Modular forms: Introduction
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- เผยแพร่เมื่อ 20 ก.ค. 2024
- This lecture is part of an online graduate course on modular forms.
We introduce modular forms, and give several examples of how they were used to solve problems in apparently unrelated areas of mathematics.
I will not be following any particular book, but if anyone wants a suggestion for a book I recommend "A course on arithmetic" by J.-P. Serre (Chapter VII) and "Modular functions and Dirichlet series in number theory" by T. Apostol.
Typo: at 7:05 the definition of q at the bottom is missing a minus sign in the exponent.
For the other lectures in the course see • Modular forms
This man is so generous with his time - it's great to know (by listening to his Q&A) that he also enjoys making these videos too - it's a great time to be alive - thank you Prof Borcherds - I hope your research is going well. Thanks again for going to all this trouble - I find mathematics of all levels significantly more exciting when it is being explored in good company!! I cannot thank you enough.
People who give thanks in such a wonderful way should receive some recognition.
I share your sentiments and am glad to see a comment like yours.
Me: sucks at high school math
Also me : oh yeah, let's check out modular forms
Just found this series! As someone who’s mostly self taught in mathematics, modular forms have always been the final frontier of my understanding.
Almost everything I’ve learned before has these neat and concise solutions that I can grasp but when I look at problems related to modular forms all these strange numbers and identities come up that I just have a very tough time grasping but that just makes me want to understand them even more.
Can’t wait to watch the rest of your series!
Yes! I'm super stoked for this course!
Same!
This is outstanding! I’ll be watching this series closely and recommending it to all my math friends
That totally looks like magic
This is phenomenal. I watched your series on Lie Groups, and found it fascinating. This is fascinating cubed. Thank you so much, sir!
It shows that Prof has a deep love for this topic... I believe this is the ideal intro to modular forms
I'm really looking forward to this lecture, especially bc I'm taking a course in Eisenstein series, so thank you for covering this topic!
I just learned about this channel a few days ago. Great idea to do all these online lectures + lots of fun to follow. I am a Master student specializying heavily in representation theory and it would be nice to understand the connections coming from Hecke algebras and theory surrounding the Kazdhan Lusztig Conjecture. On first sight these modular forms look extremely complicated and horrendous.
Elliptic curves are dual to modular forms.
Ellipsoids are dual to hyperboloids -- linear algebra, matrices -- Gilbert Strang.
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
I have not got a clue what he is talking about. At school I struggled with simultaneous linear equations. He may as well have been speaking Klingon for all the sense it made to me. I can only admire people who can understand this.
Can you do a short undergrad course of Differential Geometry?
Really thanks for your contribution in open source math education
... I just want to say "thank you" for your modesty ... the world surely needs more people of your kind ... (and hopefully I - as a non-mathematician - shall understand at least something from your lectures) ...
Thanks a lot for this. Incredibly fascinating. Definitely planning to go through your courses as you keep adding to them (both graduate and some undergraduate courses)
Yes! I’m doing q-series & partitions. Techniques from modular forms seem to pop up every now and then. Really looking forward to properly learning these stuffs!
This is great! He's talking about math that he got the fields medal for working on!
Wow! Everything is connected to everything... Great Introduction!
Thank you Sir for delivering such a wonderful class on modular symmetry. Everything is explained so well.
Been waiting for this one 🙌😤
Thanks a lot for everything you post here!
Thank you very much professor. Your time and efforts are very much appreciated.
Can one of the lectures cover how to solve that exercise at the end?
Really great explanation!
After viewing professor's interview, really help. The suite on this important and exciting subject of modular forms, is natural. Thanks for earlier than expected first lecture video.
thank you so much Sir !
Around 7:32, it seems the definition of q is off, since it is not small (perhaps it should be the reciprocal of what is stated?).
Yes, I think it should be exp(-pi*sqrt(163)) instead of -exp(pi*sqrt(163)).
@@malharmanagoli It should be -exp(-pi*sqrt(163))
Yes, there is a minus sign missing from the exponent.
Thank you for this video! Monster group lives in 196883 dimensions, meaning it has a 196883-dimensional faithful representation. (Isn't this more precise?)
What a sales pitch!
I really liked Ram Murty's book on Modular forms
I am in high school currently and I want to study these apart from the math in my course books. Howevee, I need to start from basics as I am unaware ofsome of these terms.
Where should I start from?
Dr Borcherds, quick dumb question before I watch this series a second time, the periodic functions I'm familiar with all have period 2*PI so I'm wondering why for modular forms and functions we use period 1 instead of 2*PI, I'm pretty sure I'm wrong about "1", its probably any rational number since it comes from the SL2(Z) matrix, but near as I can tell the period is not irrational or transcendental. I guess I'm not sure which to be more surprised about, that modular forms were designed to have rational period, or that the periodic functions I'm familiar with have transcendental period in spite of having rational coefficients in their power series.
I think lecture #8 where you suggest that engineers get Fourier Transform wrong putting the 2*PI out front of the integral rather than in the exponent inside the integral tells me its pretty much "six of one or half dozen of another", you simply move the 2*PI around to simplify your equations as necessary or out of convenience to simplify things.
What's the professor's accent? I think I've heard it before from John Conway, but I don't know where to place it.
In order to understand what a Modular Form is , What Topics of Math should I know? What Prior Knowledge should one have ? & In what order should they be Studied ?
Be well versed in complex analysis and have decent knowledge of harmonic analysis, elliptic curves. Plus also ensure that you have sufficient knowledge on grp theory and algebraic structures in general. If you have all this, it should be easy for you to go along. But even if you are not the best at those, just pick up a modular forms book and whenever need comes, revert back to these topics
I allowed myself a small interludium in stability of maps of S^1=T^1, S^2 and T^2. I wonder if you have something along these lines in stock?
EDIT: The following comment is incorrect. See Prof. Borcherds' comment below.
16:15 I think the sum is over n >0?
If this is the case, at 15:45 the integral on RHS does converge (at least for Res large):
From the expression 16:15, when t gets large theta(it) decreases rapidly; and when Res is large, there is no problem when t-> 0 as well.
The sum is over all n in Z, and the integral does indeed not converge for any s (as it says in bright red). It needs to be regularized, which I plan to discuss in a later lecture.
Thanks for the reply! Now I see the sum should be over n>0 and n
The equality deemd "false" by the professor in 17:13 does not have an s on the right hand side, so I don't follow what it is supposed to mean.
the deeper you get into number theory, the more it turns into numerology
aaaah finally I know the meaning of 196884
well and the 24 I guess
Someone should create a discord for this channel (or at least this course).
Discords always turn into degenerate communities.... no matter what the original intention.....
Hmm, after seeing their appearance in at least seven disparate math areas shown in this video, I am beginning to think modular forms may be somewhat significant 😆
GOAT
Haha, all of this sounds so esoteric.
yeee
Submit modular forms P ≡ P (mod n)
1:17 He is very imaginary indeed