I was on the bed with flu, kind of bored, so I picked up a pen, some paper and started watching your modular forms lectures. Everything is motivated and linked to the other concepts. This is how math should be taught.
I was a bit skeptical at the beginning of the video expecting it to be a rather unpleasant pile of calculations but the video became an incredible teaser for the rest of the lectures. I am looking forward the rest of the lectures!
I believe the 1/m term in the series for pi/tan(pi*z) at 4:30 should be 1/(tau+m), and then one can either multiply by 1/2 or rewrite as 1/tau+ sum_{m=1}^infty [1/(tau+m)+1/(tau-m)].
I _feel_ that the 1728 somehow is connected to Ramanujan's observation about 1729. But I'm not equipped with enough knowledge to check if this hunch is correct.
1728=12^3 and Ramanujan observed that 1729=12^3+1^3=10^3+9^3 is the smallest number that is a sum of two cubes in two different ways. Haven't seen anything deep arise from the "taxicab" property, but it sure is neat!
@@theflaggeddragon9472 Yeah, I think you're right, it's likely got nothing much to do with Ramanujan's observation, and it's just that 1728 = 12^3. 12^3 is certainly not a surprise.
@@theflaggeddragon9472 1729 is the second Carmichael number en.wikipedia.org/wiki/Carmichael_number so who knows? there might be a relationship between sum of two cubes, Carmichael numbers and modular forms
Jubilation when you don't have to look at your notes to write the coefficient of q^2 in the j function :) I know by heart 196884 but didn't manipulate those enough to remember the next coefficient !
I find it beautiful that you use the action of SL(2, R) on the upper half plane with the element \tau -> \tau + 1 to parametrise the summation. Is there other ‘naturally important’ series that come from other subgroups of SL(2,R)?
I'm not even a math student but I'm finding myself excited each day for a new video on this channel. It's this and the latest Marvel episode.
I haven't watched the first episode of the Falcon and the Winter Soldier, but I've watched all three videos of modular forms so far
I dont want to spoil for you but future episodes of modular form will be great.
@@justanotherman1114 Haha, looking forward to them!
I was on the bed with flu, kind of bored, so I picked up a pen, some paper and started watching your modular forms lectures. Everything is motivated and linked to the other concepts. This is how math should be taught.
Prof Borcherds is a class act.
Best explanation for j-invariant I have seen so far! Weight 0 actually turns it into an invariant.
Thank you Dr. Borcherds!
It is possible to find more information about this "historical mistake"? 22:58
Check the last video in the series, "Modular forms: Petersson inner product" at 8 min.
I was a bit skeptical at the beginning of the video expecting it to be a rather unpleasant pile of calculations but the video became an incredible teaser for the rest of the lectures. I am looking forward the rest of the lectures!
I believe the 1/m term in the series for pi/tan(pi*z) at 4:30 should be 1/(tau+m), and then one can either multiply by 1/2 or rewrite as 1/tau+ sum_{m=1}^infty [1/(tau+m)+1/(tau-m)].
13:26 minor typo - (-1)^{k/2} + 1 and no 2 in the denominator
Fantastic.
9:50 minor typo - sin and cos should be reversed. Rest of expression ok.
Also e^{i \tau} should be e^{i \pi \tau}, and 2 pi i should be pi i. Although it really doesn't matter...
I _feel_ that the 1728 somehow is connected to Ramanujan's observation about 1729. But I'm not equipped with enough knowledge to check if this hunch is correct.
1728=12^3 and Ramanujan observed that 1729=12^3+1^3=10^3+9^3 is the smallest number that is a sum of two cubes in two different ways. Haven't seen anything deep arise from the "taxicab" property, but it sure is neat!
@@theflaggeddragon9472 Yeah, I think you're right, it's likely got nothing much to do with Ramanujan's observation, and it's just that 1728 = 12^3. 12^3 is certainly not a surprise.
@@theflaggeddragon9472 1729 is the second Carmichael number en.wikipedia.org/wiki/Carmichael_number so who knows? there might be a relationship between sum of two cubes, Carmichael numbers and modular forms
Jubilation when you don't have to look at your notes to write the coefficient of q^2 in the j function :)
I know by heart 196884 but didn't manipulate those enough to remember the next coefficient !
19:50 This is obtained by equating coefficient of q^2? Then it should be 240 sigma_3(2) * 2 + 240^2 = 480 sigma_7(2).
I find it beautiful that you use the action of SL(2, R) on the upper half plane with the element \tau -> \tau + 1 to parametrise the summation. Is there other ‘naturally important’ series that come from other subgroups of SL(2,R)?
There are! But I think you mean SL_2(Z). Not to plug myself, but I have a new series of lectures up that explain some of these things.
@@Entropize1 Woah! Cool channel. Subscribed!
@@mathmo I appreciate you!
May I know what is your camera set up? This is a very efficient way of teaching.
You can find it in the description of the channel. I think he made a community post about this as well.
I wonder what other funcion should have been in the numerator to avoid the historical mistake? E_6^2? E_12?
Must have for analytic number theory!
ye
Sir please make videos on real analysis please sir