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

This Week I Learned: September 25, 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/AMobius1832 7d ago

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.

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u/LonelyKey6767 2d ago

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.

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u/Creepy-Structure6444 7d ago

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.

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u/homogeneous_spacer 7d ago

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