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So, until now I have studied a fair amount of linear algebra(matrices and determinants mostly, but some vector spaces too), abstract algebra(groups-up to topics such as the Sylow Theorems, FTFGAG, some rings, division rings and fields- nothing too fancy, mostly the introductory material) and real analysis(up to Calculus 3 if we are to use the US system as a metric and maybe a bit more since I don't think that they cover things like integral inequalities and interchanging the limit and the integral). Examples of problems that I have enjoyed solving and thinking about are : Matrix problem similar to Problem 3, SEEMOUS 2019, Problem from the shortlist of the Romanian Mathematical olympiad and Prove that $\int_0^1 \big(1-x^2\big) \big(f'(x)\big)^2\,dx \ge 24 \left(\int_0^1 xf(x)\,dx\right)^{\!2}$ (I decided to choose some that are on this website).
Given this background and interests (by the way, I am not really a big fan of epsilon proofs in analysis, but I guess that I can put up with them when they are really necessary), what do you think that would be a good continuation for my studies? I am just looking for some suggestions since there seem to be so many options, but I dont't know which would be close to the things I have already done and I liked. Anyway, I am looking forward to hearing your thoughts.
EDIT: I forgot to mention, I am enrolled in a math major.

user69503
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    Try the representation theory of finite groups! Super fun topic: https://en.wikipedia.org/wiki/Representation_theory_of_finite_groups – Qiaochu Yuan Aug 30 '20 at 23:33

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Generally, people like you would really enjoy preparing for Putnam. Here's a link to a very widely known book: Putnam and Beyond. There are some other great books, such as a compilation of the problems and solutions to Putnam on the web.

You don't necessarily have to take the competition. Whether or not you can take it, or are eligible to take it, is irrelevant--the questions are top-notch and really interesting. If you go through all the Putnam material, you may also find IMO shortlisted questions interesting (although IMO has much less to do with university mathematics than Putnam). Happy problem solving!

Albert
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Based on what you said you enjoyed studying, I think we like similar topics and problems, and I'm now in grad school doing a PhD (hopefully) in an area under operator algebras. So I would say if we do have similar tastes, you might like to study some functional analysis (check out https://users.math.msu.edu/users/banelson/conferences/GOALS/notes/KU%20Leuven.pdf) and some basic operator theory as a next step! In fact, you could glance over these notes: https://users.math.msu.edu/users/banelson/conferences/GOALS/notes/prereq.pdf and see if you want to dig into any of those topics. They should be accessible shortly after you've done real analysis and linear algebra, which you have!

metricforbees
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  • This is really nice, thank you! I was wondering, do these operator algebras also make use of algebraic techniques or are they mostly functional analysis? – user69503 Aug 31 '20 at 08:07
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    Oh they definitely use algebraic techniques! I was just doing old-school proofs about ideals the other day so it certainly still comes up! :) – metricforbees Aug 31 '20 at 14:08