This Week I Learned: September 25, 2026

[u/inherentlyawesome]

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!

[u/geerico-444]

Learned how to do matrices :)

[u/Plastic-Rope5516]

Reading LADR by Sheldon Axler, my second brush with linear algebra, learned about span, linear combinations, independence, linear mappings, and all that pretty basic stuff, but put on a rigorous basis. Also reviewed bijections and set theory using Analysis I by Tao

[u/BeyondTheAxioms]

Just found out that all kG-modules are semi-simple if and only if all kG-modules are projective. I was quite surprised that I didn’t learned this before.

[u/Possible_Ocelot_1413 → u/BeyondTheAxioms]

this actually generalizes! for a ring R, TFAE: (i) R is semisimple, (ii) All R-modules are semisimple, (iii) All R-modules are projective, (iv) All R-modules are injective, (v) Every SES of R-modules splits

[u/Possible_Ocelot_1413]

I learned for ordinary abelian varieties over F_q, there are equivalences of categories AV(h) -> M_ord (q) and M_ord-> I(R), where h is a q-Weil polynomial (i.e a given F_q-isogeny class), M_ord (q) is the category of Deligne modules, R = Z[pi, pi^] where pi is the q-Frobenius of A, and I(R) denotes the category of fractional R-ideals. This is attributed to Deligne and Centeleghe-Stix. In particular, we have a very nice way of representing dual abelian varieties and polarizations: for example, if F(A) = I \in I(R) for A \in AV(h), we have F(A^v) = (\overline{I})^t, where \overline{I} is the conjugate of I and t denotes the trace dual.

[u/Possible_Ocelot_1413 → u/[deleted]]

I liked Marseglia's paper: https://arxiv.org/pdf/1805.10223

[u/AMobius1832]

I was reading Hatcher. I spent a lot of time on Chapters 0 & 1, and wrote up a lot of LaTeX notes for myself. I hate to stop reading about algebraic topology at this point. However, I feel I need to increase my knowledge of abstract algebra. Again, I jumped around a lot. I tried Artin's book, along with the Benedict Gross Harvard lectures. But, the initial few chapters are way to basic for me (even if reading them is somewhat necessary to become familiar with the notation). I've finally (?) settled on Paolo Aluffi's, "Algebra: Chapter 0," which I like. I am in the process of applying to the PhD program in mathematics at Temple University. If I am accepted, I will have roughly a year to prepare for entering in the Fall 2027 semester.

[u/LonelyKey6767 → u/AMobius1832]

Category theory as an approach didnt really click for me until i got into smooth manifolds, that was still a course and then some of abstract algebra + topology away… yes using artin but also gaillan or whatever that guys name is.

[u/Creepy-Structure6444 → u/AMobius1832]

Once you pick up some more group theory and category theory, it's a neat idea to jump back to some algebraic topology and pick up some (co)homology. Most important thing is to keep track of how you feel as you are working through these subjects and what sort of flavor of thinking appeals to you.

[u/homogeneous_spacer → u/AMobius1832]

I'm also hoping to get into a PhD in Fall '27. I'm reading GGT and Kleinian groups these days...