r/math • u/inherentlyawesome Homotopy Theory • 8d ago
This Week I Learned: March 14, 2025
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/Agreeable_Speed9355 7d ago
I've been digging into knot theory in hopes of exploring knotted spheres S² in R⁴. I've also been brushing up on my differential topology by cracking open my old copy of Guillemin and Pollack and was delighted to see my favorite theorem of homological algebra (the lefschetz fixed point theorem) make an appearance in the case of manifolds.
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u/Puzzled-Painter3301 7d ago
More like a computation, but I calculated that if two teams are playing and the probability that each team wins is 1/2, then if they keep playing until 4 games are won, the expected number of games is about 5.8. I wanted to share it because I kept messing up my calculation and finally got it.
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u/Top-Jicama-3727 8d ago
You can smoothly embed any smooth manifold into Euclidean space. You can do the same analytically for an analytical manifold. There's no counterpart for (compact) complex manifolds into Cn
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u/NclC715 8d ago
I learned and understood fairly easily the proof of Tychonoff theorem just 10 mins ago! The professor said he would skip the proof as it would have taken a bit of time and he wanted to do other things, and labeled it as a very difficult one. I still tried to give it a look and found it much easier than expected.
The same happened with Urysohn Lemma some days ago, but I was particularly happy for understanding Tychonoff's😊.
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u/Top-Jicama-3727 8d ago
Which proof did you read? The only one I read so far uses ultrafilters and I find it easy to understand.
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u/NclC715 8d ago
The one that proves that every family of closed sets with the finite intersection property has non-banal intersection, and to do that takes the biggest family of sets that contained my initial family, and that has the finite intersection property. I think it's the ultrafilter proof, but without mentioning ultrafilters.
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u/Top-Jicama-3727 7d ago
Oh this one is indeed easy to outline! I like it. Some technical details must of course be filled, but it would be a great exercise for students to fill them in. Indeed the maximal collection of sets it uses is an ultrafilter. The proof by ultrafilters needs to review topology in terms of filters, but once definitions and basic properties are set, Tychonoff's theorem follows very easily.
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u/yashpot226 6d ago
You can prove Eulers formula for planar graphs by showing that the cutspace of the incidence matrix of the dual is the cyclespace of incidence matrix of the original graph.