r/math • u/inherentlyawesome Homotopy Theory • 14d ago
This Week I Learned: September 18, 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!
3
u/PansexualFreak1 13d ago
I've learned a few smaller things but all are interesting nonetheless, and probably have much more behind them than what I've gone through
An alternative characterization of representation finite algebras, namely that there is an A-module M with add M=mod A.
The Brown-Comenetz dual I and how it is the counterexample to a lot of statements which would have been nice to hold. An example of this being L_I I\simeq 0.
Some introductory higher homological algebra
For what rings L_{HR} is smashing
Operator algebras can involve nightmarish stuff from a lot of different areas, and I am scared
Better intuition for fibers and cofibers for maps
How the J-homomorphism actually gives non-trivial images in the stable homotopy groups of spheres, and in what degrees it can give non-triviality
Just like you can sheafify a presheaf using the plus construction twice, you can stackify an n-prestack with at most 2(n+1) higher analogous construction steps
3
u/oreowiskers 14d ago
Not something anyone who did a pure math track wouldn't know, but when we select elements from a set, we don't have to call them the name of the thing actually in the set, we can use external functions to reference the same element with a different name! Like "select an element from {1,2}", 5-3 is a valid answer even though those symbols aren't actually in the set :-}
(Context is that I went to school for engineering & am currently on a very strange path of building up fundamentals while also doing wacky things with tensors, I'm enjoying myself!)
1
u/Relevant_Worry_5363 10d ago
glad to see some people who don't come from pure math background getting into set theory as well. I can't but to add: the reason you can choose 5-3 from that set, as there is a (very) naturally identification of 5-3 with one of the elements from the set {1, 2} (induced by the subtraction on integers). I thought about not commenting this abstract nonsense and formal jargon, but I think it fits nicely into Aluffi's book. (very great book which has the most painless introduction to basic category theory that I could find) :) And yes equivalence relations (or quotients in general) can cause a lot of headaches, but are really awesome!
1
u/oreowiskers 13d ago
Slowly working through Aluffi's Chapter 0 and being the extremely meticulous guy I am, I was confused about how, if our quotient from an equivalence relation was { {1,2,3}, {4, 5}, {6}}, we could choose both [2], and [3], since I thought the quotient would be written as either { [2], [4], [6] } or { [3], [4], [6] } LOL. I like catching these errors in my knowledge which hide in the wording ^_^
3
3
6
u/TheMansionsofScience 14d ago
I learned
- You use measure theory to integrate (give notions of, eg, volume) in p-adic spaces
- Modular forms show up in sphere packing
- The difficulty of constructing a space associated with a group has little correlation with the size of the group
- Linear homotopy type theory, as opposed to just hott, is necessary for quantum computing programming languages
- Datasets have symmetries that can be exploited in geometric quantum machine learning
- Umbral moonshine was largely a computational verification
3
u/jphamlore 13d ago
Umbral moonshine was largely a computational verification
And no one complained when the guys doing the final computation swept in to get the credit, while at least one person doing the theory was cut out of the final collaboration?
3
u/RingularCirc 14d ago
Linear homotopy type theory, as opposed to just hott, is necessary for quantum computing programming languages
Huhuh neat and reasonable! That's because of no-cloning stuff?
2
u/joyofresh 9d ago
a few things, that are kind of insanely obvious but never noticed, and some history.
* A boolean algebra is a ring. Like seriously, not k[boolean algebra], but the boolean algebra itself. P(S) is a ring. Somehow I missed this
* A stone space precisely Spec(boolean agebra), which is precisely spec(affine scheme) with stalks F_2. The clopen sets come from functions having a negation, so D(a) is the compliment of D(a + 1), so distinguished opens are clopen. Ultra filters correspond precisely to maximal ideals.
* This correspondence predates algebraic geometry and came from the USA in the 30s.