphism is not a binary operation, it's a mapping from one set to another. So the idea of commutativity does not apply to it. But then it gets a bit confusing, because category theory sometimes uses the phrase "commuting diagram" to talk about diagrams that "close up" in a nice way. And the definition of a homomorphism happens to be such a diagram. Please don't confuse this phrase "commuting diagram" with the commutativity property for operations. Yeah, I know, mathematicians are not always the best at coming up with good names for things...
@@AllAnglesMath thanks for the info btw i think i meant to ask about the commutativity of composition of homomorphisms and not the homomorphisms themselves, oops sorry about that
I think that the fact that you can "know" the internals of a group from its homomorphisms isn't so magic once you realize it's the groups own internal structure the determines which homorphisms are possible from it to other groups.
Is there any math structures above the rings where you add extra N binary operation with new identity element and the identity element of n-1 binop becomes void for nth binop?
You could, though rings are motivated by the existence of the real numbers (it basically acts like a number system) so if you could show that such a structure has use then by all means, go ahead!
@@enpeacemusic192 flips and turns not really motivated by numbers, but simply isomorphic to them. I've seen the dude who wrote his own math solver which helped him to discover existence of semirings with 2+2=5 and if I remember correctly up to size 6 there are over 100 such weird things. So my question was is it possible in principle or there is some contadiction arise at some point, like it happens with solving quintics in general form. So I wanted to cut the corner and find anyone who prolly tried to construct such hyperthing.
This series has helped me actually understand some ideas I've been wrestling with for some time. Sincerely, thank you.
Glad I could help.
I look forward to seeing your videos as much as I look forward to videos from 3Blue1Brown and StandUpMaths. Excellence work! Please keep making awesome maths content!
(If you ever felt up to making videos about homotopic type theory, category theory, or geometric algebra, I would welcome them wholeheartedly, too. 😁 )
Thank you for the nice comments.
You're in luck: A series on geometric algebra is planned for next year.
For category theory, I would need a lot of assistance because I don't understand it very well. And homotopic type theory? No idea what that is. Is it related to type theory as in computer science?
i can hardly wait for the next subjects!
I love how it's obviously MS paint from 6:00 onwards haha. But seriously, it's great that you provide so many visuals along with your detailed voice narration, even if the visuals are not super fancy. They're clear enough to explain the point but probably simple enough it allows you to produce your videos fast enough.
That's exactly right. Some youtube channels have amazing graphics, but they publish only every 2 months. I want to go faster than that, so I can't always make the visuals super fancy.
the zero proof is craaazy 🤯🤯
This has been fantastic series, as a particle physicist, I dream you extended this to group decomposition, Young Tableaux and weight diagrams... Those things scare me
Yeah, they scare me too 😆
Maybe you can get some useful information from this playlist by John Baez: ruclips.net/p/PLuAO-1XXEh0a4UCA-iOqPilVmiqyXTkdJ
The first few videos talk about Young tableaux and their role in group classification. (Careful: The videos in the playlist may be in reverse order).
I hope this helps!
You are doing an incredible job, great work!
Thank you!
20:10 I wonder how this relates to the generalized Stokes' theorem: which written in LaTeX is ```(\int_{\partial \Omega} \omega = \int_{\Omega} d\omega)```. It essentially says that in Calculus on smooth manifolds, the boundary is the opposite of the interior, and if you know one, you can figure out the other.
Is homomorphism the same as morphism? or do they mean different things
Good question!
A homomorphism is a mapping between 2 monoids or groups. It satisfies an important property as explained in the video.
An isomorphism is a homomorphism that goes both ways, so it's a bijection (which means it maps 1-on-1 in both directions).
The word "morphism" is just a term that covers both homomorphisms and isomorphisms. This term is not often used in group theory, I stole it from category theory.
I hope that clarifies things!
Thank you!
this is good fucking stuff
I found this series incredibly easy to follow, so thank you, but I always find it difficult to understand the "why" of group theory.
With e.g. calculus or linear algebra, I can really intuit why it's useful from its applications - but with group theory I'm not sure what its applications are. Were the pioneers of group theory trying to solve a certain class of problems that demanded the theory? Or is this form of abstraction more about trying to understand the fundamentals of something?
I'm not trying to impose an application of the theory if its not merited, but I always find it hard to motivate myself to learn more when I can't see them clearly.
Thanks again!
Thank you for this excellent question!
Group theory has many applications, and most of them are centered around symmetries. For example, a simple physical or chemical system like a molecule can exhibit a number of symmetries. Think of a hexagonal benzene molecule. If you rotate it over the correct angle, it stays exactly the same. These symmetries can be used to simplify equations and models, greatly reducing their complexity.
But here is an even deeper application: the symmetries of the universe dictate certain properties of the laws of nature. See the series on linear algebra, and specifically video number 6, for some very good examples.
Sorry for the bother but, would it be feasible for you to provide proper subtitling? Asking since the autogenerated ones often fail to transcribe what was said properly, or just drops entire sentences, and consequently I'm forced to try to reconstruct what might have been said based on what the visuals are and what remains of the text, if any of it does.
I appreciate your candor, and I have empathy for your situation.
The problem is that creating explicit subtitles is a lot of extra work, and we just don't have the resources to do that at the moment. We're only a small team and we already have our hands full with the research, writing the scripts, producing the animations, recording the audio, etc.
Maybe we could figure out a way to outsource the production of the subtitles, but it will probably not happen anytime soon.
Hey there, I'm willing to volunteer to write subtitles. How can I help?@@AllAnglesMath
@@paradox9551 Click on video settings gear -> Subtitles -> Add translation. For some people on some videos it might be absent and I don't know the logic behind it. So if you are lucky enough you could provide translation and if author of video approves it it will be displayed to the rest of the viewers.
@@paradox9551 I have looked into the process of subtitling in a bit more detail, but I still think it would take too much work on our side, even if someone else provides the input text. Still, I really appreciate your offer. Thank you. We may get back to this in the future when we have more "room" in our process.
@@AllAnglesMath you can actually just take the autogenerated subtitles as a base and then fix any errors. this takes like ~an hour of work so it's not much effort
I noticed: you said matrix multiplication isn't commutative, then later at the end of the video you said linear transformations and matrices act like homomorphisms for vector spaces.
Does this imply that homomorphisms are not commutative, in the case of monoids and groups?
That's a good question.
First of all, the only similarity between matrices and morphisms is that they both preserve some kind of structure. But the structure of a monoid is very different from the structure of a vector space. So you shouldn't think that matrices "are" homomorphisms in the formal sense. It's only an analogy.
Commutativity only applies to binary operations such as addition or multiplication (of numbers, matrices, or other objects). The operation must be binary, which means that it must take 2 inputs. It is commutative if you can swap those 2 inputs around.
A homomorphism is not a binary operation, it's a mapping from one set to another. So the idea of commutativity does not apply to it.
But then it gets a bit confusing, because category theory sometimes uses the phrase "commuting diagram" to talk about diagrams that "close up" in a nice way. And the definition of a homomorphism happens to be such a diagram. Please don't confuse this phrase "commuting diagram" with the commutativity property for operations. Yeah, I know, mathematicians are not always the best at coming up with good names for things...
@@AllAnglesMath thanks for the info
btw i think i meant to ask about the commutativity of composition of homomorphisms and not the homomorphisms themselves, oops
sorry about that
I think that the fact that you can "know" the internals of a group from its homomorphisms isn't so magic once you realize it's the groups own internal structure the determines which homorphisms are possible from it to other groups.
Is there any math structures above the rings where you add extra N binary operation with new identity element and the identity element of n-1 binop becomes void for nth binop?
So the next level up would be an operation with 1 as its absorbing element? Well, 1 to any power is 1 so maybe powers?
@@koenvandamme9409 it fails the requirement to have different identity for new operation though.
@@DeathSugar True. I don't have an alternative idea yet.
You could, though rings are motivated by the existence of the real numbers (it basically acts like a number system) so if you could show that such a structure has use then by all means, go ahead!
@@enpeacemusic192 flips and turns not really motivated by numbers, but simply isomorphic to them. I've seen the dude who wrote his own math solver which helped him to discover existence of semirings with 2+2=5 and if I remember correctly up to size 6 there are over 100 such weird things. So my question was is it possible in principle or there is some contadiction arise at some point, like it happens with solving quintics in general form. So I wanted to cut the corner and find anyone who prolly tried to construct such hyperthing.
video seems to lack geometry.