There is an audience for this kind of content - people like me who vaguely remember this stuff from university, but have been in the real world for a while and like being reminded
Honestly, you should feel even more encouraged to make a video on an unpopular or niche topic. There are a billion resources out there for competition math or Calc 2 help, but with under-represented stuff, you're really meaningfully breaking open ground for new minds to explore.
These videos are at the perfect level imo, introducing something new and abstract but at such a level that anyone who is okay with (e.g.) linear algebra or polynomials can dig in. One suggestion: it would be nice to end with a recommend resource (e.g. standard textbook) for those inspired to find out more. Anyway, extremely here for any follow up videos - but regardless, focus on making content you enjoy so you don't get burnt out!
This is called proximal learning technique. Not too easy and not too difficult either. You are in the proximity of that border separating easy / difficult.
I love abstract and pure mathematics and your extraordinary clean presentation makes even complex topics accessible by rather short videos. I appreciate whatever you do!
As a student applying to PhD programs, videos like this help keep me motivated to learn mathematics. I would definitely enjoy watching more of these videos
As a maths graduate who is now navigating life outside of academia, I just wanted to say I really appreciate these kinds of videos for keeping me in touch with my passion for mathematics. Thank you so much!
I really like these videos! There’s very few high quality introductory materials to advanced math concepts on the internet. For those of us who aren’t currently at a college (say working in industry or still in high school) this type of content is invaluable. Please do make a follow-up video!
I am a PhD student in math (particularly Algebraic Number Theory) and I greatly appreciate videos such as these. However, I understand if they don’t feel worthwhile to make if they require some effort and if the audience for this is a very small minority of your viewers. I’d be up for seeing a follow up on the central extension of this algebra. Do you intend to make anything (or perhaps have you already made something) on representation theory?
Often I, and probably many other people, are afraid they're not going to understand a video like this, but I was pleasantly surprised with how well I understood. Definitely interested in a sequel to this, particularly one about showing if there's linearity with the lie operator here
I love your advanced videos. They are definitely filling some gaps in TH-cam materials. Also your advanced videos are especially useful for me because my university doesn't offer much algebraic topics and your videos show me this beautiful branch of Maths that I know nothing about. But you also keep your videos on such a level that they are understandable for viewers that do not have much algebraic knowledge but have some maths intuitions.
This kind of video is cool. Please post more like this! I actually found it easier to follow than a lot of the tricksy Olympiad type problems: more concept heavy, sure, but more interesting and less like some kind of newspaper puzzle.
One of my professors specialties, if I recall his CV correctly, was on "Witt Rings". My undergrad education didn't cover this type of Algebraic structure; so, thank you for covering it. I
I think Witt rings are actually unrelated to the Witt algebra (mathematicians are horrible at naming things...). Witt rings come from number theory, and they're always commutative rings (whereas the Witt algebra isn't even associative). I think the name relation is just because they were both first thoroughly studied by the same person.
I learned a bit about Lie algebras when studying Quantum Field Theory but this was a nice refreshment. I would like to see a continuation into the Virasoro theorem.
I like your videos on topics like this. I find you to be an extremely clear orator, especially for more "abstract" topics. For example, your videos on the tensor product and free vector spaces were extremely helpful for understanding those objects.
I love videos like this. I'm only a first year undergrad but I could follow along pretty easily, I don't know if there's a big audience for it but those of us who did like it seem to like it a lot. myself included
I enjoy your usual videos, but this is the stuff I'm really here for! I'm sorry they don't get as many views, because they really are great. I would love to here you talk more about the Virasoro Algebra!
It’s a shame the audience for these kinds of videos is so small. As a student who’s self-studying linear algebra and recently learnt about some simpler applications of polynomial rings like Gaussian numbers to number theory, these sorts of videos that give exposure to neat things in math really pique my interest for the possibilities of what I could study in college. Thank you so much!
one of the best videos on this channel so far. i'm on my 3rd year of uni learning physics and just started learning complex analysis, so this was fascinating. would love to see a followup video and more videos like this in general
This really takes me back to my bachelors studies into mathematical physics as you mention in the end of the video :) I would love to see more abstract algebra content like this from time to time. This was really great!
I love the recent content you've been putting out. It's a nice refresher from your usual content around solving problems. (And I'm learning about Lie algebras, representations, etc. right now so the timing is perfect!)
I'm currently an undergraduate mathematics major, so these topics are a little above my head but provide challenging math, which is always appreciated!
This is a great video on a topic I really enjoyed learning about. Please make the follow up video on the virasaro algebra. I am really interested on how this math applies to the physical applications that you mentioned at the end of the video. More on these topics please.
Please keep these coming because they help when your studying this stuff from a textbook to get a good understanding of the subject so the really difficult material in my texts is more digestible! Thank you!!!!!
This video is GREAT! You take a topic i didn't know anything about and just using a bit of linear algebra you give a lot of interesting ideas, in just half an hour! Also really smooth to follow.
I believe in minute 34:40 you forgot to pull the - sign in front of the (m-n) into the L_{m+n}, since the L_m elements were defined with a negative sign. And I personally would love a follow up, it sounds really interesting.
This video was amazing. I'd love to see a continuation on this, I'm a theoretical physics PhD student and I'm super enthusiastic about mathematical physics, although I lack some formal introductions on advanced math (such as rings, topology and so on), so this kind o video reeeeeally gets me going, so I can be able to look out for more topics to study. Thank you!!
I greatly appreciate yours videos and thank you for making them. As for the 'Virasoro Algebra' video, I would certainly appreciate it and encourage you to make it. Thanks again for making all the videos.
You ought to mention whether it satisfies the Jacobi identity. And is there a reason why there is a minus sign in the basis definition? Does it make some future calculation neater? And in answer to your question t the end, I would like to see a follow-up video on the Virasoro algebra.
I am so thrilled your return video is abstract algebra!! I wish there was more abstract algebra content out there, and yours is more understandable than anything I've seen (i sent your video to a friend who's a chemist and now she wants to learn abstract algebra). Please please please please PLEASE release more, I've already run out of your videos!
I would love to see more algebra stuff! I’m a second year grad student with a background in analysis. My advisor does stuff in harmonic analysis on groups, so I have been trying to learn Lie algebra and representation theory stuff, which has been brutal.
I have a PhD in discrete math and this type of video is great to watch during breakfast. Learn a new definition, see some trivial stuff (until about 25:00), learn something a little less trivial. And learning about applications of Witt algebra would be great, even though it sounds much more challenging video than this one
I took a mathematical methods class last semester were we used Laurent polynomials a lot for complex integration. This video showed me another perspective of them that I hadn’t seen before, and that’s pretty cool.
You remind me of a professor I had at UofM who taught a 1-credit (honors) course in simply solving problems. I thought it was silly and dropped it, and you are making up for the presumptive ignorance I had in my late teens. Cheers.
Absolutely More algebra and video like this one!! You're so gifted in explaining complex concepts, that's why the web was invented, to spread the knowledge beyond physical libraries or classrooms
I would definitely like to see more abstract algebra topics like this. Especially the Virasoro algebra and more videos on vertex operator algebras and their orbifolds. I watched the previous videos on those topics, but I lacked familiarity with the graded structure of the vector spaces therein, so I'd love to see more on that
I am super excited by these more abstract and higher algebra videos !!! I'd be super pumped if you do a follow-up on the Virasoro algebra. It sounds super interesting ^_^ If you ever have any ideas for representation theory in positive characteristic I'd be incredibly grateful too XP
Hi Michael - love the content and especially this type - noticed the derivation and was reminded of Fox differential calculus on groups with finite representations the derivation is on a possibly non communicative ring algebra rather than a polynomial ring - and so the derivation is into an ideal and the ideal being one sided requires a triviality adjustment to the derivation definition. It is very interesting and certain invariants emerge as the principle ideal component of the ideals generated. It is worth looking fox up - i think and it allows derivations of discrete group elements g -> g-1 - worth looking at (the Alexander polynomial for knots can emerge this way)
This is great. I encountered Viasoro algebra on reading String theory, this provides a nice context. Videos of this type are appreciated as I know there is a lot of work. Thanks
This post was full of Witt and wisdom, from which I derived much enjoyment. I quite like these more in-depth systems, even if, as a physicist, they feel silly. (in a good way)
![ประสบการณ์ จ้าง Developer ที่ผมจะไม่มีวันลืม [ภาค1]](http://i.ytimg.com/vi/bzivN4KDg-k/mqdefault.jpg)
We highly appreciate the exploration into the more difficult math.
Absolutely
Yes sir !
Not neccessarily more difficult but more advanced. Some of the olympiad problems he does are really hard but they're not this advanced
@@santiagoarce5672 Lmao i was literally going to reply the exact same thing. "not more difficult but more advanced"
Are you Bulgarian?
There is an audience for this kind of content - people like me who vaguely remember this stuff from university, but have been in the real world for a while and like being reminded
Beautiful comment..
EXACTLY!!!!!
Please keep making these videos
Well said! I can totally relate ...
This is me 🖐️
100% - I’ve not even been in the real world that long and I already miss it
+1 for both
- a follow-up on the Virasoro algebra &
- generally, more videos like this
Thank you very much! 🙏🏼
Yes more videos about Algebras, Lie groups, Representations, Manifolds, Hilbert Spaces, Operators, Operator Algebras,............
I agree.
Honestly, you should feel even more encouraged to make a video on an unpopular or niche topic. There are a billion resources out there for competition math or Calc 2 help, but with under-represented stuff, you're really meaningfully breaking open ground for new minds to explore.
Agree completely!
Agreed!!
These videos are at the perfect level imo, introducing something new and abstract but at such a level that anyone who is okay with (e.g.) linear algebra or polynomials can dig in. One suggestion: it would be nice to end with a recommend resource (e.g. standard textbook) for those inspired to find out more. Anyway, extremely here for any follow up videos - but regardless, focus on making content you enjoy so you don't get burnt out!
Yes! A reccomendation for some further reading materials would be great :)
This is called proximal learning technique. Not too easy and not too difficult either. You are in the proximity of that border separating easy / difficult.
9:00 Really Good Place To Start
36:35 Good Place to Stop
No way
Didn't think prof P could say those words 😎😎
I love abstract and pure mathematics and your extraordinary clean presentation makes even complex topics accessible by rather short videos. I appreciate whatever you do!
As a student applying to PhD programs, videos like this help keep me motivated to learn mathematics. I would definitely enjoy watching more of these videos
I would love to see a follow up on the virasoro algebra.
Love it!!! Yes to follow up
As a maths graduate who is now navigating life outside of academia, I just wanted to say I really appreciate these kinds of videos for keeping me in touch with my passion for mathematics. Thank you so much!
I really like these videos! There’s very few high quality introductory materials to advanced math concepts on the internet. For those of us who aren’t currently at a college (say working in industry or still in high school) this type of content is invaluable. Please do make a follow-up video!
I am a PhD student in math (particularly Algebraic Number Theory) and I greatly appreciate videos such as these. However, I understand if they don’t feel worthwhile to make if they require some effort and if the audience for this is a very small minority of your viewers.
I’d be up for seeing a follow up on the central extension of this algebra. Do you intend to make anything (or perhaps have you already made something) on representation theory?
Michael started a series on representation theory but sadly discontinued it due to low viewership
This video requires very low effort lol
Love these more advanced/theoretical video's, more like this please!
Often I, and probably many other people, are afraid they're not going to understand a video like this, but I was pleasantly surprised with how well I understood. Definitely interested in a sequel to this, particularly one about showing if there's linearity with the lie operator here
I love your advanced videos. They are definitely filling some gaps in TH-cam materials. Also your advanced videos are especially useful for me because my university doesn't offer much algebraic topics and your videos show me this beautiful branch of Maths that I know nothing about. But you also keep your videos on such a level that they are understandable for viewers that do not have much algebraic knowledge but have some maths intuitions.
Loved the clarity of your explanation. I’d also really like more content on how algebras like this relate to physics, particularly quantum mechanics.
This kind of video is cool. Please post more like this! I actually found it easier to follow than a lot of the tricksy Olympiad type problems: more concept heavy, sure, but more interesting and less like some kind of newspaper puzzle.
More video like this. I really like it.
One of my professors specialties, if I recall his CV correctly, was on "Witt Rings". My undergrad education didn't cover this type of Algebraic structure; so, thank you for covering it. I
I think Witt rings are actually unrelated to the Witt algebra (mathematicians are horrible at naming things...). Witt rings come from number theory, and they're always commutative rings (whereas the Witt algebra isn't even associative). I think the name relation is just because they were both first thoroughly studied by the same person.
Yes! I would appreciate more videos like this one!
I learned a bit about Lie algebras when studying Quantum Field Theory but this was a nice refreshment. I would like to see a continuation into the Virasoro theorem.
I stand for this kind of content!
I really enjoy these videos about more complicated topics and definitely would like to see more. Thanks for your content.
More videos like this, please! I had encountered the Witt algebra from loops of S^1 but this finally filled in the gaps for me. Thank you!
yes, please more of those. Great addition to the usual math contests!
I like your videos on topics like this. I find you to be an extremely clear orator, especially for more "abstract" topics. For example, your videos on the tensor product and free vector spaces were extremely helpful for understanding those objects.
Make more videos like this! They are awesome.
I love videos like this. I'm only a first year undergrad but I could follow along pretty easily, I don't know if there's a big audience for it but those of us who did like it seem to like it a lot. myself included
Loved the video!
All of this can be derived just from linearity and the product rule?! I'm 10 minutes in and already blown away. Please make more videos like this!
I enjoy your usual videos, but this is the stuff I'm really here for! I'm sorry they don't get as many views, because they really are great.
I would love to here you talk more about the Virasoro Algebra!
I would highly appreciate a sequel about Virasoro algebras!
I also like these more advanced topics. The explanation is so clear that it's easy to follow. Highly interesting stuff!
These kinds of videos are something I'd want to see more of on your channel so please make more videos like this!
Many thanks for that. It is a very clear exposition and I would love to see a follow up on the Virasoro algebra
Yes please! Give me more! Love this!
This is fantastic. I'd be very curious to see the Virasoro algebra stuff.
It’s a shame the audience for these kinds of videos is so small. As a student who’s self-studying linear algebra and recently learnt about some simpler applications of polynomial rings like Gaussian numbers to number theory, these sorts of videos that give exposure to neat things in math really pique my interest for the possibilities of what I could study in college. Thank you so much!
one of the best videos on this channel so far. i'm on my 3rd year of uni learning physics and just started learning complex analysis, so this was fascinating. would love to see a followup video and more videos like this in general
This really takes me back to my bachelors studies into mathematical physics as you mention in the end of the video :)
I would love to see more abstract algebra content like this from time to time. This was really great!
I love the recent content you've been putting out. It's a nice refresher from your usual content around solving problems. (And I'm learning about Lie algebras, representations, etc. right now so the timing is perfect!)
I would love to see some more of these in depth videos into abstract topics. You Michael would probably enjoy making them too.
I'm currently an undergraduate mathematics major, so these topics are a little above my head but provide challenging math, which is always appreciated!
I would like more like this.
Thank you . Of course there is an audience to know the difficult and different areas of maths. Pls keep on doing your good job.
Yes and yes! More videos on abstract/niche math, and more videos following up to this one. 🙏🏻
I would absolutely love a follow-up on the Virasoro algebra!
Very fun! I study related stuff for a living but still enjoyed your exposition. Would love to see the Virasoro algebra followup.
I would definitely enjoy watching more of these videos!
I would love to see more videos like this!
This is a great video on a topic I really enjoyed learning about. Please make the follow up video on the virasaro algebra. I am really interested on how this math applies to the physical applications that you mentioned at the end of the video. More on these topics please.
Loved studying math in undergrad but needed $$ so switched to data science but I love running into content like this. Makes me want to go back to math
I would love more stuff like this!!
Please keep these coming because they help when your studying this stuff from a textbook to get a good understanding of the subject so the really difficult material in my texts is more digestible!
Thank you!!!!!
Make more of these! The tensor product video was very nice
Follow up video about Virasoro Algebra would be a must watch. Thanks for the great content.
This video is GREAT!
You take a topic i didn't know anything about and just using a bit of linear algebra you give a lot of interesting ideas, in just half an hour!
Also really smooth to follow.
I believe in minute 34:40 you forgot to pull the - sign in front of the (m-n) into the L_{m+n}, since the L_m elements were defined with a negative sign. And I personally would love a follow up, it sounds really interesting.
This is a great topic. Thanks for the video. I love these advanced videos.
This video was amazing. I'd love to see a continuation on this, I'm a theoretical physics PhD student and I'm super enthusiastic about mathematical physics, although I lack some formal introductions on advanced math (such as rings, topology and so on), so this kind o video reeeeeally gets me going, so I can be able to look out for more topics to study. Thank you!!
PLEASE MAKE MORE VIDEOES LIKE THESE!!!! Trust me there's a huge demand
yes more of this please ! and thank you
this sort of stuff is what i am interested in tbh. i would definitely appreciate a follow up or something similarly abstract in the future.
These videos are appreciated. Thanks
I greatly appreciate yours videos and thank you for making them. As for the 'Virasoro Algebra' video, I would certainly appreciate it and encourage you to make it.
Thanks again for making all the videos.
Please, make more of these!!!
Much more abstract than I learned in engineering - but you presented it really clearly! I definitely wouldn't mind watching abstract stuff like this.
Please keep making these videos. They are challenging but very rewarding.
I'd love to see a follow up with the Virasoro algebra!
You ought to mention whether it satisfies the Jacobi identity. And is there a reason why there is a minus sign in the basis definition? Does it make some future calculation neater?
And in answer to your question t the end, I would like to see a follow-up video on the Virasoro algebra.
I am so thrilled your return video is abstract algebra!! I wish there was more abstract algebra content out there, and yours is more understandable than anything I've seen (i sent your video to a friend who's a chemist and now she wants to learn abstract algebra). Please please please please PLEASE release more, I've already run out of your videos!
this was very interesting, thanks michael
I would love to see more algebra stuff! I’m a second year grad student with a background in analysis. My advisor does stuff in harmonic analysis on groups, so I have been trying to learn Lie algebra and representation theory stuff, which has been brutal.
More advanced content like this, please. Onward and upward to the Virasoro Algebra indeed!
I have a PhD in discrete math and this type of video is great to watch during breakfast. Learn a new definition, see some trivial stuff (until about 25:00), learn something a little less trivial. And learning about applications of Witt algebra would be great, even though it sounds much more challenging video than this one
I took a mathematical methods class last semester were we used Laurent polynomials a lot for complex integration. This video showed me another perspective of them that I hadn’t seen before, and that’s pretty cool.
yes i enjoy this kind of videos
I think this type of topics should be explored more by everyone.
You remind me of a professor I had at UofM who taught a 1-credit (honors) course in simply solving problems. I thought it was silly and dropped it, and you are making up for the presumptive ignorance I had in my late teens. Cheers.
I loved this video! I would like to see stuff like this about Clifford Algebras and Geometric Algebra in particular.
Absolutely More algebra and video like this one!! You're so gifted in explaining complex concepts, that's why the web was invented, to spread the knowledge beyond physical libraries or classrooms
I learned so much from this video. It inspired me to run down a number of tangential rabbit holes. I’d certainly appreciate more like this!
I would definitely like to see more abstract algebra topics like this. Especially the Virasoro algebra and more videos on vertex operator algebras and their orbifolds. I watched the previous videos on those topics, but I lacked familiarity with the graded structure of the vector spaces therein, so I'd love to see more on that
please do more videos like this one!!!
Yes, please, do a video on the Virasoro algebra!
I love the more theoretical math videos. This was a great watch!
This video was great, please make more on topics like this one.
I am super excited by these more abstract and higher algebra videos !!!
I'd be super pumped if you do a follow-up on the Virasoro algebra. It sounds super interesting ^_^
If you ever have any ideas for representation theory in positive characteristic I'd be incredibly grateful too XP
Excellent video. Very interesting, informative and worthwhile video. I hope you make sequel videos and other videos on similar topics.
Hi Michael - love the content and especially this type - noticed the derivation and was reminded of Fox differential calculus on groups with finite representations the derivation is on a possibly non communicative ring algebra rather than a polynomial ring - and so the derivation is into an ideal and the ideal being one sided requires a triviality adjustment to the derivation definition. It is very interesting and certain invariants emerge as the principle ideal component of the ideals generated. It is worth looking fox up - i think and it allows derivations of discrete group elements g -> g-1 - worth looking at (the Alexander polynomial for knots can emerge this way)
I like this type of video more than your usual videos, this one was amazing! I'm looking forward to the followup video
This is great. I encountered Viasoro algebra on reading String theory, this provides a nice context. Videos of this type are appreciated as I know there is a lot of work. Thanks
Watching to haze myself with confusion. Good introduction to the feeling you get from seminars.
Yes, more. Please.
This post was full of Witt and wisdom, from which I derived much enjoyment.
I quite like these more in-depth systems, even if, as a physicist, they feel silly. (in a good way)
Yeah definitely videos like this
Really enjoyed this video - the presentation is very clear and intuitive. Would love to see one on the Virasoro algebra!
I would like for you to make more videos like this!!