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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):

  1. What are some good study tips when tackling beasts like the ones above?
  2. If this exists, is there a good order to keep in mind when handling which subject at a time?
  3. 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.
  4. 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! :)

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closed as off-topic by Matthew Towers, Lee David Chung Lin, Lord Shark the Unknown, Cesareo, Jean-Claude Arbaut Jun 2 at 13:52

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    $\begingroup$ This post may be closed as opinion-based, but.. +1, for I am facing similar problems. $\endgroup$ – lisyarus Apr 21 '16 at 6:21
  • $\begingroup$ I understand. Just curious, since you and I are in the same boat, have you tried consulting anyone? What are some things you have tried on your own? For me, my gut really tells me to just really surgically dissect Rudin first and get a good grasp on that. I am still in contact with my professors so I may ask them for a word of advice or two. $\endgroup$ – Anthony Apr 21 '16 at 6:33
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    $\begingroup$ I really like math, and I'm sure a lot of other people on this site do too. :) Anyway, to answer your question, just pick up a textbook on the subject and start reading. If you don't like it or get stuck, find a different one. It's easier if the goal is to learn about the subject rather than to learn the contents of a specific curriculum. $\endgroup$ – anomaly Apr 21 '16 at 6:37
  • $\begingroup$ I figured. It's just that I'm so used to "curriculum based" learning (under time constraint and selected topics given that) that I haven't really sat down and tried to appreciate things a little more. :( $\endgroup$ – Anthony Apr 21 '16 at 6:50
  • $\begingroup$ I don't see abstract algebra anywhere on your list, and it would encompass "advanced linear algebra." $\endgroup$ – Matt Samuel Apr 22 '16 at 2:21
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I'm open to make some math friends too $:)$

A partial answer. I will elaborate on some more points if you ask me to!

Study tips for tackling the beasts above:

$1)$ Set up an good schedule and stick to it! What I like to do is set aside $2-4$ hours to work alone with nobody bothering me. This is really important in today's world where there are so many distractions. Use a calendar, a real one, or on your phone, or both.

$2)$ When you get stuck on a problem in one of those book or get confused with what the author is explaining, come ask a question on math stack exchange, or browse similar questions on here. It's likely someone on here has asked a very similar question if not the exact same one!

$3)$ Go on a road trip and visit as many universities as you want! I usually bring a few friends and crash professor's office hours. They don't mind at all! You can talk math with the experts for free (besides gas money). It's really fun, please try it.

$4)$ Do not think of them as beasts! Think of them as little tiny beasts! Don't let yourself get overwhelmed by thinking how much you have to cover. Just take it one step at a time and learn at a pace that is comfortable to you.

$5)$ Make it fun. This is crucial because you will get burnt out if it feels like work and no play. You also have some catching up to do it seems! This is how I would make it fun: Get a large blackboard or whiteboard and put it in your work area (room, study, work area). Invite your friends over to discuss math every Friday or Saturday night and bring snacks. Put on classical music or whatever music helps you do math the best. Just talk about ideas and do interesting problems that are not necessarily in any textbook. Create your own cool problems. Solve them in your own unique ways. You will begin to develop a mojo and niche that know one else quite has. Think about other fields of mathematics and the world in terms of your specialty. Feed everything you know to be true into your models and watch your models morph and change alongside your understanding.

An adept mathematician has multiple options to tackle every problem with. This is similar to a quaterback in the National Football League. He has a first receiver option. If that guy is not open he quickly shifts his attention to another option and if that guy's not open then he goes to his third option. Having the flexibility to choose the option that fits the given problem is a good thing to have as a mathematician.

How should I approach the proofs in Analysis or Topology?

I am not qualified to answer this. I hope someone else is!

If I'd like to study higher mathematics after this... I really want to know what Representation Theory or Lie Groups are. Which should come first? Or does that even matter? (I'm really pushing it, aren't I?)

Yep! You really are pushing it! My fingers are getting tired $:)$ Just kidding. If you are interested in those things go for it! Make sure you've got some good prerequisites down at least :)

Note $1$: Peter Scholze said he never took linear algebra, he just assimilated all that knowledge from working on problems that gave him (energy, inspiration, motivation) to learn anything and everything that was required to win!

Note $2:$ Peter Scholze is really really really good at math. I'm not saying be exactly like him.

But...

If big problems give you uplifting energy go work on those problems and fill in the gaps when you have to. This is one of the most important tips for you specifically because your window could be closing soon? $:$P Just kidding. So have fun, you only live once.

I wish you the best of luck!

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