# How to build a strong foundation for university mathematics?

My objective: To study pure mathematics at university level next year. I find Abstract Algebra, Number Theory and Foundations of Maths(Set Theory, FOL, etc.) to be intriguing yet mostly inaccessible with my current mathematical maturity, but I would love to learn more about them in the time to come. I want to make sure when that time comes, I can digest most of the material without getting bogged down. Ultimately, I would like to make a meaningful contribution to the field of pure mathematics :)

My background: I recently finished high school, covering single variable calculus(Calculus 1), some matrices-determinants(mostly computational problems) and some elementary notions of sets, relations, functions, combinatorics, basics of vector algebra and discrete probability.

I happen to have a year of time at hand, and I can devote multiple hours of study consistently. However, I've tried learning a variety of topics for some time now, but disorganised learning makes me lose track of my progress. I want to utilise the time to come gainfully, so as to get a better insight into what mathematics is about, while also building a strong foundation. I tried organising a broad outline of how I could possibly study during this time, divided into 3 tracks:

Track 1-Continuation of school mathematics: Continuing with Calculus 1, I could start diving into Calculus 2 and 3; similarly extend my knowledge of matrices-determinants to basic linear algebra. While I do this, I could lay some more emphasis on proving and understanding results as opposed to just performing mechanical computations.

Track 2-Studying for high-school maths olympiads: This doesn't imply that I would be enrolling for any olympiad; rather, I'd be covering maths that is usually not taught at school but constitutes questions in maths olympiads geared towards high school students. I'll try to cover topics such as elementary number theory, Euclidean geometry, functional equations, inequalities, theory of equations, combinatorics and probability, etc.

Track 3-Start diving into undergraduate mathematics: Due to the currently prevailing circumstances, there has been a flurry of online learning resources for all levels of learners. MOOCs on higher mathematics are no exception. Thus, I could start studying some basic real analysis, introductory linear and abstract algebra, set theory and logic. I like to prove things, but I cannot figure out how to hone this skill.

I have ample learning resources available with me(a plethora of maths textbooks such as Analysis 1 and 2 by T.Tao, Contemporary Abstract Algebra by Gallian, Ordinary Differential Equations by M.Tenenbaum to name a few- I wouldn't shy away from buying more reasonably priced textbooks which are necessary to further my objective). However, here's what I'm confused about:

1)Which of the aforementioned tracks is best suited to achieve my objective? I most certainly don't expect to become a jack of all trades within a year, but I want a firm footing in later years of my maths education. Detailed suggestions about any other track are also welcome.

2) In the absence of an instructor, how do I evaluate and monitor my progress in a time-bound manner? Of course, nothing can substitute for actually studying maths at university, but what's the least I could do to evaluate my work? I want to be sure that I don't get bogged down in the middle, not sure where I'm going with my studies.

My preference in these tracks goes as $$3>2>1$$. I'm enthralled at the perspective of learning higher mathematics(I did some basic group theory while in high school), but eventually gave up because even though I used to frame a single proof in 40-50 minutes of time, I wasn't sure if it was correct at all in the end. Further, topics at higher levels are interconnected and hence require some background/prerequisites(part of the reason I stopped studying group theory was my lacking background in modular arithmetic) along with mathematical maturity, which sometimes become a barrier to learning. Nevertheless, I welcome any and every suggestion which comes from a community of maths students, teachers and professionals. Sidenote: I have already checked out several questions asked on related themes on MSE and elsewhere, but could not reasonably relate to any of them.

• I am using the book "How to Prove It" by Daniel Velleman to, well, learn how to prove stuff. I feel that this book has improved my mathematical maturity and framework for problem solving. I think it lays a good foundation to write proofs and understand higher level mathematics. – C Squared Aug 2 at 3:03
• @CSquared Thank you! I've heard a lot of praise for that book, so I'll definitely look into it. – Manan Aug 2 at 3:07

I would strongly recommend you to go for "Track 3". To say why, Olympiad aren't the best choice for you right now, you have already finished high school and for good foundation in high school olympiads you need to devote at least more than a year. And also continuing with high school exercises (rather than problems) isn't really a good choice If you are curious for learning more and more and thinking more abstract in mathematics.

I would recommend you start with real analysis, like i did. Its been 4 months I started with real analysis I from Tao's book and due to holidays/etc right now a lot of free time is available, for which I completed Tao I,parts of Bartle Sherbert's real analysis,did linear algebra and am now with Metric Spaces (in these 4 months and self study). Personally I did math most of the day, 5 days in a week. So If you can, try starting as soon as possible and within the next year, you can develop that mathematical maturity for University maths and moreover, self study on your own topics which attract you more.

• Thank you! I really like Tao's analysis, however, due to constantly shifting objectives I never went beyond his excellent explanation of the Peano Axioms. I'll take your suggestions into consideration. Further, how do you evaluate your work(seeing if your proofs are correct, etc.)? – Manan Aug 2 at 3:15
• @Manan you can always post solutions to exercises here under the proof-verification tag to see if they are correct! This is often a simple way to get feedback – csch2 Aug 2 at 3:17
• Well you make a plan and stay with one topic as much as possible. Also I try most exercises to check whether the basics are cleàr and try to prove the theorems I come across myself (if I know some basic tools beforehand). For exercises, when I do one, it's mostly understandable if it's right or wrong, if it's not I post it here in MSE (you can check my profile) – user732848 Aug 2 at 3:18
• Taken into account, Shamim and csch2. Thank you! – Manan Aug 2 at 3:24