3:57 "The connection between elliptic functions and modular forms is the key part of the proof of Fermat's Last Theorem": there is a confusion here between elliptic CURVES and elliptic functions (they are totally different). The modularity theorem which implies FLT is about the relationship between elliptic CURVES and modular forms.
"There are two ways of doing anything, the smart way and the dumb way. When you do it the smart way, that's mathematics." ~ Definition attributed to a kid in a fifth grade Lego class in Lexington, Massachusetts.
that's exactly me right now. started out hanging in there and by mid video was overwhelmed. my approach to these videos is find the place where i got lost first and start taking copious notes there after. i then go through my notes and try to research these problem areas until i either get it or give up for lack of prerequisites.
This is an appetizing introduction to the topic. Easily accessible overview to undergraduate students, with a vertiginous glimpse into the bottomless depths of modern analytic number theory.
I watched a couple of videos giving high level overview of the process and what I got out of it is that it combines many many different fields of mathematics and not just surface level of those fields either. So basically you'll probably need a degree in mathematics or years of amateur experience with all those fields to actually get it
Thank you so much. Great video. I tried to read Weil's introduction to his proof and was overwhelmed by how many things I couldn't understand. I still make my students read it, to see the amazing emotional journey, but I enjoy steadily plugging away at learning the bits of terminology. Your videos are wonderful!
12:04 Great attention to small design elements. If you pause a video in landscape mode at 12:04, the third circle matches exactly with the play/pause gray UI circle of the UA-cam app (at least on the iPhone).
no words can describe what i'm feeling. HOW IN THE WORLDS NOBODY EXPLAINED THE "why" OF THE DEFINITION ! I always wondered why we asked the function to change (accordingly to the weight) with SL2 matrices and now, with this approach, the need of matrices of determinant 1 come as evident. Truly wonderful +1 :))
I’ve just been watching Kimberly Brehm’s linear algebra series, and this is the first group theory video I’ve watched since then, and you have finally given me the necessary insight to have an intuition on what groups are.
Really awesome, and appreciate your work. From my childhood, I always dream, one day i will understand proof of FLT. Now I may hope, it is easy to complete my dream through your videos. Thank you 😊
I'm curious what your sources were for the definition. In my studies, we define the modularity as f(gamma(z)) = (cz +d)^k f(z), where gamma is some SL(2,Z) matrix. You had a negative k in your definition.
Proof of Fermat's Last Theorem for Village Idiots (works for the case of n=2 as well) To show: c^n a^n + b^n for all natural numbers, a,b,c,n, n >1 c = a + b c^n = (a + b)^n = [a^n + b^n] + f(a,b,n) Binomial Expansion c^n = [a^n + b^n] iff f(a,b,n) = 0 f(a,b,n) 0 c^n [a^n + b^n] QED n=2 "rectangular coordinates" c^2 = a^2 + b^2 + 2ab Note that 2ab = 4[(1/2)ab] represents the areas of four right triangles) "radial coordinates" Lete p:= pi, n= 2 multiply by pi pc^2 = pa^2 + pb^2 + p2ab Note that pc^2, pa^2, and pb^2 represent areas of circles, wile p2ab = a(2pb) is the product of a radius (a) and a circumference (2pb). This proof also works for multi-nomial functions. Note: every number is prime relative to its own base: a = a(a/a) = a(1_a) a + a = 2a (Godbach's Conjecture (now Theorem.... :) (Wiles' proof) used modular functions defined on the upper half of the complex plane. Trying to equate the two models is trying to square the circle. c = a + ib c* - a - ib cc* = a^2 + b^2 #^2 But #^2 = [cc*] +[2ab] = [a^2 + b^2] + [2ab] so complex numbers are irrelevant. Note: there are no positive numbers: - c = a-b, b>a iff b-c = a, a + 0 = a, a-a=0, a+a =2a Every number is prime relative to its own base: n = n(n/n), n + n = 2n (Goldbach) 1^2 1 (Russell's Paradox) In particular the group operation of multiplication requires the existence of both elements as a precondition, meaning there is no such multiplication as a group operation) (Clifford Algebras are much ado about nothing) Remember, you read it here first) There is much more to this story, but I don't have the spacetime to write it here.
Modular functions, modularity condition and modular forms are interesting as is modular group. Is tau the same as Ramanujan's tau? Modular action leads to comparing a batsman's performance with a bowler. I think if we could compare them, Muralitharan may turn out to be better than Tendulkar.
Just wanted to provide some feedback, it's a general pet peeve of mine, don't take it personally: Math has this way of defining things where all the whys? and the ideas get lost as they get written as some 1 letter-long matrices, groups, sets and some redefined + and * operators. Look how nice the function at 4:14 is: you know exactly what happens and what each thing is, then you call it "g" like every damn function that exists in a math textbook. Then the Bases become (base1, base2), (Bx,By), (b1, b2)? NO w1 and w2. You used a programming language for this video, can you imagine naming every class A,B,C,... and every variable a,b,c... it quickly becomes unreadable and wouldn't pass any code review, yet math seems to get away with it. 4:14 is also the point where you started losing me, it all quickly became a soup of random letters. It's a shame because you seem you have researched this topic well and wanted to share your intuition and understanding.
It's just that in math you should always have a way of disconnecting the object from how you are writing it. So for every object introduced you should pause the video and just think about it so it becomes instinctive. When you see (w1,w2) you should be seeing a lattice not some "random" variables. If every function looked like this modular_form(t), it would be so much more tedious to write maths.
Thank you very much for this. I'm a physicist, not a mathematician, but have become more interested in learning about this topic for a long time. Your explanation of how the weight appears in the definition of a modular form is the first time I've been able to understand this intuitively. You've really helped my understanding. Thank you.
Imagine seeing a classic and simple video on the subject compared to this ... and understanding even more ... given the unnecessary addition of causal and undefined and unproven informations
3:57 "The connection between elliptic functions and modular forms is the key part of the proof of Fermat's Last Theorem": there is a confusion here between elliptic CURVES and elliptic functions (they are totally different).
The modularity theorem which implies FLT is about the relationship between elliptic CURVES and modular forms.
"There are two ways of doing anything, the smart way and the dumb way. When you do it the smart way, that's mathematics." ~ Definition attributed to a kid in a fifth grade Lego class in Lexington, Massachusetts.
Dumb way is the method of learning and smart way is the result of it
How about the Max Power way?
Lego class?
Made it halfway through before I got lost - but that was much further than I expected to get! Great video!
that's exactly me right now. started out hanging in there and by mid video was overwhelmed. my approach to these videos is find the place where i got lost first and start taking copious notes there after. i then go through my notes and try to research these problem areas until i either get it or give up for lack of prerequisites.
This is an appetizing introduction to the topic. Easily accessible overview to undergraduate students, with a vertiginous glimpse into the bottomless depths of modern analytic number theory.
This is actually genius. I hope you explain the entire proof of FLT
I watched a couple of videos giving high level overview of the process and what I got out of it is that it combines many many different fields of mathematics and not just surface level of those fields either. So basically you'll probably need a degree in mathematics or years of amateur experience with all those fields to actually get it
These are very excellent videos. I really have to concentrate and hit pause and rewind a lot and take time to make notes, There's no fluff. Enjoy!
Very nice, thank you! Simple explanation of very non trivial things. Would love to see a sequel about Fermat's theorem. Subscribed.
Thank you so much. Great video. I tried to read Weil's introduction to his proof and was overwhelmed by how many things I couldn't understand. I still make my students read it, to see the amazing emotional journey, but I enjoy steadily plugging away at learning the bits of terminology. Your videos are wonderful!
12:04 Great attention to small design elements. If you pause a video in landscape mode at 12:04, the third circle matches exactly with the play/pause gray UI circle of the UA-cam app (at least on the iPhone).
Thank you. This was concise and clear.
no words can describe what i'm feeling.
HOW IN THE WORLDS NOBODY EXPLAINED THE "why" OF THE DEFINITION !
I always wondered why we asked the function to change (accordingly to the weight) with SL2 matrices and now, with this approach, the need of matrices of determinant 1 come as evident.
Truly wonderful +1 :))
I’ve just been watching Kimberly Brehm’s linear algebra series, and this is the first group theory video I’ve watched since then, and you have finally given me the necessary insight to have an intuition on what groups are.
Great video! Would love a sequel on the relation of modular forms to FLT
How beautiful! Well done, such a great presentation! I had never seen modular forms explained like this before. I am very grateful!
It is really nice to see a video on modularity:) I really enjoyed your video! Thanks!
Subscribed:)
Great video
Really awesome, and appreciate your work. From my childhood, I always dream, one day i will understand proof of FLT. Now I may hope, it is easy to complete my dream through your videos. Thank you 😊
I'm curious what your sources were for the definition. In my studies, we define the modularity as f(gamma(z)) = (cz +d)^k f(z), where gamma is some SL(2,Z) matrix. You had a negative k in your definition.
Thank you for this excellent video about Modular Forms which I learnt a lot just by viewing this excellent video.
at 9:26 the S and T should exchange?
Thank you!
Proof of Fermat's Last Theorem for Village Idiots
(works for the case of n=2 as well)
To show: c^n a^n + b^n for all natural numbers, a,b,c,n, n >1
c = a + b
c^n = (a + b)^n = [a^n + b^n] + f(a,b,n) Binomial Expansion
c^n = [a^n + b^n] iff f(a,b,n) = 0
f(a,b,n) 0
c^n [a^n + b^n] QED
n=2
"rectangular coordinates"
c^2 = a^2 + b^2 + 2ab
Note that 2ab = 4[(1/2)ab] represents the areas of four right triangles)
"radial coordinates"
Lete p:= pi, n= 2
multiply by pi
pc^2 = pa^2 + pb^2 + p2ab
Note that pc^2, pa^2, and pb^2 represent areas of circles, wile p2ab = a(2pb) is the product of a radius (a) and a circumference (2pb).
This proof also works for multi-nomial functions.
Note: every number is prime relative to its own base: a = a(a/a) = a(1_a)
a + a = 2a (Godbach's Conjecture (now Theorem.... :)
(Wiles' proof) used modular functions defined on the upper half of the complex plane. Trying to equate the two models is trying to square the circle.
c = a + ib
c* - a - ib
cc* = a^2 + b^2 #^2
But #^2 = [cc*] +[2ab] = [a^2 + b^2] + [2ab] so complex numbers are irrelevant.
Note: there are no positive numbers: - c = a-b, b>a iff b-c = a, a + 0 = a, a-a=0, a+a =2a
Every number is prime relative to its own base: n = n(n/n), n + n = 2n (Goldbach)
1^2 1 (Russell's Paradox)
In particular the group operation of multiplication requires the existence of both elements as a precondition, meaning there is no such multiplication as a group operation)
(Clifford Algebras are much ado about nothing)
Remember, you read it here first)
There is much more to this story, but I don't have the spacetime to write it here.
One question related to the code. The domain colouring was done in manim? Or you have imported an image in the background?
Thanks for this! btw the T^{-1} at 9:10 is wrong though... keep this great work going!
also what is white sum of device function sigma? (it is shown in subtitle) at 12:53
Modular functions, modularity condition and modular forms are interesting as is modular group. Is tau the same as Ramanujan's tau? Modular action leads to comparing a batsman's performance with a bowler. I think if we could compare them, Muralitharan may turn out to be better than Tendulkar.
Today I saw the video from Wiles. This video explains his ppt😃😃😃 thank you💪🏻
Personally, I am waiting for chat GPT 6, then I am going to ask it to summarize Wiles proof in a way a high school student would understand.
Great video
Subscribed!
You have a nice voice :D
Just wanted to provide some feedback, it's a general pet peeve of mine, don't take it personally:
Math has this way of defining things where all the whys? and the ideas get lost as they get written as some 1 letter-long matrices, groups, sets and some redefined + and * operators.
Look how nice the function at 4:14 is: you know exactly what happens and what each thing is, then you call it "g" like every damn function that exists in a math textbook. Then the Bases become (base1, base2), (Bx,By), (b1, b2)? NO w1 and w2.
You used a programming language for this video, can you imagine naming every class A,B,C,... and every variable a,b,c... it quickly becomes unreadable and wouldn't pass any code review, yet math seems to get away with it.
4:14 is also the point where you started losing me, it all quickly became a soup of random letters. It's a shame because you seem you have researched this topic well and wanted to share your intuition and understanding.
Do you have any idea how it can be made better? If "Yes!", please provide it... Otherwise - ...
You could find the why's if you looked up the original research papers.
It's just that in math you should always have a way of disconnecting the object from how you are writing it.
So for every object introduced you should pause the video and just think about it so it becomes instinctive.
When you see (w1,w2) you should be seeing a lattice not some "random" variables.
If every function looked like this modular_form(t), it would be so much more tedious to write maths.
Thank you very much for this. I'm a physicist, not a mathematician, but have become more interested in learning about this topic for a long time. Your explanation of how the weight appears in the definition of a modular form is the first time I've been able to understand this intuitively. You've really helped my understanding. Thank you.
Beauty!
I had one of these when i was
I am so confused
Great
A transcript that actually matches the content of this video would have been useful.
Modular forms Submit
Imagine seeing a classic and simple video on the subject compared to this ... and understanding even more ... given the unnecessary addition of causal and undefined and unproven informations