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

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:

(Paper on MIT) Poisson summation on lattices - https://math.mit.edu/~brubaker/Math78...

Duals of lattices - https://cseweb.ucsd.edu/classes/wi12/...
Modular forms - https://wstein.org/books/modform/modf...

Modular form graph was made using Sage with code adapted from this StackExchange answer:
https://math.stackexchange.com/a/4309925

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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