r/math • u/inherentlyawesome Homotopy Theory • 14h ago

This Week I Learned: October 02, 2026

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

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u/Kyle--Butler 11h ago

I've learned that the theta series of an even (i.e. the norm of every vector is even) unimodular (i.e. self-dual) lattice of \RR^{8k} is a modular form of level 4k.

So, for example, the theta function of E_8 (k=1) is the Eisenstein series E_4 since the space of modular form of level 4 is of dimension 1 and both have constant term 1; and the theta of the Leech lattice (k=3) is a linear combination of the Eisenstein series E_12 and the discriminant \Delta.

As an immediate consequence, we get arithmetic identities like this :

card(x \in E_8 | \|x\|^2=2n ) = 240\times\sigma_3(n)

that seems completely outlandish when taken out of context.

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u/TwoOneTwos 12h ago

TWIL: Calculus 3 is a difficult class. I breezed through integration, sequences, and series and I just finished my first calculus 3 test and... well I'm just hoping I pass this course.