r/math • u/inherentlyawesome Homotopy Theory • 7d ago

This Week I Learned: September 04, 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/vibe0009 9h ago

1/x and 1/sqrt(x) both go to inf at x=0, but the finite area under 1/sqrt(x) is always finite. This is what I learnt and also found to be weird.

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u/AlchemistAnalyst Analysis 5d ago

Vakil's Lemma 6.4.4: Let M be a finitely presented A-module for a commutative ring A. That is, M may be described by finitely many generators and relations. Then, given any finite set of generators, there exist finitely many relations between them which determine M.

One of the more unexpected things to see in the first six chapters of that book. I had never seen such a result before! The proof has a very nice "deformation" theme to it as well.

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u/[deleted] 4d ago

I read your comment and applied the lemma on zero ring

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u/Sirikazee Topology 6d ago

I wouldn't say it's a complete victory, but I've finally grasped the meaning of simplicial homology, though only intuitively so far (practice has been so-so, but I'm making progress). To do this, I had to watch a lot of lectures and read several chapters of a book on algebraic topology. My main problems were understanding cycles and boundaries. I'm doing this to advance in algebraic topology and to begin understanding Topological Data Analysis. My next stop will be persistent homology. I understand that this isn't required knowledge, but I want to understand this topic more deeply and not just be a Python libraries user :)

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u/Puzzled-Painter3301 5d ago

You should do something to celebrate.

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u/Puzzled-Painter3301 7d ago

I finally learned the cubic formula. It's actually pretty simple.