One of the most important algebras -- The Witt Algebra

[@michaelslack8900]

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

[@Hyakurin_]

Beautiful comment..

[@DiegoTuzzolo]

EXACTLY!!!!! Please keep making these videos

[@MichaelFJ1969]

Well said! I can totally relate ...

[@briandennehy6380]

This is me 🖐️

[@Trythisone264]

100% - I’ve not even been in the real world that long and I already miss it

[@frozenmoon998]

We highly appreciate the exploration into the more difficult math.

[@void7366]

Absolutely

[@rounaksinha5309]

Yes sir !

[@santiagoarce5672]

Not neccessarily more difficult but more advanced. Some of the olympiad problems he does are really hard but they're not this advanced

[@mastershooter64]

@@santiagoarce5672 Lmao i was literally going to reply the exact same thing. "not more difficult but more advanced"

[@kosed7041]

Are you Bulgarian?

[@jkid1134]

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.

[@maurocruz1824]

Agree completely!

[@erictao8396]

Agreed!!

[@Alex_Deam]

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!

[@thomasreid2609]

Yes! A reccomendation for some further reading materials would be great :)

[@aadilansari5997]

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.

[@STCB]

+1 for both - a follow-up on the Virasoro algebra & - generally, more videos like this Thank you very much! 🙏🏼

[@mehrdadassar2542]

Yes more videos about Algebras, Lie groups, Representations, Manifolds, Hilbert Spaces, Operators, Operator Algebras,............

[@habermasnyc]

I agree.

[@goodplacetostop2973]

9:00 Really Good Place To Start 36:35 Good Place to Stop

[@aweebthatlovesmath4220]

No way

[@perappelgren948]

Didn't think prof P could say those words 😎😎

[@woody442]

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!

[@murmol444]

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

[@ericvazquez8639]

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

[@erikdavis8901]

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?

[@schweinmachtbree1013]

Michael started a series on representation theory but sadly discontinued it due to low viewership

[@mikhailmikhailov8781]

This video requires very low effort lol

[@andrewmichel2525]

I think you underestimate how much interest there is for these type of videos. There are tons of channels that do challenging integrals and Olympiad problems, which are fun, but that niche is definitely filled. There aren't that many channels presenting this type of higher level material with such high production value while still being digestible to people with less mathematical experience (such as myself). I would definitely like to see sequel to this video and really wouldn't mind at all if you steered your channel more into the realm of your mathematical interests and research. It's your channel, do what you want and I'm sure many of us will support you.

[@captainsnake8515]

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!

[@romajimamulo]

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

[@TheOneThreeSeven]

Love these more advanced/theoretical video's, more like this please!

[@thomasreid2609]

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!

[@Holital]

Love it!!! Yes to follow up

[@dansheppard2965]

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.

[@evanknowles4780]

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!

[@ДенисЛогвинов-з6е]

More video like this. I really like it.

[@djsmeguk]

I would love to see a follow up on the virasoro algebra.

[@aspiringmathmatologer3445]

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!

[@greivinalfaro8339]

I stand for this kind of content!

[@maxdominate2481]

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

[@ChronusZed]

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.

[@zuzaaa1998]

I love your advanced videos. They are definitely filling some gaps in KZbin 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.

[@rishilohar2114]

Loved the clarity of your explanation. I’d also really like more content on how algebras like this relate to physics, particularly quantum mechanics.

[@officiallyaninja]

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

[@daveaasen]

Make more videos like this! They are awesome.

[@EtienneSturm1]

yes, please more of those. Great addition to the usual math contests!

[@ttausend1106]

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.

[@alepel792]

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

[@valeriobertoncello1809]

What do you mean there's no audience?? THIS is the content I'm subscribed for!

[@radadadadee]

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.

[@bobdowling6932]

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.

[@dcqin]

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.

[@patrickcrehan355]

Many thanks for that. It is a very clear exposition and I would love to see a follow up on the Virasoro algebra

[@SlipperyTeeth]

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!

[@dimalukashov8148]

I really enjoy these videos about more complicated topics and definitely would like to see more. Thanks for your content.

[@beatrizbrandaor8]

Yes! I would appreciate more videos like this one!

[@kiga14]

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!

[@arthursteenkist2595]

I also like these more advanced topics. The explanation is so clear that it's easy to follow. Highly interesting stuff!

[@m1323fj]

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.

[@TimDuncanofSoccer]

This is fantastic. I'd be very curious to see the Virasoro algebra stuff.

[@YanickSP]

Yes please! Give me more! Love this!

[@charlesconnors1066]

Loved the video!

[@alexnagle575]

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.

[@renpnal229]

These kinds of videos are something I'd want to see more of on your channel so please make more videos like this!

[@masonholcombe3327]

I'm currently an undergraduate mathematics major, so these topics are a little above my head but provide challenging math, which is always appreciated!

[@sp_danger1729]

I would love to see some more of these in depth videos into abstract topics. You Michael would probably enjoy making them too.

[@iconoclasticistherr9965]

Very fun! I study related stuff for a living but still enjoyed your exposition. Would love to see the Virasoro algebra followup.

[@mohanbuvan]

Thank you . Of course there is an audience to know the difficult and different areas of maths. Pls keep on doing your good job.

[@BrutishLearner4]

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!

[@Nikolas_Davis]

29:29 as a physicist, I could see the commutator making an appearance from *miles* away :-) (The answer to your question "should I make another video?" is *always* going to be 'yes' :-D )

[@paulkohl9267]

I like elaborate integral videos; but getting a video on more abstract topics like this is also very much appreciated. Too bad the numbers are so low. There are those of us in the small minority who exist and can follow along. For those brave few, we warrant your all of videos as enjoyable. 🙂

[@ivanklimov7078]

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

[@leandrocarg]

Yes and yes! More videos on abstract/niche math, and more videos following up to this one. 🙏🏻

[@levanjikiya4454]

I would like more like this.

[@jewishjewom12ify]

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.

[@nossahalf119]

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!!

[@mictecacihuat665]

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.

[@josephmellor7641]

9:00 "That's a good place to start." I feel like I'm in a mirror universe.

[@harmindersandhu2750]

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.

[@andreben6224]

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

[@lorentzianmanifold718]

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!!!!!

[@jaykazmi]

PLEASE MAKE MORE VIDEOES LIKE THESE!!!! Trust me there's a huge demand

[@aryanrahbari1411]

I would highly appreciate a sequel about Virasoro algebras!

[@Ro-zanc]

Please keep making these videos. They are challenging but very rewarding.

[@notfeelin6610]

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.

[@francoisamman2620]

I would definitely enjoy watching more of these videos!

[@speeshers]

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!)

[@johnsalkeld1088]

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)

[@Jacob-ky4vk]

I would absolutely love a follow-up on the Virasoro algebra!

[@MichaelMavros]

I would love to see more videos like this!

[@matthewsalvas4538]

Make more of these! The tensor product video was very nice

[@charlottedarroch]

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

[@danielwilckens578]

Follow up video about Virasoro Algebra would be a must watch. Thanks for the great content.

[@MELONOLDER101]

yes i enjoy this kind of videos

[@buxeessingh2571]

This is a great topic. Thanks for the video. I love these advanced videos.

[@scipionedelferro]

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

[@tomctutor]

@34:47 you forgot to remove the final -ve (I think!). Commutators are used in physics Quantum Mechanics a lot. It is good to remind undergraduates that simple arithmetic is not the only structure in maths and maybe talk about why associativity, commutivity and so on are important properties and what the deeper meaning is if these properties fail (as in this example).

[@heygooooooooo]

Much more abstract than I learned in engineering - but you presented it really clearly! I definitely wouldn't mind watching abstract stuff like this.

[@riccardofiori828]

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.

[@driesceuppens7623]

I love the more theoretical math videos. This was a great watch!

[@jrb0580]

I loved this video! I would like to see stuff like this about Clifford Algebras and Geometric Algebra in particular.

[@biggiemac42]

Huge fan of this video! If you make more like it I will continue to be a huge fan!

[@AdrianBoyko]

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!

[@kushaldey3003]

I think this type of topics should be explored more by everyone.

[@AdrienLegendre]

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

[@cebiclock]

please do more videos like this one!!!

[@lumpi806]

yes more of this please ! and thank you

[@lexinwonderland5741]

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!

[@ethandole2218]

I would love more stuff like this!!

[@mMaximus56789]

I would love to see a follow up. I also think that as long as there's a buildup there'll be an audience (I certainly enjoy these videos). If you'd like to continue showcasing more complicated algebras I'd like to propose Clifford\Geometric algebra and maybe buildup till geometric calculus as I haven't seen an easy introduction to the calculus part yet in the internet; plus it's heavely used in physics and computer graphics!

[@Budha3773]

This video was great, please make more on topics like this one.

[@Blitnock]

Yes, please, do a video on the Virasoro algebra!

[@francescobollini4254]

These videos are appreciated. Thanks

[@caseyglick5957]

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)

[@statisticserinokripperino]

Please, make more of these!!!

[@robertschlesinger1342]

Excellent video. Very interesting, informative and worthwhile video. I hope you make sequel videos and other videos on similar topics.

[@MrSamwise25]

This was a fun and very clean video, and I'd definitely like to see the follow-up. If you make more of this kind of video, I'd also really like to see some motivation regarding why these things are studied in the first place.