Richard gave 3 references, but was too modest to give the fourth. Why is 196883+1=196884, where the 196883 is the lowest dimension of an irreducible representation of the monster group and 196884 is the coefficient of q in the j function? What’s the connection between these things? When this was first discovered, it was so weird it was labelled “monsterous moonshine” by John Conway and Simon Norton, and it remained a mystery until RIchard Borcherds finally proved it, using a new invention of his called Vertex Algebras. I do not have a book recommendation beyond those on Wikipedia, but now that I’ve broken the ice, maybe Richard would like to chip in with his recommendation?
I knew the name Richard Borcherds was familiar, I just couldn't pin it down! (For clarification, I am new to the channel, coming here via the MegFavNumber series.)
J. S. Milne lecture notes on Group Theory highly appreciates professor's quotes. Also cites a quote by John Conway in 2014 interview on Numberphile: "The one thing I would really like to know before I die is why the monster group exists".
Thank you for providing my new fav MegaFav video! I was already thrilled when 3Blue1Brown mentioned Moonshine, but this is even better. Just found your channel and I already love it🤩
I've been studying modular forms for the past few months and seeing your name appear in connection with Moonshine and here I was having no clue you had a RUclips channel! Incredible to find this! Thank you for the great content Professor Borcherds.
Ok, I saw the numberphile video about the Heegner numbers (covering the numbers 1, 2, 3, 7, 11, 19, 43, 67 and 163), but the fact you've now linked this to the number of dimensions the monster group lives in is just... insane. Mind blown.
Thank you for reminding me of "Time Travel and Other Mathematical Bewilderments!" It was one of the very few good math books I was able to read back in elementary/middle school, as it was one of the very few good math books they had at the town library. I had largely forgotten the title, and didn't know how to find it. As soon as I saw it open by chance to the four-color theorem page, I recognized the diagram. Being a kid, I thought the text was saying that that map was actually not colorable, after which I tried and failed to color it. 🤣 It also had the problem about the ant trying to traverse the 1cm bridge at a rate of 1cm/s while the bridge grew uniformly at 1km/s. Very interesting!
A book that helped me a lot is Gannon's "Moonshine beyond the Monster". Gannon visits many of the topics mentioned in the video and provides a rather broad background.
I've known about the Euler quadratics that produced a lot of prime numbers. I've known about Ramanujan finding these "almost integers" by calculating e^pi*sqrt{167} and similar numbers. I've known about the Monster Group and how it's the biggest Group that doesn't fit into any other description. I had ABSOLUTELY ZERO idea they were connected! If this doesn't make you feel some sort of deep emotion inside your soul, then you just aren't human.
Thanks, you answered a question that has bothered for some time... and opened new questions to which I seek answers.... you have my subscription, Thanks again..
I know that the numbers you mentioned 43, 67,163 are related to imaginary quadratic fields with class number one. So, you have brought together very interesting topics in mathematics through the observations of values of the exponential function.
On a side note, I can echo a strong recommendation for the free42 app, available at the URL you mention, for numerous platforms - pc, Mac, iOS, Android, at least. I have it on my iPhone and my iMac, and, as a very long-time hp RPN calculator enthusiast, I can confirm that it's tremendous! Kudos to Tom Okken for developing it! It is also the system used in an actual physical calculator made by SwissMicros, the DM42. Unlike the app, however, the DM42 is *not* free. Be advised that there are two versions of the app to choose from - binary and decimal. The binary version is faster, but is limited to 16-digit precision. The decimal version is still pretty fast, and has 34-digit precision. Also take note that, if you get the decimal version, and you want to see all the digits of a long number, just do Shift-. (decimal pt), which is the "SHOW" function. Hold it down for as long as you wish to see the full-digit display. Fred
Thank you for sharing this and providing a gateway into your channel. There is some fantastic content on here that’s masterfully explaining some really delicate and difficult concepts.
A nice thing about finite groups (and compact Lie groups) is the "unitarization trick" by which you can construct an invariant scalar product on any representation (i.e. a vector space with a group action by linear maps). So whenever you have a representation V of such a group, you can think of it as a subgroup of O(V) (or U(V)) with respect to a suitable scalar product.
"they've got the order wrong, they've got the dimension wrong and they've got the picture wrong - apparently comic books are not a reliable source of information about mathematics" - Richard E. Borcherds I can't stop laughing:)))
Thank you for giving us those views of some of Ramanujan's paper. I'd never seen this. It explains something I was going to ask -- how come exp(pi sqrt 58) is so near an integer, seeing as 58 isn't a Heegner number. And it turns out there's a 12th-power analogue of the 24th-power formula for j.
That 1, 2 and 3 are Heegner numbers is perhaps just the strong law of small numbers? Anyway, the Heegner numbers h>2 are the only h where there is exactly one class of binary quadratic form of discriminant -h. More simply, the Heegner numbers h>3 are the first 6 primes of the form 4p-1 where p is prime. There are plenty of other primes of this form, e.g. 211=4*53-1, but by then there is scope for many more classes of binary quadratic form of discriminant -h, and the magic vanishes.
I never made that connection with "poles" before! EDIT: Should that j value, that's exactly an integer, have a minus sign in front? Otherwise your next few equations don't make sense
Yes, but it's a single *complex* variable which means two real variables. If you wanted the whole function you'd need four dimensions. that's why the graph shows the absolute value of the output rather than the output itself.
There's either a mistake in the title or in the number you write onscreen at the start. A 4 or an 8? And thanks for the vid, now I'll watch the rest :)
@@richarde.borcherds7998 After watching it's obvious the title is the correct one so, no worries :) Just caught my attention right away :) Bet it will slip under many people's radar :p
Yup! The "8" in the title, description, and video at 8m47, is wrong; the "4" in many of the book references in the video, is right. It's 262537412640768000. Fred
The symmetry group of j is SL_2(Z), the 2 by 2 matrices with determinant one and integral entries. There is a certain way to describe the action described in the first book(or just wikipedia).
One could even go to a faithful action and take PSL_2(Z) as the group. PSL_2(Z) is the quotient obtained from SL_2(Z) by identifying the matrices +1 and -1. This group is generated by the two elements written down in the video: The reflection S: τ -> -1/τ and the shift T: τ -> 1 + τ. Seen this way, the answer is already implicit in the video.
I guess I'm pretty credulous, because I'd have assumed that the elliptic modular function was real precisely because it was so messy and chaotic looking. Like how "real life" based equations are super messy compared to the idealized versions.
I just realised the book was written by one of the people who came up with the tanyama-shimura theorem which was used to prove Fermat's last theorem. It seems this topic has applications everywhere.
Yes, it does look like maths and physics are both hinting at some very deep symmetrical ideas underlying how reality is. It's amazing stuff. I'd like to dip my toe, but am a tiny bit worried about disappearing down the rabbit hole and never coming back.
@@johntate6537 the problem is that these symmetrical ideas come from different fields I guess. If there was a person who had a deep understanding of all of the separate links in the chain, then maybe we could get answers?
Funnily enough there are a lot of connections with elliptic curves here, for example the j invariant function can be interpreted to be defined on the set of elliptic curves. Then two elliptic curves will be basically the same if they have the same j-invariant. Also solutions to quadratics show up when you want to find elliptic curves with certain Endomorphism rings (maps to itself) (ones that are not Z), these elliptic curves are said to have complex multiplication, another buzz word you might have heard.
You start with 262537412640768000, which is indeed 640320^3 and indeed approximates e^(pi*sqrt(163))-744, but when describing the elliptic modular function (and when writing the title and description) you change the four in the ten-millions place to an eight.
@@ophello I said if it's a normal number. That means that all combinations of numbers exist at some point in it, which means that despite them not all being consecutive, there would be infinitely many 9. However, it has not been proven to be normal.
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Richard gave 3 references, but was too modest to give the fourth. Why is 196883+1=196884, where the 196883 is the lowest dimension of an irreducible representation of the monster group and 196884 is the coefficient of q in the j function? What’s the connection between these things? When this was first discovered, it was so weird it was labelled “monsterous moonshine” by John Conway and Simon Norton, and it remained a mystery until RIchard Borcherds finally proved it, using a new invention of his called Vertex Algebras. I do not have a book recommendation beyond those on Wikipedia, but now that I’ve broken the ice, maybe Richard would like to chip in with his recommendation?
I knew the name Richard Borcherds was familiar, I just couldn't pin it down! (For clarification, I am new to the channel, coming here via the MegFavNumber series.)
I knew that I recognized the name! Hahaha! That was the best surprise ever!
J. S. Milne lecture notes on Group Theory highly appreciates professor's quotes. Also cites a quote by John Conway in 2014 interview on Numberphile: "The one thing I would really like to know before I die is why the monster group exists".
Brilliant
Thank you for providing my new fav MegaFav video! I was already thrilled when 3Blue1Brown mentioned Moonshine, but this is even better. Just found your channel and I already love it🤩
Added bonus is that you can be sure Prof. Borcherds speaks with some expertise in the subject. (He's won a Fields medal for his work in the area!)
I don’t mean to be disrespectful but was your father really a famous theorem?
I've been studying modular forms for the past few months and seeing your name appear in connection with Moonshine and here I was having no clue you had a RUclips channel! Incredible to find this! Thank you for the great content Professor Borcherds.
Nice, wasn't actually expecting you make a video on the theme.
Thank you so much for all your work. Truly, thank you !
“Comic book are not a reliable source of mathematical informations” 😂
To be fair it's never going to be easy to draw a 196883 dimensional object on a 2 dimensional page.
You are at 64032. You get at 6412172087. This says you are in. 103125. Also. That there is a 21875 that is now 218935625.
I can’t get it closer than 3149135988
Ok, I saw the numberphile video about the Heegner numbers (covering the numbers 1, 2, 3, 7, 11, 19, 43, 67 and 163), but the fact you've now linked this to the number of dimensions the monster group lives in is just... insane. Mind blown.
In my head Professor Borcherds has become a sort of David Attenborough of mathematical creatures
Today I Learned: why poles are so called! (and other bits)
Absolutely unfathomable. The connections. Thank you. Fantastic video. Also, I don't believe any of this. Frickin' nerds.
Came from 3B1B videos, this is insanely good content, thank you very much
Thank you for reminding me of "Time Travel and Other Mathematical Bewilderments!" It was one of the very few good math books I was able to read back in elementary/middle school, as it was one of the very few good math books they had at the town library. I had largely forgotten the title, and didn't know how to find it.
As soon as I saw it open by chance to the four-color theorem page, I recognized the diagram. Being a kid, I thought the text was saying that that map was actually not colorable, after which I tried and failed to color it. 🤣
It also had the problem about the ant trying to traverse the 1cm bridge at a rate of 1cm/s while the bridge grew uniformly at 1km/s. Very interesting!
I think another great reference for this is Cox, 'primes of the form x^2+ny^2', which is more approachable at the undergraduate level.
Super interesting topic! By the way, he missed a minus @ 8:56; this large integer is negative.
Thanks. I was looking for such a comment in order to check.
This is an absolutely wonderful and very interesting video
A book that helped me a lot is Gannon's "Moonshine beyond the Monster". Gannon visits many of the topics mentioned in the video and provides a rather broad background.
Are you by any chance a differential geometer?
@@dmr11235 Maybe eventually if things go well :D
Currently I'm just a grad student in mathematical physics who uses his favourite theorem as an alias.
I've known about the Euler quadratics that produced a lot of prime numbers.
I've known about Ramanujan finding these "almost integers" by calculating e^pi*sqrt{167} and similar numbers.
I've known about the Monster Group and how it's the biggest Group that doesn't fit into any other description.
I had ABSOLUTELY ZERO idea they were connected! If this doesn't make you feel some sort of deep emotion inside your soul, then you just aren't human.
Definitely glad to see this number show up.
21:03 And now this number has a name, it's called "The number whatever".
😂😂😂😂
Thanks, you answered a question that has bothered for some time... and opened new questions to which I seek answers.... you have my subscription, Thanks again..
I know that the numbers you mentioned 43, 67,163 are related to imaginary quadratic fields with class number one. So, you have brought together very interesting topics in mathematics through the observations of values of the exponential function.
On a side note, I can echo a strong recommendation for the free42 app, available at the URL you mention, for numerous platforms - pc, Mac, iOS, Android, at least.
I have it on my iPhone and my iMac, and, as a very long-time hp RPN calculator enthusiast, I can confirm that it's tremendous!
Kudos to Tom Okken for developing it! It is also the system used in an actual physical calculator made by SwissMicros, the DM42. Unlike the app, however, the DM42 is *not* free.
Be advised that there are two versions of the app to choose from - binary and decimal.
The binary version is faster, but is limited to 16-digit precision.
The decimal version is still pretty fast, and has 34-digit precision.
Also take note that, if you get the decimal version, and you want to see all the digits of a long number, just do Shift-. (decimal pt), which is the "SHOW" function. Hold it down for as long as you wish to see the full-digit display.
Fred
Thank you for sharing this and providing a gateway into your channel. There is some fantastic content on here that’s masterfully explaining some really delicate and difficult concepts.
Is the monster group a subgroup of SO(196884)? If not, even the term "rotations" used in the comic book is mathematically dubious. :)
Yes, the monster is a group of rotations. So they did get one thing right.
A nice thing about finite groups (and compact Lie groups) is the "unitarization trick" by which you can construct an invariant scalar product on any representation (i.e. a vector space with a group action by linear maps). So whenever you have a representation V of such a group, you can think of it as a subgroup of O(V) (or U(V)) with respect to a suitable scalar product.
@@lukiatiyah-singer5100 That's a good point!
Really amazing video. Enjoyed it very much
"they've got the order wrong, they've got the dimension wrong and they've got the picture wrong - apparently comic books are not a reliable source of information about mathematics" - Richard E. Borcherds
I can't stop laughing:)))
15:11 my mind is blown! All of a sudden, these quadratics with lots of primes are related to near integers involving e and pi? So crazy
Why? e and pi, whatever base we use to represent them, appear all over the place.
found you while looking for the videos on the theme. excellent video, sir
Thank you for giving us those views of some of Ramanujan's paper. I'd never seen this. It explains something I was going to ask -- how come exp(pi sqrt 58) is so near an integer, seeing as 58 isn't a Heegner number. And it turns out there's a 12th-power analogue of the 24th-power formula for j.
That 1, 2 and 3 are Heegner numbers is perhaps just the strong law of small numbers? Anyway, the Heegner numbers h>2 are the only h where there is exactly one class of binary quadratic form of discriminant -h.
More simply, the Heegner numbers h>3 are the first 6 primes of the form 4p-1 where p is prime. There are plenty of other primes of this form, e.g. 211=4*53-1, but by then there is scope for many more classes of binary quadratic form of discriminant -h, and the magic vanishes.
I never made that connection with "poles" before! EDIT: Should that j value, that's exactly an integer, have a minus sign in front? Otherwise your next few equations don't make sense
Currently 3 March 2021, 9.30am, AEST. 32051 views. How long until 196884 views
I recollect when the Martin Gardner hoax on Ramanujan's number was published in Scientific American. The issue came out on April 1st.
Amazing video :) Great work.
Great video
Yey! Thanks Rich!
At 3:06 for 93 I get very approximately exp(30.3), which has integer part of 14 digits.
And why are these numbers perfect cubes?
262537412640768000 = 640320^3
147197952000 = 5280^3
884736000 = 960^3
I think your right cylinder thumbnail is cool.
Amazing. Thank you
Why is j(τ) * q being expanded as a power series rather than j(τ)? That is, why is there a q⁻¹ term in the expansion?
Sir I envy your library....
So great...
What's the book in 12:37
Pi: A Source Book - J.L. Berggren, Jonathan Borwein, Peter Borwein
Hold up, where did the three dimensional graph come from? We were talking about a single variable function I thought
Yes, but it's a single *complex* variable which means two real variables.
If you wanted the whole function you'd need four dimensions. that's why the graph shows the absolute value of the output rather than the output itself.
Its called ramanujan constant ❣️
windows 10 built in calculator has high enough precision for e^(pi*sqrt(163))
Great !! No need to use Python !!!
@@vishalmishra3046 indeed
I did it on my phone (moto g5 +)
Thank you
163 and Ramanujan Constant - Numberphile from different angle.
There's either a mistake in the title or in the number you write onscreen at the start. A 4 or an 8? And thanks for the vid, now I'll watch the rest :)
I guess I should have done the number 111111111111
@@richarde.borcherds7998 After watching it's obvious the title is the correct one so, no worries :) Just caught my attention right away :) Bet it will slip under many people's radar :p
@@Godwinsname No, the 4 on the first page is correct. The title, description, and page at 8:47 are wrong.
@@ravi12346 Ok. I am not familiar with the number itself. It seemed unlikely, but I now also see the book at 1:56 has a 4 indeed.
Yup! The "8" in the title, description, and video at 8m47, is wrong; the "4" in many of the book references in the video, is right.
It's 262537412640768000.
Fred
This reinforces my belief that we are living in a simulation created by some alien playing with elliptic functions on his personal quantum computer
You have to remember that mathematics does not depend on your universe, so it cannot tell you anything about it.
@@stighemmer maybe it does, if logic necessity somehow varies by universes.
mind. blown.
At 7 minutes in, this sounds like a Lovecraft story except it's about obscure math instead of obscure history.
Very interesting!
What is the group structure generated by the symmetry in j? Is it something like R/Z?
The symmetry group of j is SL_2(Z), the 2 by 2 matrices with determinant one and integral entries. There is a certain way to describe the action described in the first book(or just wikipedia).
@@justanotherman1114 Amazing, thank you!
One could even go to a faithful action and take PSL_2(Z) as the group. PSL_2(Z) is the quotient obtained from SL_2(Z) by identifying the matrices +1 and -1. This group is generated by the two elements written down in the video:
The reflection S: τ -> -1/τ
and the shift T: τ -> 1 + τ.
Seen this way, the answer is already implicit in the video.
@@lukiatiyah-singer5100 Yes, I forgot this fact so in reality the symmetery group is PSL_2(Z).
Sir is there any conection between this and Heegner number ? DrRahul Rohtak Haryana India
Yes, there is a close connection to Heegner numbers; see en.wikipedia.org/wiki/Heegner_number
Hello Dr Gupta, is there a close relationship between professor Ramanujan and yourself?😂
Thankyou sir 👍, u r amazing
Martin Gardner's column was the main reason I used to buy Scientific American.
看到最后查了一下原来他就是证明相关定理的guy直接惊呆了😱
Thankyou
World of wonders...
I guess I'm pretty credulous, because I'd have assumed that the elliptic modular function was real precisely because it was so messy and chaotic looking. Like how "real life" based equations are super messy compared to the idealized versions.
0:45
Germans: DOCH!
OH!
Great video
Feel like these videos could be dubbed over with the x files theme
The truth is 'in' there! (Inside all the connections between the various maths!)
I just realised the book was written by one of the people who came up with the tanyama-shimura theorem which was used to prove Fermat's last theorem. It seems this topic has applications everywhere.
Yes, it does look like maths and physics are both hinting at some very deep symmetrical ideas underlying how reality is. It's amazing stuff. I'd like to dip my toe, but am a tiny bit worried about disappearing down the rabbit hole and never coming back.
@@johntate6537 the problem is that these symmetrical ideas come from different fields I guess. If there was a person who had a deep understanding of all of the separate links in the chain, then maybe we could get answers?
Funnily enough there are a lot of connections with elliptic curves here, for example the j invariant function can be interpreted to be defined on the set of elliptic curves. Then two elliptic curves will be basically the same if they have the same j-invariant. Also solutions to quadratics show up when you want to find elliptic curves with certain Endomorphism rings (maps to itself) (ones that are not Z), these elliptic curves are said to have complex multiplication, another buzz word you might have heard.
Now I get it completely how the Neanderthals must have felt when they met a Homo Sapiens...😮
20:53
The whole chapter is a hoax,,as an April fools prank...please be aware while referring to that book
Win10 standard calculator gives 262.537.412.640.768.743,99999999999925
The number in the title is different
You start with 262537412640768000, which is indeed 640320^3 and indeed approximates e^(pi*sqrt(163))-744, but when describing the elliptic modular function (and when writing the title and description) you change the four in the ten-millions place to an eight.
was hoping to see a quote from an episode of the simpsons (mathologer)
"the number whatever" 😂
Wow
Imagine a world where this man has 1000’s more followers than Jay Z.
I actually kinda like the low framerate
wow
Glad you lot enjoyed it! I did A level maths back in 1968 and I didn't follow much of this video. Hey-ho...
와 이걸 이렇게 하네
이게 왜 추천 영상에... 여기서 한국인을 보니 반갑네요
waitaminute! Description and title say 262537412680768000, in the video he writes 262537412640768000 .. What kind of trickery is this?
If it is found to be a normal number, you can can say that it is ...
.999999999999 followed by infinite 9s, which rounds up to that whole number.
But it’s not infinite nines.
@@ophello I said if it's a normal number. That means that all combinations of numbers exist at some point in it, which means that despite them not all being consecutive, there would be infinitely many 9. However, it has not been proven to be normal.
The number in the title is incorrect
4:35 He get progressively more annoyed as time goes on.
The number in the video title is incorrect, it should have a 4 where there's an 8
91
2*pi*tau should be simplified to tau^2
Tau, in this case, is a variable, not a constant which equals 2*pi
Bruno Moreira joke went over someone’s head!
It's an integer, proven last year
Low fps
Incredible video but quality is so poor... Maybe presentation would be better than live hand write?
anyway will gladly watch more
Wow. The production 'quality' is like 2001. Is it meant to look vintage or does he just not care?
It gets the point across just fine as is
I don't care. It's not every day that someone with a Fields medal makes his knowledge more accessible.
Good talk let down by your video skills
21 minutes of almost incoherent rambling. Truly a blessing to listen to, I'm going to watch a few more of your videos now.
It isn’t incoherent, nor is it rambling.
wow