I am a student who has just completed their degree in pure math. Unfortunately, my undergrad was a very... Unpleasant time for me due to personal reasons. Although math is accepted as a very "poorly-marketed" and just overall "atrocious" monster of a subject nobody really likes, despite all this, I still appreciate and enjoy it to this day. However, during my undergrad, I really felt like I missed out on a lot (seriously). I am looking to do a lot of self-study and perhaps write notes/record videos on my progress to not only educate myself, but other enthusiasts. I still feel like an amateur in some sense... Simply because I feel like my knowledge and skills are inadequate and what I currently possess could always be better.
Moving forward, some areas I have an "okay" time working through are usually group theory, ring/field theory, galois theory, linear algebra, and number theory. This is just what I can think of at the top of my head. I'd like to get better in the following courses
- Real and complex analysis - texts used: (Principles of Mathematical Analysis (3rd ed.) by Rudin + Real Analysis by Carothers, Complex Analysis (8th ed.) with Applications by Brown-Churchill
- Advanced linear algebra (dual spaces, multilinear algebra, etc) - text used: Linear Algebra (3rd ed.) by Serge Lang
- Manifolds... (forgive me if this is vague... I took a course called "Calculus on Manifolds" and we looked at topics such as basic properties of manifolds, differential forms, Stokes' Thm, DeRham Cohomology, Hodge Star operator, Mayer-Vietoris sequences)... - text used: Intro. to Manifolds (2nd ed.) by Loring Tu
- Topology (undergraduate level preferably...) - text used: Topology by Munkres (2nd ed.)
So far this seems like a huge stretch and I got my work cut out for me, but I'd really like to improve on these areas as much as it'll be a painful. In the mean time, I'll still review what I'm already somewhat good at because I like refreshing my knowledge every now and then.
Last but not least, I want to ask the main question(s):
- What are some good study tips when tackling beasts like the ones above?
- If this exists, is there a good order to keep in mind when handling which subject at a time?
- How should I approach the proofs in say... Analysis or Topology? These are the ones I find myself having the most difficulty generally. I always feel bad sometimes reading the proof before trying it on my own. I probably need to stop this.
- If I'd like to study higher mathematics after this... I really want to know what Representation Theory or Lie Groups is. Which should come first? Or does that even matter? (I'm really pushing it, aren't I?)
I'm seriously asking for a lot here and I am sorry for the long post, but I appreciate your time in reading. Math has been a subject I hold pretty dear considering I'm even writing here. I don't talk to nobody, but professors about maths usually simply because I'm not very fond of the cocky attitudes my colleagues in my program had. It was an awful experience and I just felt out of place and looked down upon at times for being slow. I don't dislike my learning style because I feel happy every time I understand something difficult.
Anyways, any feed back would be appreciated! If you'd like to talk about math with me sometime that would be cool too. I'm open to make some math friends! :)