While I frequent many other math channels, this is the one I credit with re-energizing me about my mathematical growth rather then just feeling entertained
I'm a string theorist so this video excited me but I had already learnt about the main topic. What I did learn however is how to write \mathfrak{a} on a blackboard. This is gamechanger for my future talks!
I understood almost all of the Witt algebra video but this one left me puzzled almost instantly. I think you'd have to break down a video like this even more so that a general audience of this channel could understand this topic.
Felt the exact same way. The Witt Algebra video was great, but this one seemed to require a much deeper background in abstract algebra than that video. It's a shame too because it seems really interesting!
Me too. I know (knew) most of the things in the first few minutes from regular finite linear (i.e. matrix) algebra, like kernel and in/surjection, but a bit more motivation where we're heading and reminder what it all is (non-native speaker here), and maybe some illustrations would have helped; in contrast, the various rapid-fire notations (fraktur font, arrows, iota) did not help me. 😢
I agree, interesting topic, but way too many memorized definitions needed without an example to get an idea of what he is talking about. Its just gibberish on the first pass, not good.
I think it is a nice goal do come back to this video some years from now and see how much more things makes sense. Really love the effort put into bringing that of an advanced topic to a larger audience. I couldn’t follow almost anything but learning has to be a challenge sometimes. I hope that I will learn enough to one fully absorb this video
this is absolutely wonderful!! I've been looking for content like this for YEARS-- I am *begging* you, please, keep up the series on abstract algebras like these!! I can't get enough!!
I've always felt that my school class feels like it's remedial, and so I extremely enjoy the way this introduces me new topics that I have to research.
I have ADHD, so I tend to do homework/ect. more slowly*, even if I understand it better. *on average I also have autism, but, to the extent that I have it, I consider it to be a personality trait rather than a "disability", unlike my ADHD. A lot of the people (~~50%-~~70%, I've heard) who are autistic, also have ADHD.
In bosonic string theory the commutator of the constraints satisfy the virosoro algebra with λ equal to D/12 where D is the dimensions of space-time ,which are 26 for bosonic string theory
Nice! After your first video I couldn't wait for the derivation of the Virasoro algebra, so I went ahead and studied the construction of central extensions using 2-cocycles on the underlying Lie algebra and ended up doing something equivalent to what you presented here. Your video gave me a better intuition on that stuff, and I'm definitely interested in seeing a followup video! Will you talk about Kac-Moody and current algebras at some point? That would be awesome! Thanks for your work!
Great video with a lot more substance than your usual ones! The audience might be more limited, but I definitely think this type of videos also belongs on your channel.
This is a really cool structure! Definitely had to watch the video twice though to get everything.. definitely excited about the follow up video. I did take intro to conformal field theory, but never learned too much about where the mathematical details of the structures we were using came from. So this video really let some puzzle pieces fall into place :D
Really big Thx, that's two really nice videos, I would guess for extra videos in this line: OPEs, Vertex algebras.. , but I would add also something in the direction of harmonic analysis e.g. how the conformal transform compares to fourier transforms, and how quantization can be compared. I see also your back ground on orbifolds can you do some stuff on moebius / modular /fucshian / klein / bianchi groups and how they attach back to harmonic analysis and quantization ? Finally of course still from your background some stuff about heisenberg algebras / Weyl quantization etc...
This video proved that I have not mastered the prerequisites. The only thing that would have made this understandable to me was if he said April’s Fools at the end. Congratulations to those that were able to understand this.
39:47 I could sense a subtle passive aggressive reference to numberphile when you said "very very famous formula" and "popularly written as" like that 😂
I would love a video on regularization. It seems like a really weird trick when it comes up in quantum mechanics and it confuses me how we can just ignore convergence.
Hi Prof Penn..Thanks for another great video endeavouring to explain very advanced concepts..I really enjoyed your Number Theory course..and currently going through the Abstract Linear Algebra..and Complex Analysis courses..Looking forward to when u make a video course on one of the Graduate courses that you teach..Thanks Again !
Awesome video! Definitely much more challenging to follow than most of the others here on the channel, but this is what math is all about. Can't wait for the next one especially due to the regularized sum of series of Naturals mentioned at the end. Remember myself encountering this sum for the first time in one of the third year pregraduated physics courses, and the shocked it deployed on me when I realized that a clearly diverging series not only has some application in physics but also has been attached a finite value, which is also a negative.
@@gennaroponsiglione1098 Yes, if you look at the full scale of difficulty, it might look like that: primary school < secondary school < undergraduate lecture < graduate lecture < expert research talk. You can find videos at all these levels on TH-cam. What is quite rare in my experience, however, is such a video about a topic that is interesting to people up to the research level but presented at an intermediate, say undergraduate, level of difficulty.
@edwin steiner Oh yes, if you mean an undergraduate level of an advanced topic I understand perfectly, is what makes this channel great, even if this time I had some difficulties
I'd be lying if i say I understood every step since I've never studied Lie algebra, but I got the rough idea of what was going on, which is cool. Main confusing bit was the tensor product/direct sums steps, since I'm not really sure how all that works. Interesting thanks.
Hey Michael, it seems to me that when you’re replacing f with f’, you’re subtracting off multiples of the basis vectors L_m from your c. And in that way you can be sure that your algebra is still isomorphic. But it seems like you’re making an infinite number of choices for g, which means you’re subtracting off an infinite number of vectors from c to get it into that standard form. My question is: how is this allowed? We’re just working with a plain infinite-dimensional space, no topology, infinite sums aren’t well-defined. Is there some dependence condition that forces only finitely many of the L_m to need to be subtracted?
I don't think your construction gives an isomorphism: if you subtract (multiples of) Lm from the central charge c, the resulting element won't be in the center. Instead I imagine that a multiple of c is subtracted from every Lm (something along the lines of Km=Lm±g(Lm)c). Since every equation features finitely many Lm no infinite sums would arise.
@@ichtusvis Ah, this makes sense. I should admit that I didn’t do the math (obviously), but it does make sense that we’re choosing lifts of the L_m to g’, instead of choosing the c differently. I also agree that changing c would jeopardize the centrality condition as well.
As a PhD student on mathematical string theory I love this video, even though I have already learn most of this stuff, i am really looking forward on you going even deeper on the algebra of superstrings or even M-theory and beyond! :)
Fantastic video! However, unless I've misunderstood something, the case lambda=0 is not isomorphic to other cases. So should the definition of Virasoro algebra be the unique _non-trivial_ central extension of the Witt algebra?
I really enjoyed this video, is the follow-up available? Thanks a lot for all the effort you are putting in your channel and sharing your knowledge with the rest of the world
Normally, I can get behind how most your videos work, but this is still too complex for me, I will be happy to wait for the next video instead of watching this to the end
you wrote "The.".. on the first entry. then I was lost in the language I did not understand. But your explanation was so convincing and obvious is was very interesting..
Edit: I forgot the impulse function has _area_ 1, not magnitude 1. Disregard. -Wait, we use the Dirac delta function instead of the impulse function, even though we're comparing to zero? I'm guessing that's because the Dirac delta is used all over the place and the impulse function isn't?-
One question. I studied short exact sequences of modules and it's not true in general that the middle element is a direct sum of the other two. When that happens, the sequence is called split exact. In your case, because you're working with Lie algebras, every short exact sequence splits because they are vector spaces? Is my reasoning correct?
From what I could tell, he’s just saying that every SES of VECTOR SPACES splits. It gives him a nice way to write the vectors of the extension, as v1+v2 where v1 is in the ker and v2 is in the coker, so as to be able to define the Lie bracket, because you thus only need to define [v1,w2] (as [v1,w1] and [v2,w2] already are defined from the original algebras).
Another great video on a deep topic! Is there a reason that one-dimensional central extensions are of particular interest instead of other kinds of extensions, or higher-dimensional central extensions?
I don't know if you will do the sequel, which I would like to see, but can't you please add some literature reference when doing the kinda advanced stuff? That would be great :)
12:30 why is multiple of new vector depending on f(x, y)? Also, more than lambda being 1/12 , I want to understand, what do you mean by it being absorbed in the basis vectors of C?
I really love your videos. Thank you for making them. I don't know if you like to get questions here. In this video, you are talking about constructing a Lie algebra structure on g' given algebras a and g but if the sequence is a short exact sequence of Lie algebras, then g' must already have a bracket, right?
I recently came to know to know that Ramanujan's theta function plays a huge role in string theory .Can you make a video about that ???.It will really be nice if you can make one .
I am interested in string theory and want to know how Ramanujan used the q-series and derived 24th power of q-series, please. If you can please derive the partition function showing how to count all the photons in the universe. Thank you.
I really like that this channel is making excursions into more "exotic" topics. Amount of knowledge and ideas I got from it is invaluable.
Facts
While I frequent many other math channels, this is the one I credit with re-energizing me about my mathematical growth rather then just feeling entertained
I kove these videos! I shall return to this video later, when I have the time to throughly investigate and research the nuances, topics, and tangents.
I'm a string theorist so this video excited me but I had already learnt about the main topic. What I did learn however is how to write \mathfrak{a} on a blackboard. This is gamechanger for my future talks!
Yes, we would like a video on Kac-Moody algebras! Great video btw.
Finally someone is going to explain how the - 1/12 thing gets used!
I understood almost all of the Witt algebra video but this one left me puzzled almost instantly. I think you'd have to break down a video like this even more so that a general audience of this channel could understand this topic.
Felt the exact same way. The Witt Algebra video was great, but this one seemed to require a much deeper background in abstract algebra than that video. It's a shame too because it seems really interesting!
Me too. I know (knew) most of the things in the first few minutes from regular finite linear (i.e. matrix) algebra, like kernel and in/surjection, but a bit more motivation where we're heading and reminder what it all is (non-native speaker here), and maybe some illustrations would have helped; in contrast, the various rapid-fire notations (fraktur font, arrows, iota) did not help me.
😢
I agree, interesting topic, but way too many memorized definitions needed without an example to get an idea of what he is talking about. Its just gibberish on the first pass, not good.
Agreed. And ending on sum natural numbers = 1/12 and throwing that into the equation... is that what string theory is based on???
@vincent button To be honest, I never gave this video a second thought, or made any attempt to make sense of it. The video kind of failed I guess.
It might be interesting to make a video detailing how this Virasano algebra relates to vertex algebras.
I have big plans for VOAs on youtube in the future...
@@MichaelPennMath eyes emoji
I think it is a nice goal do come back to this video some years from now and see how much more things makes sense.
Really love the effort put into bringing that of an advanced topic to a larger audience. I couldn’t follow almost anything but learning has to be a challenge sometimes. I hope that I will learn enough to one fully absorb this video
This has to be one of the best videos on TH-cam. It's certainly one of the best maths/physics videos. I'd love to see the follow-up!
this is absolutely wonderful!! I've been looking for content like this for YEARS-- I am *begging* you, please, keep up the series on abstract algebras like these!! I can't get enough!!
I've always felt that my school class feels like it's remedial, and so I extremely enjoy the way this introduces me new topics that I have to research.
I have ADHD, so I tend to do homework/ect. more slowly*, even if I understand it better.
*on average
I also have autism, but, to the extent that I have it, I consider it to be a personality trait rather than a "disability", unlike my ADHD. A lot of the people (~~50%-~~70%, I've heard) who are autistic, also have ADHD.
In bosonic string theory the commutator of the constraints satisfy the virosoro algebra with λ equal to D/12 where D is the dimensions of space-time ,which are 26 for bosonic string theory
Loved the Witt algebra video and this video, think it's safe to say I'm happy with any further videos of this kind of stuff!
Nice! After your first video I couldn't wait for the derivation of the Virasoro algebra, so I went ahead and studied the construction of central extensions using 2-cocycles on the underlying Lie algebra and ended up doing something equivalent to what you presented here. Your video gave me a better intuition on that stuff, and I'm definitely interested in seeing a followup video!
Will you talk about Kac-Moody and current algebras at some point? That would be awesome!
Thanks for your work!
I really love these videos that dive into more deep topics of Algebra. I would love a Video on Lie Algebras, to get to know them a little more.
Most definitely want to watch that follow-up video!
Likewise!
Fascinating stuff. Loved the teaser at the end. I'm going to be fascinated how zeta pops out when you publish that.
Fancy seeing cpw here!
This really puts into perspective how advanced you are as a professor. I understand each term separately, and even still that's not enough to keep up
Great video with a lot more substance than your usual ones! The audience might be more limited, but I definitely think this type of videos also belongs on your channel.
This is a really cool structure! Definitely had to watch the video twice though to get everything.. definitely excited about the follow up video. I did take intro to conformal field theory, but never learned too much about where the mathematical details of the structures we were using came from. So this video really let some puzzle pieces fall into place :D
To put it in Yoda's own words : "An injective map, I am!"
Im enjoying this series so much! Very tough topic for yt, and awesome video. Id love to see A LOT more!
I am so excited for the follow up!
39:50 Angry Mathologer noises
40:07 Wow, what a topic for a video. I hope it gets a lot of views 👍
Damn
I'm so happy you're uploading this kind of content
Really big Thx, that's two really nice videos, I would guess for extra videos in this line: OPEs, Vertex algebras.. , but I would add also something in the direction of harmonic analysis e.g. how the conformal transform compares to fourier transforms, and how quantization can be compared.
I see also your back ground on orbifolds can you do some stuff on moebius / modular /fucshian / klein / bianchi groups and how they attach back to harmonic analysis and quantization ? Finally of course still from your background some stuff about heisenberg algebras / Weyl quantization etc...
This video proved that I have not mastered the prerequisites. The only thing that would have made this understandable to me was if he said April’s Fools at the end. Congratulations to those that were able to understand this.
I did look at the date to make sure this wasn't a video from last April.
36:00 I think that should be f`(L1,L-1) but maybe I'm wrong..
anyway that cancels out because L-1=0
Thanks for the video, I love these topics
Yes. The second L_1 should be L_-1.
I would like to see the video on Kac-Moody algebras. I never got very far in abstract algebra for my degree so these are very nice to watch.
Keep going with this, I love this setup
Please do the follow up!
39:47 I could sense a subtle passive aggressive reference to numberphile when you said "very very famous formula" and "popularly written as" like that 😂
Would love a follow-up, thanks so much!!
I would love a video on regularization. It seems like a really weird trick when it comes up in quantum mechanics and it confuses me how we can just ignore convergence.
Hi Prof Penn..Thanks for another great video endeavouring to explain very advanced concepts..I really enjoyed your Number Theory course..and currently going through the Abstract Linear Algebra..and Complex Analysis courses..Looking forward to when u make a video course on one of the Graduate courses that you teach..Thanks Again !
Awesome video! Definitely much more challenging to follow than most of the others here on the channel, but this is what math is all about.
Can't wait for the next one especially due to the regularized sum of series of Naturals mentioned at the end.
Remember myself encountering this sum for the first time in one of the third year pregraduated physics courses, and the shocked it deployed on me when I realized that a clearly diverging series not only has some application in physics but also has been attached a finite value, which is also a negative.
Great video! Yes, please make a video about affine Kac-Moody algebras, too! Videos at such an intermediate level are too rare on TH-cam.
Intermediate?
@@gennaroponsiglione1098 Yes, if you look at the full scale of difficulty, it might look like that:
primary school < secondary school < undergraduate lecture < graduate lecture < expert research talk.
You can find videos at all these levels on TH-cam. What is quite rare in my experience, however, is such a video about a topic that is interesting to people up to the research level but presented at an intermediate, say undergraduate, level of difficulty.
@edwin steiner Oh yes, if you mean an undergraduate level of an advanced topic I understand perfectly, is what makes this channel great, even if this time I had some difficulties
More algebras videos like this please
Really love this video, and the one on the Witt algebra! Please make more videos on topics like these!
I would love to see more videos like this one although I must admit that I didn't understand some of the things presented in the video.
Very excited for the follow up video.
More of this please
It has been so many years since I took group theory. This is bringing back good memories.
I'd be lying if i say I understood every step since I've never studied Lie algebra, but I got the rough idea of what was going on, which is cool. Main confusing bit was the tensor product/direct sums steps, since I'm not really sure how all that works. Interesting thanks.
Please do more of these kind of videos!!
Awesome!! I am really looking forward for that follow up video:)
OMG!!! So awesome!! :) Gonna need another video on that regularzation of the numbers and stuff. :)
Hey Michael, it seems to me that when you’re replacing f with f’, you’re subtracting off multiples of the basis vectors L_m from your c. And in that way you can be sure that your algebra is still isomorphic. But it seems like you’re making an infinite number of choices for g, which means you’re subtracting off an infinite number of vectors from c to get it into that standard form.
My question is: how is this allowed? We’re just working with a plain infinite-dimensional space, no topology, infinite sums aren’t well-defined. Is there some dependence condition that forces only finitely many of the L_m to need to be subtracted?
I don't think your construction gives an isomorphism: if you subtract (multiples of) Lm from the central charge c, the resulting element won't be in the center. Instead I imagine that a multiple of c is subtracted from every Lm (something along the lines of Km=Lm±g(Lm)c). Since every equation features finitely many Lm no infinite sums would arise.
@@ichtusvis Ah, this makes sense. I should admit that I didn’t do the math (obviously), but it does make sense that we’re choosing lifts of the L_m to g’, instead of choosing the c differently. I also agree that changing c would jeopardize the centrality condition as well.
Was just going to ask for this video when I saw you already made it
Great video! Love physics topics, would love to see more physics content on the channel!
As a PhD student on mathematical string theory I love this video, even though I have already learn most of this stuff, i am really looking forward on you going even deeper on the algebra of superstrings or even M-theory and beyond! :)
Fantastic video! However, unless I've misunderstood something, the case lambda=0 is not isomorphic to other cases. So should the definition of Virasoro algebra be the unique _non-trivial_ central extension of the Witt algebra?
I would love to see the follow-up!
The kernel of the function mapping to zero is "everything", that just blew my mind
I really enjoyed this video, is the follow-up available? Thanks a lot for all the effort you are putting in your channel and sharing your knowledge with the rest of the world
Please make the follow up video!
it'd be cool if you made a video about affine kac-moody algebras
I can't wait to watch this
Challenge homework: take a shot every time Michael says "L"
a lot of the terms used here had to be extensively looked up to continue watching, but very informational!
You can do formal delta function distribution on Laurent Polynomials (a la Kac) (or series if you are a Physicist)
very excited for the follow-up video! in the mean time ill have to study my algebra so im not caught off guard.
definitely want to see the next video. And could you comment on why this extension is important?
Have you done Frobenius algebras from knot theory/TQFT yet?
Please keep up with this series 😄
Amazing video!!
In physics you usually call something that takes a vector space to the space over which it's defined a _covector_ (or sometimes a one-form).
This is really interesting, can I still hope for that follow up video you mentioned or was that idea thrown away?
Normally, I can get behind how most your videos work, but this is still too complex for me, I will be happy to wait for the next video instead of watching this to the end
the interest for calculations seem to prevail over explaining the significance and applicability of these definitions
Please make a follow up
Would very much like to see the follow up thank you very much
Really interesting video, but I would like to know how it is applied to physical problems and why.
you wrote "The.".. on the first entry. then I was lost in the language I did not understand. But your explanation was so convincing and obvious is was very interesting..
i wanna see that followup video!
Edit: I forgot the impulse function has _area_ 1, not magnitude 1. Disregard. -Wait, we use the Dirac delta function instead of the impulse function, even though we're comparing to zero? I'm guessing that's because the Dirac delta is used all over the place and the impulse function isn't?-
i like the 'exotic' topics like physics based math...
One question. I studied short exact sequences of modules and it's not true in general that the middle element is a direct sum of the other two. When that happens, the sequence is called split exact. In your case, because you're working with Lie algebras, every short exact sequence splits because they are vector spaces? Is my reasoning correct?
From what I could tell, he’s just saying that every SES of VECTOR SPACES splits. It gives him a nice way to write the vectors of the extension, as v1+v2 where v1 is in the ker and v2 is in the coker, so as to be able to define the Lie bracket, because you thus only need to define [v1,w2] (as [v1,w1] and [v2,w2] already are defined from the original algebras).
Yes i think the reason is that any exact sequence over free modules splits.
Sweet lord I want to see that new video! Please.
Another great video on a deep topic! Is there a reason that one-dimensional central extensions are of particular interest instead of other kinds of extensions, or higher-dimensional central extensions?
As soon as you got to 1/12, I knew Riemann was going to make an appearance. However, I don't know why, so please do make the follow-up video.
-1/12 alert! Mathologer and Michael Penn Collab would be fantastic!
Great stuff.
Really have to dig deep into the undergrad memory tank for this one. I unfortunately can't keep up on this one.
oh wow, what a great video
I don't know if you will do the sequel, which I would like to see, but can't you please add some literature reference when doing the kinda advanced stuff? That would be great :)
This is soo cool!
12:30 why is multiple of new vector depending on f(x, y)? Also, more than lambda being 1/12 , I want to understand, what do you mean by it being absorbed in the basis vectors of C?
I really love your videos. Thank you for making them. I don't know if you like to get questions here. In this video, you are talking about constructing a Lie algebra structure on g' given algebras a and g but if the sequence is a short exact sequence of Lie algebras, then g' must already have a bracket, right?
hopefully the followup will happen someday!
hey can you make a video on Multilinear Algebra
I was going to guess lambda was doing to relate to reimann zeta when you placed 1/12, but I guess you let us know before hand.
I recently came to know to know that Ramanujan's theta function plays a huge role in string theory .Can you make a video about that ???.It will really be nice if you can make one .
I am interested in string theory and want to know how Ramanujan used the q-series and derived 24th power of q-series, please. If you can please derive the partition function showing how to count all the photons in the universe. Thank you.
No no trust me, the first series is good but it really picks up in the algebra arc
Have you done a video about lie agebras: G2 F4 E6 E7 E8 ?
Yes Affine , Kac-Moody lie algebras please.
Nice ending
At 12:49 shouldn’t it be “ f is a function from g cross g to C “..?? ….. not g’ I guess.
i would like the next video please
Did we eve3r get that follow up video?!