Group theory, abstraction, and the 196,883-dimensional monster
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- čas přidán 2. 05. 2024
- An introduction to group theory (Minor error corrections below)
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Special thanks to these supporters: 3b1b.co/monster-thanks
Timestamps:
0:00 - The size of the monster
0:50 - What is a group?
7:06 - What is an abstract group?
13:27 - Classifying groups
18:31 - About the monster
Errors:
*Typo on the "hard problem" at 14:11, it should be a/(b+c) + b/(a+c) + c/(a+b) = 4
*Typo-turned-speako: The classification of quasithin groups is 1221 pages long, not 12,000. The full collection of papers proving the CFSG theorem do comprise tens of thousands of pages, but no one paper was quite that crazy.
Thanks to Richard Borcherds for his helpful comments while putting this video together. He has a wonderful hidden gem of a channel: • MegaFavNumbers 2625374...
You may also enjoy this brief article giving an overview of this monster:
www.ams.org/notices/200209/wha...
If you want to learn more about group theory, check out the expository papers here:
kconrad.math.uconn.edu/blurbs/
Videos with John Conway talking about the Monster:
• Monster Group (John Co...
• The Monster Group - Jo...
More on Noether's Theorem:
• The most beautiful ide...
• Noether's Theorem and ...
The symmetry ambigram was designed by Punya Mishra:
punyamishra.com/2013/05/31/sy...
The Monster image comes from the Noun Project, via Nicky Knicky
This video is part of the #MegaFavNumbers project: • MegaFavNumbers
To join the gang, upload your own video on your own favorite number over 1,000,000 with the hashtag #MegaFavNumbers, and the word MegaFavNumbers in the title by September 2nd, 2020, and it'll be added to the playlist above.
Thanks to these viewers for their contributions to translations
German: dlatikaynen
Hebrew: Omer Tuchfeld
Italian: mulstato
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These animations are largely made using manim, a scrappy open-source python library: github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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the difference between fiction and reality is that fiction has to make sense.
Would that I could like a comment twice.
That is a beautiful quote
Stories have a beginning, a middle, and an end. Reality doesn’t.
Lawrence D’Oliveiro technically, reality does have those, but only 1 of each. (hint: scope = universe)
@@maxlife459 It’s not clear that the universe has a beginning or an end.
I'm a simple man. I see an outrageously large number in the video title, I click.
how's the KFC austin
ayy it's Austin love your vids
Ayyy Austin how's that birdwatching going
Heyyyyy! I heard your sister in law got married.
Eyyy
It feels like with "the monster" humanity has learned the answer to something without being able to even ask the question. Almost as if we've stepped into territory that isn't meant for us yet. Of course the given name adds to the lovecraftian feeling of it and I love stuff like this.
Your comment genuinely made me teary-eyed, ngl
You share the same number as the monster, Jupiter. Seems sus to me.
"humanity has learned the answer to something without being able to even ask the question" literally the plot of hitchhiker's guide
"Forty-two," said Deep Thought, with infinite majesty and calm.
Given that Horrible Problems Lovecraft himself "didn't have the constitution for math", do we have the "constitution" for this?
"The universe doesn't really care if its final answers look clean; they are what they are by logical necessity, with no concern over how easily we'll be able to understand them."
Elegantly stated. As a grad student in mathematical physics, this definitely lines up with my experience!
For some reason this sounds like a happy 70 year old grandpa who studied maths
Wow, seems incredible, because math independently can feel so elegant and beautiful , but the most amazing thing is when it’s applied on the natural ways of reality, you can actually see it shift, and basically “become a huge mess”.
Really tells you something about the way math serves us , and the way it explains reality simultaneously
Wish you best of luck!
@@amitbenjam Thats the thing. The field in question here has nothi g to do with physical reality.
This is a result of first order logic.
Its purely mathematical.
@@ineednochannelyoutube5384 incredible
I as a student in an entirely different field, hate solving equations in mathematics....but something always draws me into the theorems that it provides.
Just got new earbuds and while trying them on, suddenly my google assistant starts reading my notifications. Of course, it had to read this never-ending title for me lmao
Weird flex but ok...
@@kavinbharathirm9478 You're meme-flexing, and it's misplaced. Earbuds can be had for a few bucks.
Which earbuds btw?
wf 1000xm3, so not exactly cheap though
Hahaha that's awesome. Did it work out the name of the number (with variations on duodecicillion etc... maybe somebody can give us the actual name), or just read out the digits?
15:54 reminds me of the "how to draw an owl" joke:
step 1: draw a circle
step 2: draw the rest of the owl
This is the essence of LISP
Its TRIVIAL and BASIC
@@benjaminschocket-greene2536 my proof: 23 pages long
Homestuck..
@@tsanguine Damn crazy how the comment had nothing to do with that. Gonna have to send you to the shadow realm for that one
Honestly, there's something beautiful about the way 3b1b explains things. At around 4:28, he explains that the permutations of 101 different objects would amount to 9x10^159. However, instead of simply saying 'this is roughly the same as the number of atoms in the universe squared', he says 'if every atom in the universe had a mini universe inside of it, that would be how many sub-atoms there would be.' Take the time to appreciate the time he took to make these numbers just a bit more interesting!
It's a nice metaphor since humans suck at understanding multiplication intuitively. A big number multiplied by a big number just equates to a bigger number for us, we're bad at telling the difference.
I'm glad you were here too, though, so I understood he meant atoms squared.
1:47 identity
the way you explain the concept of group as the concept of number "3" is really mind opening and important. A lot of people having trouble with math because it's seem so conceptual and they always try to link it to something more grounded, but to be good at math they need to approach math as how conceptual it is. Eventually of course math is used to help real life problem but it's not always straight forward, so you need to think about it in the world of math itself. It's like when I started coding and at first my mind will work only with what the final UI or graphic display on the screen, but slowly my mind would think purely of what happen data wise and not really the final representation.
I feel like CZcams is robbing me of your comment cuz it ends on "on the screen, but slowly my mind ..."
@@user-lz1yb6qk3fbut slowly my mind would think purely of what happens data wise and not really the final representation.
I doubt this helps but I typed it now lol
"We always consider the action of doing nothing to be part of the group" - 1:41
My favorite quote
Story of every group project! Now you have the excuse: "I am the group's identity element, I act on nothing and I am necessary" :)
@@alexanderschafer8979 "Next time, how about we have a semigroup project!"
That one truly hits home ;)
I'm part of some organized activity. There's this dude that definitely is our groups identity element. He's nice though. Zero initiative, but there's not a bad bone in his body.
Part of the group project more like
Other Mathematicians: "Polynomials, Permutations, Quaternions"
John Conway: "Monster, Baby Monster, Happy Family"
You are short of pariahs!
We need more John Conway's. Why make something so dense and complex even more difficult with letter-number combinations when you can just name stuff BABY MONSTER
Monstrous moonshine!
CoronaVirus took away one of the greatest man of our time
dog Need more help naming polytopes, there are weird shortenings like “griddip” or “groh”
A quote from a Pratchett novel comes to mind, when a wizard tries to explain how a mysterious cabinet works.
'Yes. The box exists in ten or possibly eleven dimensions. Practically anything may be possible.'
'Why only eleven dimensions?'
'We don't know,' said Ponder. 'It might be simply that more would be silly.'
do you know what book thats from? it seems interesting.
@@jetison333 "Making Money" by Terry Pratchett. Although most of the book is not related to that quote. Still a fantastic book. It's the sequel to "Going Postal" so that may be a good place to start.
Pratchett is my absolute favorite.
Me: how did you get so strong?
Mathematician: every time I find a new dimension, do 1 push-up
Me: Jesus Christ
Yes but
This is IMO the best video 3b1b has produced by far. Amazing explanations, visualizations, stories, everything. At first everything went over my head, but after learning a bit about group theory, this video is so cool.
Imo the best vid was the Alice and Bob one.
He did the math, he did the monster math.
I have no idea who you are or where you are, but I want you to know I love you, just for this comment. Stay beautiful, my friend.
@@pcdm43145 Awww. Thanks! You too!
yes monster math should be a thing
Nice.
It was a graveyard graph!
Someone should give him a medal for making such an abstract theory so beautiful and entertaining, and yet extremely educational! ❤️
Т
He is the one who more than deserves it !
I like your final quote "Fundamental objects are not necessarily simple. The universe doesn't really care if it's final answers look clean; they are what they are by logical necessity, with no concern over how easily we'll be able to understand them."
I remember reading "Symmetry and the monster" about 15 years ago, and fell in love with the monster group -- and due to one of the later chapters in it, the next video queued up after this one is now "Hamming codes and error correction
". Classic work.
Thanks for the book mention, I think I'll check it out.
@@jherbranson 🙂
Conclusion: "The universe is under no obligation to make sense to you!"
Sigma approves this message
hmm it would appear that the universe is singing to me
What is that melody?
Evidence suggests there's Math ahead
A Unexpected But Not unwelcomed Outcome
"It's the size... of the monster" is such a scary way to put it. What the hell, math?!
And then you get to the end of the video and there's a monster, a baby monster, and a happy family.
"The Monster At The End Of This Video"
The scary part, as far as I can tell, is not how BIG the Monster is, but the fact that it is NOT infinite... At some point, it just... stops.
And that's where the grouping stops. It can't go any further than The Monster boss, because calculation past that is impossible. Wow man, just epic.
This has got to be one of my favorite 3Blue1Brown videos. I love the way you present just how fundamental groups are. One line in particular I just love: "This is asking something more fundamental than 'what are all the symmetric things?' It's a way of asking, "what are all the ways that something can be symmetric?'".
I love your videos. I’ve always been a little intimidated for the level of abstraction of some mathematical concepts, but you can explain many of them more intuitively, with elegance and also generating more interest. Thank you and please keep doing this great work. :)
Typical horror film technique: don’t show the monster to preserve suspense.
Well, how does one "show" it, exactly? Constructing it, or even defining it, is not really straight forward. For a long time, the only result or description of it was a proof of its existence.
no plz cuz Grant really likes it, let him appear the monster plz
I might be wrong but I think photons are in 5D space, as far as special/general relativity is concerned anyway. The basic idea which when you go into detail is totally different (going off memory, may be wrong) is photons reveal what is in 3d space, travel through time (4d) and they are in the 5d. Surely, much much more could be unknown about em and time is controversial to begin with but im sure thats the consensus in modern physics so far; although anyone could come along and massively correct me
plus, time is apparently, according to some, just a result of the universe expanding or whatever.
So, photons apparently dont actually move, they stay static in one part of space time, but space time moves really fast and its as if the light travels as fast as is possible in space, but its just travelling at the fastest (known) possible speed in the universe; the rate at which it 'expands'.
I think some people claim that in one dimension, there is a pulling force, like a gravity, to the universe and then in another dimension theres like an opposite pull - not so much a push but pulling in the other, or another, direction - and basically, the waves that photons make are a result of one dimension pulling, causing a peak in the wave (the light wave travelling 'up'), while the other dimension pulls the other way, causing a trough in the wave (the light wave travelling 'down'). Idk if one dimension, would be space, or time, or electromagnetism.
Now, it's a bit like when you grab two edges of a piece of paper and pull them apart. The paper rips into two pieces in around 0.27 of a second. On the macro level (watching in real time) it's as if it's a clean break, it was pulled near enough equally in both directions and torn apart.
Yet, if you were to slow it down to an extreme level, like 20000 times slower or something and magnify by about the same, you would see the left hand pulls the paper diagonally to the left, then the right hand pulls it to the right, very quickly.
These pulls to the left and to the right happen very frequently, very quickly and there are very much of them, as the tear travels down the paper; from whence the tear started and whence the tear must end (the other end of the paper).
and when you observe time at a very slow instance, almost freezing time in effect, you can see the up, down, up, down, up, down, up, down of the photon. Really, if you consider the photon is staying still, and this is all happening very fast, perhaps at the speed of which the universe is 'expanding', then it's like you can see the different pulling dimensions acting as the 'left hand' and the 'right hand'. But, it's not 2d like the paper, it's 3d or 4d or some higher amount of dimensions.
The light wave, travelling in one direction, is kinda like the tear in the paper. You can see how the universe is tearing, being pulled left to right in the patterns of the light wave.
Yet, just like how the paper was near enough a clean break, torn apart in 0.27s, this pulling and tearing of the universe is near enough a clean break (we'll assume; scientists have found that the universe is likely expanding at a faster rate in one direction than another direction(saw Zach Star mention an article about it in a hypersphere universe video) but that's a story for another day haha) so whether or not we see the pattern, the pulling left to right on the microscopic, micro-timed level, the result is still near enough the same. We see the light, pretty much in an instant, before we can even notice we see it.
If there is a plentiful amount of photons, which they're most usually is, we see a constant stream of photons and we visualize the object that emits the photons.
@Grant Jacobson to answer your question: even if light is in a lower dimension, a 1d point could travel through 3d space and hit near enough exactly the vertex(corner) of a cube. This 1D point would then be reflected along another path in 3D space(4D spacetime if you wanna get technical).
In a similar way, whether photons are 3D, 5D or More Dimensional, if a photon was to travel directly into a vertex, a corner of the monster, it would be deflected and reflected along a different path in it's own dimensions; or perhaps the photon would be reflected into different dimensions to the ones that it was originally in, then it's path would continue there.
I can only assume The Monster, being some form of physical object, in space, would have oodles and oodles, trillions of corners.
I can also only assume, that these corners must reach the edge of many other dimensions because... they're corners. With it being such a highly dimensional being, these corners must crossover into many and lots of different dimensions, so it goes to suggest that they must probably crossover every, or most, dimensions, including the ones that light travels in.
Now, if enough light hits enough corners, travelling from (starting from) enough different dimensions (assuming it can even be reflected/deflected into hitting the corners of the monster; covering all grounds)(and assuming light could even act the same within it's presumed finite dimensional range in any set of dimensions, that's a pondering for another day;[*]), then, some of the monster could be illuminated and visualized. Maybe the light could travel along edges, not just corners, or perhaps even it would be hard for us to distinguish between corners and edges at that high level of dimensions; this way we could possibly see more of it.
Through seeing the corners we could get some kind of glimpse of what the shape of it could be, from some presumably very limited angle; yet, we could estimate a full picture of the monster by calculating symmetry patterns in what we do see.
* -[perhaps the higher the dimension the light travels in, the higher range of dimensions it occupies. Wherein, the range of dimensions it occupies as it occupies higher dimensions could expand at a proportional/formulaic rate as the range of dimensions it occupies gets higher]
TL;DR: To the one person who may have actually been interested in this: Thank you. You're welcome!
@Pybro by the way, I like this use of the definition here, very refreshing. thanks
21:15 "The universe doesn't really care if its final answers look clean, they are what they are by logical necessity with no concern over how easily we'll be able to understand them" - I love this sentence!
This video was amazing. It's just so cool that structure arises when studying symmetries themselves. Your videos always inspire in me a sense of awe and wonder, but this video was something else. Thank you for all the time and effort you put into this.
Indeed, it WAS something else. One of his very best; and the bar was sky high to begin with
Let's not discuss platonism vs constructivism.
It looks possibly to have taken inspiration from Niel Degrasse Tyson's quotation, something to the effect of, "the universe is under no obligation to make sense to you."
@Dr Deuteron He looks like a gentle giant to me, so maybe he cares about our universe. Let's just be careful not to go near baby monster lol
@Dr Deuteron no, it is just you who don't want to think about the monster so you make it sound scary, even though the monster has always been part of reality & doesn't hurt us
This is absolutely brilliant! I have been looking for an introduction to Group theory that would help me understand some of the foundations of Galois representations to try to grasp even a very general understanding of the maths underlying the proof of Fermat’s last theorem. But most video content on Galois groups assume so much knowledge already that I couldn’t make any headway, until I found this one, so thanks! I loved your channel anyway, as an amateur maths enthusiast :)
This is truly one of the best videos describing overview of group theory and its recent developments in the past few decades. Thank you!
Any chance of an "Essence of Group Theory" series? I would love that!
underrated comment
As a physics student I really need that
I would really love that!
plhease
Sounds like fun
Thanks to the math community for having good and reasonable events often, it's always a pleasure to see math collabs.
Lmao you say math collabs like it’s a song! I can’t wait for “Monstrous Moonshine Conjecture” by Mc-Kay feat. John Conway
@@carlsagan1377 Lol
@@carlsagan1377 You'll be waiting the rest of your life, he died in April.
@@maxwellsequation4887 Oftenly isn't a word, you can just use often.
Xeridanus I know, and to COVID, too. It’s a damn shame. RIP
So informative! The fact that these videos are so synoptic makes them all the easier to understand. So often lecturers can be rather reductionistic in their approach, and students suffer as a result.
I've already concluded with the Euler constant that no matter how abstract we get we will never get rid of arbitrary constants. If anything, I hope that one day group theory will be able to derive geometrical and algebraic constants from fundamental logic.
I think you're mixing math and physics - all constants in math are very well defined, and we know how they arise. In physics, constants and principles are usually made up to correct the formulas into observations and nobody knows why exactly those numbers, nor do they apply everywhere or just here (anthropic principle)
@@HorukAI This whole video id about a mathematical constant and nobody knows why it's exactly that. I mean, we can calculate it, but that doesn't explain much.
@@tacticalassaultanteater9678 sorry this was not a constant, but a cardinality of a particular finite simple group. My point is that constants in mathematic indeed do arise from deduction process (logic) while in physics it's often an equation scalar based on observation, or mathematical necessity added so that it fits to hypothesis (think inflation)
@@HorukAI A cardinality is a number and a number not dependent on any variable is a constant.
@@HorukAI Numbers like the gravitational constants aren't even proper constants, they are derived from the quantities we pick. When I say constant I mean precisely values like π, Euler's number or the cardinality of the monster group. Artifacts of logic, independent of our circumstances, universal, yet hard to explain why their values are exactly what they are.
This was like watching anime without subtitles. I didn’t understand a thing but it was gorgeous.
Observing this creature vlog about their life's passion is like watching an A.G.I. install itself onto a biological framework, that just happens to be named Grant Sanderson for convenience. But I see you. You go, you wild fractal-made-conscious, you.
Right it was frustrating I watched the whole thing and was like wait that’s it
yeah the video is so interesting and it's so annoying that i still don't really understand it also my dyslexia doesn't help either...
can relate
"""""""anime""""""
"Nothing has given me the feeling that I understand why the monster is there."
That quote is honestly viscerally terrifying.
That's some Cthulu shit
I was worried since I saw the Numberphile video with John Conway, and I'm still worried after watching this.
@@staglomagnifico5711 the way he said "oh no, it's not arbitrary" has sort of haunted me since. RIP Sir John Conway
i honestly love it; it's a big indication of how the universe doesn't really care if it makes sense to us. if anything, to me it seems to actively resist being understood, like it doesn't like humans poking and prodding at it so it conjures up stuff like quantum physics and the monster group as if to say "ha! explain THIS". it's amusing and oddly charming, even if it is just me personifying reality itself.
Math always felt so incredibly clean, structured and sterile to me. The idea that something so bizarre could exist in it, with nobody understanding why, kinda terrifies me too. It feels anomalous, like something that shouldn't even exist. Gives me SCP vibes to be honest. Remember Theta Prime?
I lack the words to express how blown my mind is, and I'm amazed at how well you explained it conceptually... it's just so much to understand...
God, as a high school student, this one of the most challenging videos I’ve ever watched. My brain hurts.
That’s making space for more Info 😳
You should watch the videos on 'Tree 3' from Numberphile's channel. I've never felt such a sense of doom. Not as challenging, just bizarre.
Sometimes I lull myself into believing that Grant is a normal human being, and then I see a video like this, and I remember that we are speaking with higher-dimensional beings.
@Juho Grohn Actually the representation was 196,883^2. So he is much bigger than just 196,883.
Yeah, I learned a "lot" of math during my CS/info-engineering studies, and I loved group theory and Galois fields, but maaan - the Monstrous Moonshine Conjecture
It is comments like this that put a smile on my face every time I open CZcams. Thanks.
LOL!!! :D Same here! A lingo that lower-dimensional beings like us are not destined to understand!
The higher-dimensional beings only speak to you, you can never speak to them ;)
Physicists: 11 dimensions... That's a lot...
Mathematicians: Haha dimensions go brrrrrrrrr
69 dimensions is pretty fun
Grahams number be like
in case you dont get it, grahams number was originally created as a number of dimensions involved in a math problem. Mathematicians had to define that massive number of dimensions to solve the problem.
@@lilapela haha
@@rhealastname266 nice
That 11 dimensions relate to 1+1+1+8, and the 196,560 of the Leech lattice relates to 24, where bosonic M-theory is in 27=1+1+1+24 dimensions. There are physical theories with 196,884 degrees of freedom that contain both the 196,883 and the 196,560 of the Leech.
I just noticed the preview in the thumbnail (when hovering over the video before clicking it) is a perfect loop of rotating the wood puzzle. Definitely seems like an easily overlooked, intentional detail, so good job!
this is one of my all-time favorite videos on youtube, i keep coming back to rewatch it
"I'm the smartest guy alive and I just finished solving this 50 year-old problem, and I'm going to name this group what I want, dammit!"
John Conway dgaf
"I'm the smartest guy *alive*"
@@bored_person rip
Griess is still alive... but both Fischer and Conway passed away this year :/
hes like cave johnson fron portal 2
I'm not THAT much of a math person, but when you showed how the square permutations are the same as the dot permutations, my mind was blown in the best possible way
same dude same
THAT is what non math people should be seeing im math classes, not boring formulas without context
cube
Blowing your mind in the best possible way is one of the most beautiful things about learning pure mathematics imo.
It comes from a deeper and through understanding, and it's very satisfying.
Yep. It is a 1 to 1 correspondence between cube diagonals and permutations of 4 objects. Sounds scary if you don't know the jargon.
I'm at the end of my CS Ba. with maths as secondary subject and just this last semester I was told off-handedly that we know what *all* of the fields are. This blew my mind. But nothing in all my university education has come close to giving me an intuitive understanding of groups like this video. hats off to you, Mr 1Brown, my subscription is long overdue!
This is really fascinating. The more I learn about math, the more I realize the ways we can transform reality into abstract symbols that we can use to find patterns. This a pretty broad generalization, but it has huge implications. We can express just about *any* abstract concept in *any* particular facet of the entire lived experience of a human, or to *any* small detail of the entire universe in a defined, processable manner. How freaking insane is that? The more I learn, the more I can see the patterns between things I’d never thought to connect to each other, and I still don’t even know what I don’t know yet. I’m not sure I have any other words for that than “awesome” in the most literal sense.
I've been going through this process. It's a bit of a rabbit hole...
As a graduate student who studies the briefly-mentioned “theory of modular forms” and knows about the monstrous moonshine conjecture (my advisor proved a related ‘moonshine conjecture’) this video was truly wonderful and the best conceptual introduction to group theory I’ve seen. Well done!
I'm interested in understanding the statement of the 'Modularity theorem' and also getting some idea of what the Langlands program is about. Can you recommend some books or lectures? (I don't have a Mathematics degree)
@@MrAlRats
As one getting a maths degree I'm also interested in references.
commenting incase someone mentions references for me to add to my ever growing reading list.
@@MrAlRats Knapp "Elliptic curves", it's covered in the last chapter or near the end of the book.
I just graduated with a Master's degree in Mathematics, my thesis was on modular forms stuff. Who was your advisor (if you don't mind sharing)? I've attended the Automorphic Forms Workshop a couple of times, I've probably at least heard their name.
For everyone else, I think Kilford's book on Modular forms was somewhat readable with a decent background in group theory (maybe some number theory) and knowing kind of what Fourier expansions are. For anyone outside of math, you need to learn what proofs are first, and then a few more things.
Diamond and Shurman's book on Modular Forms is a beast. I kind of managed to get through the first chapter and maybe a few pieces of the second chapter, although I tried to do that before Kilford, so I think it would be easier now. Also, I think some experience with complex analysis might be necessary that I wasn't fully solid on. Modular forms was not an easy topic for me to just jump into.
Progressing from no math beyond Calculus to modular forms, here are some books I might recommend:
Learn proofs:
A Transition to Advanced Mathematics, Doud and Nielsen (free and well written, highly recommend)
Learn group theory:
A First Course in Abstract Algebra, Fraleigh
Representations and Characters of Groups, James and Liebeck (all solutions at the back of the book, it's amazing; also learn a bit about generalizations of this 196,883 thing)
Galois Theory, Cox (not strictly necessary to get to modular forms, but you do get to see what's going on with the roots of polynomials business)
Abstract Algebra, Dummit and Foote (if you really want to go hard; also, just the first fourteen chapters - after that it goes into Algebraic Geometry and other topics; also you might want to at least take a glance since Kilford touches on exact sequences and Dummit and Foote has a section on them)
Learn complex analysis:
Understanding Analysis, Stephen Abbott (probably best to start with Real Analysis)
Fundamentals of Complex Analysis with Applications to Engineering and Science, Saff and Snider
Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable, Ahlfors (this is probably the standard text, although I used the one by Stein and Shakarchi, which was alright)
Some basic ideas of Number Theory might not be bad, maybe something like Fundamentals of Number Theory by LeVeque, and maybe Introduction to Analytic Number Theory by Apostol, seeing as modular forms in Kilford are mostly treated in the sense of Analytic Number Theory.
Then Kilford.
Going through all of those books is probably not fully necessary, but since when is stopping to smell the roses a bad thing in math?
I just did an entire semester on group theory, and yet every second of this video had something for me to take away. Brilliant stuff!
you will find that you obtain gains even by taking that exact same class you took. I've learned through exp that you could take the same course and even use the exact same text and do the exact same chapters and still make gains. I can't even recall a time in my undergrad where i felt like I mastered any text book so well it was rendered useless...
@@dasmartretard Well said
@@dasmartretard Really on point. I've authored textbooks and learned much more about the subject by simply focusing so deeply on the foundations. Even new ideas come together more clearly for me on subsequent editions. I'm convinced that just a solid introductory book can make someone much more competent in applied work than the typical PhD student gets after a dozen courses.
15:04 that’s interesting, in the Galois theory course I did this year, we didn’t do composition series. Instead, we showed that insolubility of a polynomial by radicals is implied by the Galois group being insoluble. Using the fact that Sn is insoluble for n>=5, you’re basically done.
"What is it? It's the size of the monster."
Out of context, that line is completely terrifying
"What's the most important thing in math?"
"Coming up with funny names."
Google is a # and googleplex is that # squared
I wish mathematicians were more creative with naming and I didn't have to relearn the definition of "normal" and "regular" in about fifty different contexts.
@@ianvisser7899 FaK u
a sUbseT oF a mETrIC SPaCe M Is callED ClOPen WHEN its BOtH oPen AnD cLosed
@@chadpatrick6795 that is wrong
"Monstrous Moonshine Theory" would be a hell of a band name
I will be using that thank you very much
Probably a post rock band name
You have to do Math Rock with this name. :)
@@kjl3080 Keep us posted
Justin Golden ok
I'm honored to have a good friend who helped prove the umbral moonshine conjecture, but it's far, far beyond my understanding. I definitely remember group theory from linear algebra and vector spaces, and they are absolutely beautiful, but my impression from my own reading was that finite group theory gets very complicated very quickly - just as with chemistry analogy! Lovely video as always. Thank you!
Your use of teasing and suspense is masterful. These videos have some of the best pacing and production value I've ever seen. Thank you
C'mon guys. It's his favourite number and none of you even cared to read all the digits.
TOTALLY UNDERRATED COMMENT
Thanks bro. 😂❤️🙈
Eight hundred and eight sedecillion, seventeen quindecillion, four hundred and twenty four quattordecillion, seven hundred and ninety four tredecillion, five hundred and twelve duodecillion, eight hundred and seventy five undecillion, eight hundred and eighty six decillion, four hundred and fifty nine nonillion, nine hundred and four octillion, nine hundred and sixty one septillion, seven hundred and ten sextillion, seven hundred and fifty seven quintillion, five quadrillion, seven hundred and fifty four trillion and three hundred and sixty eight billion.
@@ML-xp1kp ahaha you took your time, you really took your time
I honestly don't have the energy to do so.
man I need some coffee.
This makes me feel like maths and physics is getting closer and closer to the source code of the universe
we are coded in source 8, not so buggy
Things like this studied by maths are actually more fundamental than the universe itself. If another universe were to exist with completely different physics yet housing some form of entities of sufficient intelligence, they would eventually come to these same results, since they are consequences of logical necessity rather than properties of the universe we inhabit.
@@MatthijsvanDuin what you just said blows my mind, could you, please, explain more about it? Cant logic work diferent in other universes? Truly interesting coment!
@@FnafTeamESPANA I guess it's plausible that we could live in a universe in which there were no dark energy and still have logic work the same.
@@FnafTeamESPANA In a sense, logic works independent from the universe. To reach a logical conclusion, you start with some set of assumptions, and then combine those axioms to reach some sort of conclusion. Because we exist in the universe, our starting assumptions are typically tied to the universe, but the conclusions we reach are only tied to those assumptions. We could start with an assumption that is untrue like "glass breaks when you hit it with a soft object" and reach the conclusion "if I hit a window with a feather, it will break". This conclusion is not true in our universe, but the logic is correct given the starting assumption.
Since logical conclusions seem to be independent from the universe we inhabit, it seems likely that if there were another universe with different physical laws, logic would still work independently from those laws. An intelligent species in that universe would likely still recognize Pythagoras' theorem as true, even if they didn't have any use for planar geometry, since they could still recognize that if they started with certain assumptions, Pythagoras' theorem would be a necessary result. Similarly, if these alternate universe mathematicians started studying symmetry, they would also eventually discover the Monster, since the Monster is tied only to the idea of symmetry, and not to any physical reality of symmetry.
Wow, I’m 4 weeks into my modern algebra course and this is making so much sense now. Literally the ‘click’ moment happened when you were explaining isomorphisms 🤯🤯
I'm so grateful for your work in explaining these concepts so clearly. I don't have the time to study these things in detail.
You gave essential intuition for one of the most interesting and complicated fields of modern math since it can pop up almost anywhere. Wow. Your video can actually serve as a first step into studying this branch.
I found this to be a little info-and-reference dense to be a single first step into an extremely complex field, kinda a weird flex
14:08 “This question turns out to be hard - exceedingly hard.”
Yep. I wholeheartedly and immediately believe you.
How does one even begin to ask questions like that? You'd first have to believe that there is an answer that is a finite number (instead of infinite possibilities). How do you you figure THAT out?
@@rayniac211 There are infinite possibilities. But we can still structure them.
4 is the answer I can betcha
Mathematician here. Let me assure you that this "number line of hardness" is not a walk in the park. Probably logarithmic scale. The third point, where it says "a/(b+c)+b/(c+a)+c/(b+a)=4" (btw, he made a little typo in the last denominator) as in "find (all) positive integers a,b,c that fulfill this equation", is already mind-bogglingly hard. And the fourth point is Fermat's Last Theorem. Formulated in 17. century, it took until 1994 to finally prove it.
Lone Starr yeah, I picked up on that as well. Would you say 3b1bs assessment of the difficulty in finding all the simple groups is accurate? As in is it significantly harder than Fermat’s last theorem?
I love your videos, fortunately for me my father is a mathematician so whenever I don’t understand something I can discuss it with him till I understand. Thank you for the great moments!
This is easily the best introduction to group theory out there, thank you.
Grant: "To maintain some hope of sanity..."
Me: thank you so much
Annddd... it's gone :(
*Next video: But what is the meaning of life? A visual introduction*
I know the meaning of life but I won't tell it to you :) Give me permanent citizenship
I just wanted to like your comment, but I figured the topic "the meaning of life" makes 42 a suitable number of likes.
@@skraemerLP rip it exceeded 42 :(
3blue1brown is a higher being who can understand things he can't even explain to us.
This is the first of your videos where I felt more clueless than ever, usually I feel like getting closer to God when I watch your videos but not in this one... the monster beat me xD (probably because I never read a single line of text about group theory hahahahaha while otherwise I am familiar with other branches of mathematics in varying degrees)
Please make more videos like this!!! You are truly good at digesting and telling stories!!
i remember one time i found this professor's website that i think related to this type of math, it was like this page specifically about "so you want to learn the monster" i believe. this guy had like a successive plan of steps to take, course after course with like free notes and stuff for self-learning. apparently the real work was even like years after postgrad even because he still had these courses listed that you should learn from before you could start understanding papers in this. all in all he had like over 40 textbooks recommended, and said that he was going to spend his time creating a video archive on youtube for all this stuff so someone could basically use all of the stuff in his guide plus the videos to learn the Monster on their own if they really wanted to. i can't find it anymore but that prof was an absolute legend.
This is a gorgeous explanation of the monster, presented in a way I can almost understand. My father would have loved this.
Could it be that your dad's name was John?
Your father was an amazing man
Sorry for your lost man…
Your father was most likely a cool nerd
@@Mipetz38 Did you check the last name?
17:02 "It's like the universe was designed by committee" brilliant
I thought he said "comedy"
I think this committee is really fucking up the design of the universe. God: "Hold my beer".
It's really external to this universe. This applies to all possible universes.
its a great story idea. Though, i guess its already been done, considering all the "we are in a simulation" conspiracy theories. or the quintara marathon, lol. Just an experiment by a committee of extra-dimensional scientists to see if they can make a universe capable of developing life. oh snap, isnt that a rick and morty bit too hahaha
It would be absolutely amazing if you could make a video elaborating on Noether's Theorem 😍 With your ability to deliver different concepts and ideas I have a feeling it will be outstanding
Wow I cannot express how happy I am that explanations of math like this exist; Aware of the difficulties of abstracting in math, and therefore drawing enough parallels and visualizations.
I can honestly say that without this channel my love of math would've been killed by highschool math.
The part about cycling three elements around (and ending up where you started if you keep doing it) jumped out at me, because that shows up a lot in Rubik's cube solves.
Maybe this is an urban legend, but I read that Rubik invented his cube while trying to explain group theory to his students.
@M J He was in architecture.
Yes, moves you can make on a Rubik’s cube, as well as sequences of moves, where two sequences which change the configuration in the same way are treated as equivalent, are elements of a group.
@@rchaser Hi. Want some science-recommandatios?
@@rchaser pretty sure thats not true he just wanted to make a block out of smaller blocks.
Can I just say that the abstraction analogy was so genius, it gave me the chills. It also raised a particular question for me: if our ability to understand the abstraction of numbers early on is a result of us being drilled with sheets and sheets of basic addition/multiplication homework, would it be possible for someone to grind through sheets of basic group theory problems and end up having a clear understanding of the abstraction of groups?
I'd say it's the case for those mathematicians. They manipulate groups with ease, similar to what we would do with numbers.
@@totolamenace I agree. It just seems crazy. I feel like I could never wrap my head around group theory well enough to work with them so easily, but clearly, that's what those mathematicians also thought when they were in high school (maybe).
To think that everyone at some point in their life feel the kind of confusion towards something like "3" that is similar to the kind of confusion I now feel towards something like "S5", just blows my mind.
I agree: "Genius" is an apt term for that analogy. I wonder how many eyes were opened by the care taken in explaining that the way he did
@@FareSkwareGamesFSG The important thing is to get used to abstraction but also making things understandable and fitting things in a context, like the group actions stuff that was talked about here. It turns out all groups come from symmetries of objects but how complicated those objects are depends on the group. There is a certain truth to von Neumann's quote (which I think was a sort of joke) "In mathematics you don't understand things. You just get used to them."
watched this 2 year ago thinking that i will never use this in my field, came back here because i have to rotate some stupid robot efficiently
WHat a great talk. Making group theory more approachable is indeed a laudable goal. I remember learning those same first axioms about groups, and feeling exactly that it seemed arbitrary. Conway is a maniac, i am not surprised that he talks the monster group.
RIP John Conway. He said he wanted to understand the Monster before he died.
Dont worry, no one really dies like most people think we do. The essence from him will be born again!
No one can get away from this existence. At least not so fast and easily as many of us wish we could be.
C: RUBBISH!
Says the voice from beyond.
oh! I guess that proves your point !
Maybe we'll just regroup again
God's teaching it to him right now...
@@hexagonist23 How do you know?
Man, I am the worst human being in math. I don't even remember the multiplication tables. That said, the fact that I could get a grasp of what you were saying in this video tells a lot about how good of a teacher you are! I wish my high school teachers had been a fraction as fascinating as you were in this.
Loved the video! thanks!
Wait..
So you are telling me that throwing you into a class room with 30 others and drilled constantly on outdated theories/tactics didnt interest you?!
@@jakechamberlain2206 aka school
You can pass calculus 1 without memorizing most of the multiplication table I have memorized a few multiplication problems and got fairly fast at solving the rest which was good enough for me.
Really? I had to atleast get to an introduction to proofs class before I could understand this video
dont worry im sure even the stereotypical "math wiz" would be confused by this, most of this seems pretty esoteric and you probably have to be not only smart but also pretty invested in the subject
Grant, what an amazing video! I'm watching it since I'm studying group theory right now, and even though I'm just starting, it has helped me with understanding groups a lot! Thank you so much for making it!
How are you finding this subject? Are you a math/physics major? I have this course abt group theory in sept and I am very worried cause I’ve never been good with this much level of abstraction.
@@pellaria8012 I'm a mathematics student, and I saw an introduction to group theory in my lady trimester (and it really beat me up), and seeing the course again I could understand a lot more, and Grant's explanation helped me a lot! Also, a book from a Mexican author called Felipe Zaldívar about group theory helped me a lot as well
the amount of hard work you put in your videos is simply astonishing..
15:27 'This is a super high level description of course, with about a semester worth of details missing' It made my day.
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That is the comment, I was looking for.
Might be over my head but just in case you didn't understand, he most likely meant "high level" as in "simplified, asbtracted" and not as in "advanced" :)
"Not 4 dimensions, not 5, but we'll have to go to..."
Me: Hmm the next number is six so then it should be si..."
*sees the dimension counter started to increase more than 100000
My brain: ight imma head out
I had the exact same reaction lol
Indeed. Before this part, I was following along rather nicely, understanding most of the things being said. The same can't be said of the parts after this, and I am just as confused as I ever was watching most of 3blue1brown's videos on more advanced topics.
Oh i wish i could see your reaction when you learn about hyperbolic temporal dimensions.
@@yto6095 Well I'm looking forward to it
"head out" haHAAA
All your videos are awesome, but still, this one is probably my favorite so far. Thanks so much!!!
"The universe doesn't really care if it's final answers look clean" - Sabine Hossenfelder.
This is why I love Maths and Physics. "They are what they are by logical necessity." That's mindblowingly fascinating but at the same time like super trivial. Cause of course things can't be what they can't be.
the beautiful thing is why are things that are necessitated simultaneously surprising and fascinating?
If things were different, they wouldn’t be the same
You might be slightly missing the point.
It is what it is, or else it would be different.
Looking at literally everything: It's all symmetry groups?
3B1B: Always has been.
lmao
This is why I donated the man 1000 UCO :) like I said boss do not spend until 2024 haha
What a teacher.. you remind me of someone mate keep it up this is blowing my mind how good you are I didnt sub years ago, my bad ;) but I left math and dodged programming out of lazymind. Now i be a relic :) peace all
54 digits must be one hell of a big result I cant wait to see the ending
Damn it, when my highschool math teacher asked us what our favorite mathematical theory is, I should have put "monsterousmoonshine"
Yeah I half wish I was still in school just to see my math teachers face as I talk about the Monstrous Moonshine Conjecture
This video single-handedly motivated me to buy a book abt algebraic structure and learn it. Thank you very much.
17:04 no this is exactly what chemistry did, “these all are alike, those? those are extra”
Lanthanoids and actinoids? Yeah, there are 14 of each if you count either La/Ac or Lu/Lr to be part of the main group 3, and 15 of each if you treat group 3 as only having the two elements Sc and Y.
So there's either 28 or 30 rare earths, and 26 sporadic groups. _Almost_ ...
Lantanoids and Actinoids are only written down separately to make the periodic table look more clean. In reality, they fit there perfectly. They just make the table wider.
The parts of periodic table that do not fit this nice pattern are not a part of chemistry even in uni. There is a table of isotopes that has neutron and positron counts on its axes, and then there are nuclear isomers that differ from usual atoms by the way neutrons and protons swirl inside of a nuclei (metastable isomers) or weird shape (fission isomers), even though the number of nucleons is the same. They are randomly all over the table, with no apparent pattern.
@@user-yb5cn3np5q Unstable elements can be literally anything doe, as they need not be stable.
Some chemists may be concerned with the symmetry of chemistry... while we are concerned with the chemistry of symmetry
When I grow up I wanna be a mathematician for the sole reason to be able to make names like _the monstrous moonshine conjecture._
All the best! You made being a mathematician sound even more fun!
I just wanna come up with a conjecture so I can call it Conjecturey McConjectureFace.
You actually usually don't get to name your own conjectures or theorems.
Suppose your paper contains a new result labeled Theorem 1. After you paper becomes very important and discussed a lot, people will not want to keep calling it, Farrell's 2028 Theorem 1. They'll name it something catchy and says something about the result.
For example, whoever proved the fundamental theorem of calculus didn't call it the fundamental theorem. It was the people who began using it who named it that. And clearly, it was named like this because it's a "fundamental" theorem.
Or Dimensional Ejaculation
rather than calling it « ejaculation » you can give it your name
This video is beyond beautiful. It starts with something that is quintessential, that we all have an intuition about and can feel but is hard to put into words, i.e. symmetry. "Symmetry is everywhere" is an understatement. It literally is in every group. Cayley's theorem: "Every group G is isomorphic to a subgroup of a symmetric group".
What that means for us (well, kind of) is that every form of beauty that has pleased our eye, anything and everything that we find symmetric has a mathematical representation. This is very powerful. Because of groups, we can abstract out the actions and objects across domains and study them. And because of that theorem, we can abstract out the very concept of a group by just dealing with symmetric groups. That cute little fact allows the entire subject of group theory to be introduced, talked about, and be awed at by just studying the symmetry of snowflakes (among other things).
This realization after the fact is enlightening, but to have thought of it, and create a video this beautiful is sheer genius. This conveys a very important point: The most influential pedagogies can only be devised by those that have an intimate understanding of the subject matter. We love you, Grant!
Thank you for saving my Tuesday morning, prof just started with group-theory in the exact way you mentioned.
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So I looked up the monster group on wiki, and then further to the classification of finite simple groups, and my god that is very impressive. People call this the greatest intellectual achievement of humanity and it's not an overstatement.
Yes, there are only two steps needed to do chemistry:
1. Find the periodic table
2. Do all the chemistry
Jokes aside, I was lost during the first few minutes but I followed through till the end. What kind of charm is this, Grant??
...jokes? What jokes?
@@aliince9372 Jokes aside refers to the sarcastic statements that this comment made before saying, "jokes aside." It is a relatively commonly used phrase in english.
The way Grant makes his videos:
1. Find a topic
2. Make an incredible video about it
This is amazingly well done. Kudos!
I learned group theory few times and got to say this is vastly differen view than i am used to. It's far mor relaxing :)
Just pure admiration to the way you explain such abstract concepts in a such elegant, clear and concise way. Thank you for your time and effort. Much love, appreciation and respect for you.
*The beauty of this channel is that it makes these advanced math concepts feel approachable to those with no experience. We all appreciate these videos, keep doing what you do!*
Amazing visuals. Truly makes you appreciate the beauty of mathematics. Keep up the great work Grant.
Wow. I watched this video before beginning this academic year and found it very interesting (without much understanding of it though). Now, after almost a semester of studying group theory, I watched it again. So many things started to click on my mind. Thank you SO much for this.
18:01 RIP John Conway, sadly a victim of COVID-19.
:(((((
Did he really die _of_ covid or with? Well doesn't matter he is dead anyway. *sadface*
musashi939 jeez don’t
Damn it. :(
musashi939 yes he did
me: "ayo alien, is the nummber 8x10^53 interesting?"
alien: "yea for real i love the nummber"
underrated comment
don't speak to the alien that way. you will get vaporized so fast for being that gay.
@@DrakonlordEreshkihal-vf5o You are a disgrace to the Pablo community
me: "what about number 69?"
alien: "lmao"
@@DrakonlordEreshkihal-vf5o why r u geh
Next week I'm going to teach basics of group theory, group operations and the character table for undergraduate chemistry students. I remember the first time I had contact with the subject, I got as excited as I got confused and graspless about it. I never could make sense of WTF was a character table, a representation, an irred, and stuff (at the time).
Now, as a teacher, I felt compelled to give my students the opportunity to at the least understand these things, to then show them its massive applications on chemistry. This video helped me enormously on the task! Much appreciated for such a wonderful, plain, straight to the point presentation on a subject no undergrad or grad teacher ever wondered to offer on the subject.
Excellent groups introduction, well done and thank you!
Please make an “Essence of Group Theory”. We, the undergraduate math students, need your help
Edit: "We, the undergrads of America, need your help" was purely phrased that way as a reference to old Uncle Sam Posters with similar phrasing. I absolutely was not trying to exclude undergrads outside America. I literally just thought it made for cool phrasing, but I changed it.
yeeeees, pleaseeee. I really need it and I think lots of other math students too
We, the undergrads of the fuckin planet Earth, need your help
@@AlejandroFernandez-mq3jl yes!!!
not gonna lie i can live with not seeing the essence of calculus/linear algebra as I was studying them, but not having a "the essence of group theory" as an undergrad would make me extremely envious of future undergrads that have that series
Search up for Visual Group theory on youtube, by Professor Macauley. Once you get a grasp of what Group Theory is, you can watch the playlist of
Richard Borcherds, who was mentioned here. I think the course taught by Richard Borcherds is a tad bit more complicated and I think he mentioned himself it is for very ambitious undergrad students or first-year graduate.
To make sure I understand this correctly: the chart at 18:39 reflects that the set of finite simple groups is *countable* and that it consists of 18 countable infinite subsets (with possible overlap) as well as 26 countable finite subsets?
I ask because I would have assumed that the answer to the question "how many simple groups are there" would be just "it's infinite"... Which turns out to be true, it's just that there are some groups that can be extended indefinitely by a simple algorithm and some that by their nature can only exist in spaces of certain specific dimensions-one of which is the "monster group", so named because it's the biggest of the non-infinite families.
Really really interesting. In some ways it's a baffling thing to come to, but when I think about it I realize it's befuddling in a familiar way. It makes me think of prime numbers. Prime numbers emerge so simply from basic arithmetic, and yet they exhibit behaviour that can't be fully described with arithmetic alone. The sequence of primes is in a sense "arbitrary": there's no rigid pattern to them, only some fuzzy tendencies. But in another way, the sequence of primes is anything but arbitrary: there is no accident, it is exactly what it must be.
I think that, to someone who is building mathematics from the ground up, prime numbers would be the first sign that things can't be kept in control, that there is no way to prevent simple rules from leading to surprising results. It would be the first indication that there are things like the "monster group" waiting in the mists ahead.
The fact that the set of finite simple groups is countable is pretty easy to prove. The first infinite family mentioned is the set of cyclic groups of prime order, and we know there are infinitely many primes. It's also at most countable because there can only be finitely many groups of a given size (just enumerate all the possible times-tables).
That there are 18 infinite families and 26 exceptions is much harder to prove, of course.
I think each of the 26 exceptions is not a finite set, but rather a single group. Out of all finite simple groups, there are 18 countably infinite families, plus 26 groups that don't fit into any family.
You should check Godel's Incompleteness Theorem, I think you will love it
@@mertaliyigit3288
This really doesn't apply here.
@@sarahbell180 Why not? It looks like Gödel coding might well interest the OP, given his observation of unexpected complex behaviour from simple arithmetic rules.
Wow, this might be my favourite 3b1b video so far
Looks like I just watched it second time, and it's so marvelous I couldn't resist to watch to the end again!