Counting points on the E8 lattice with modular forms (theta functions) | #SoME2

In this video, I show a use of modular forms to answer a question about the E8 lattice. This video is meant to serve as an introduction to theta functions of lattices and to modular forms for those with some knowledge of vector spaces and series.

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References:

Modular form graph was made using Sage with code adapted from this StackExchange answer:

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This is my submission for 3blue1brown's Summer of Math Exposition 2 (SoME2) and is also my first math YouTube video. I'm excited to share this with you! Let me know if there are any mistakes I made in the video, and I will make note of them in this description.

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Chapters:

0:00 - Introduction
0:34 - Lattices
5:23 - Theta Functions
8:45 - Modular Forms
12:12 - Showing Modularity
15:12 - Spaces of Modular Forms
19:35 - Coefficients and Counting
21:25 - Conclusion

[xdd]

I was a bit surprised to see in the description that this was your first math video, considering how well you managed to use a mix of edited photos, graphs, and colors to get across (not only the motivation but) a substantial amount of the content of a subject that can be very unapproachable due to both jargon and notation - amazing work!

[johnchessant]

Great video! Another cool application of modular forms is a proof of the sum of two squares theorem. The theta function at 8:25, can be rewritten as θ(τ) = sum q^(2n^2), n = -inf to inf. Then the q^(2m) coefficient of θ(τ)^2 will be the number of ways to write m as the sum of two squares. Lucky for us, θ(τ)^2 happens to be a modular form of weight 1 over the congruence subgroup Γ_1(4) (this is a certain subgroup of SL_2(Z) of index 4). By comparing to a suitable Eisenstein series, we not only classify which numbers of sums of two squares, we also get Jacobi's formula for the number of ways to write a given number m as the sum of two squares, namely 4(d_1 - d_3) where d_i is the number of divisors of m which are i (mod 4).

[whitestonejazz]

This was awesome. Very reachable for non-experts compared to other modular form introductions

[tanchienhao]

Some2 has blessed us with a video on modular forms! Great video!

[chrisatwood]

That was so informative! Who knew there was such a great application for this higher level maths? Thanks for posting the video!

[omargaber]

This is the missing puzzle in all the videos I've watched regarding Fermat's last theorem, difficult but fascinating
Thank you from my heart🥰

[abdallahchaibedra]

Brilliant videos thank you a lot, I hope you can make more videos on the application of modular like its relation with elliptic curve

[Number_Cruncher]

Thank you. This is a good appetizer for crazy stuff.

[pra.]

This was so great, was writing and experimenting on my paper while watching

[shortnotes-bds]

awesomeeee!! keep up the good work buddy

[wyboo]

i love and hate this video. i love it because it's so interesting and let's me get a glimpse at a really beautiful and surprising connection in math, but i hate it because it makes me realize i'm not as smart as i think i am

[dancingdoungnut]

I would watch your videos, please post more!!

[wyboo]

1:38 "2d spheres ... which is circles"

its a great video and you know way more than me, and i don't want to nitpick but i want to correct this. from a topological standpoint circles are 1-d spheres. a sphere is just the boundary of a ball, where a sphere is the set of all points equidistant from an origin and a ball is the set of all points with distance less than the circle. a circle is a 1d sphere, and it's interior is a 2d ball. a sphere in 3d space is a 2d surface so it's a 2d sphere, and its interior is a 3d ball

[ldman]

The only thing I understood from the title was Counting points, yet I still enjoyed the video

[juvenwang]

Thank you for all the useful materials --- all even unimodular theta function is a modular form of the "level 1." Did you explain what is level of a modular form? at 22:04

[juvenwang]

Thanks for the video -- any pedagogical summary reference for deriving equations at 20:12?

[alex_zetsu]

You know how in a 2 dimensional lattice where nearby points are equidistant can be triangular or square? I know that there are more types of lattices in higher dimensions. I wonder if there is a way to specify them. For example, since I can't visualize a 7-dimentional lattice in my head much less draw one, is there a standard notation I could use to tell someone which lattice I meant?

[gendalfgray]

9:14 is it a graph? Is it 3d graph? Where additional marks like 0, x, y to help understand whats this is.

[therealist]

Might be a dumb question, but do lattices define an algebra? Or am I understanding the concept wrong? I can see how I'd do addition, subtraction, multiplication, and division using them (and by extension exponentiation and square roots).

[TheOneMaddin]

Sorry, but this went right over my head :( And I have a math PhD in a non-related field. I can't imagine that a non-mathematician can get very far with this. However, I susbcribe nevertheless, because I still learned something new and I like videos that are not too basic.