Algebra

Steele's The Cauchy-Schwarz Masterclass is actually quite good and seemingly designed for self study. (A lingering result of this book is I heavily use inequalities even outside of analysis.) Artin's Algebra probably has had the most impact on my math thinking. The development of groups and rings while tightly linking them to linear algebra was rather brilliant.

Agreed which is why I love Artin’s Algebra. It introduces the matrix determinant early and uses it as the prototypical example of a group homomorphism and the book uses matrix groups liberally. Obviously not a beginner friendly book if you just want to understand matrix equations s. But I found the pedagogical approach great and there are some real gems in the exercises. I found it a better for learning versus the standard recommendations of Axler/Lang/Halmos for LA. (Lang’s exposition is kinda weird and Halmos’s choice of exercises isn’t my favorite).

_Algebraic Topology_ by Allen Hatcher. This is quite an advanced book, but it was the first topology book I picked up and it blew my mind. Without a serious math background you may only be able to read the first parts (some of chapters 0 and chapter 1), but even so it may amaze you. It was insane to see how much mathematical machinery can be built up to understand concepts as simple as “space” and “continuity.” Then these tools can be used to quickly prove facts about mind-boggling higher dimensional objects. _Introduction to Quantum Mechanics_ by David Griffiths. Not CS or Math, but an amazing book because if you sit down and work through it it gives you a manageable intro to a completely non-intuitive and mysterious scientific field. Prerequisites are “only” multivariable calc and linear algebra. _Algebra_ by Michael Artin. This was the book I decided to grind through at the right time to learn abstract algebra and get better at rigorous proofs, and it was worth it. Part of why I loved this book and subject is because it feels dry and mechanical at first, but if you work on it long enough you can see the beautiful bigger picture come together.…

I'm in a similar position as well - perhaps at just a wee bit higher level (am learning grad-level math by myself). Below is a list of books that I have enjoyed. Some might perhaps be at a slightly higher level than you might need right now though. 1. Analysis - The Elias Stein, Rami Shakarchi Analysis series, Meassure Theory by Terrence Tao, Lebesgue Integration on Euclidean Spaces by Frank Jones, A radical approach to Lebesgue Integration by Bressoud 2. Algebra - Can't do much better than the classic texts by Herstein & Artin here. I know Artin doesn't get much love but after a point, you really begin to appreciate it. 3. Complex Analysis - is there a better math book than Visual Complex Analysis by Needham? 4. Topology - i'm partial to a Dover book on Topology by Theodore Gamelin. I learnt the basics of point-set topology & homotopy form this book. Great exercises. 5. Number Theory - check out the 2 problem books on number theory by Ram Murty. Melvyn Nathanson's book is quite good as well. 6. Combinatorics - Combinatorics of Finite Sets by Ian Anderson. Extremal Combinatorics by Jukna. 7. Graph Theory - Diestel or Bondy/Murty. 8. Galois Theory - Rotman is great.

I'm a big fan of SICP. I'd describe it as "hard-core": requiring hard work, but extremely rewarding for the people who take the time and effort to go through it. I'd recommend the following three books to similarly cover the three main areas of higher mathematics (analysis, algebra, and topology): 1. Principles of Mathematical Analysis by Rudin (http://www.amazon.com/Principles-Mathematical-Analysis-Inter... (http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X)) 2. Algebra by Artin (http://www.amazon.com/Algebra-2nd-Featured-Titles-Abstract/d... (http://www.amazon.com/Algebra-2nd-Featured-Titles-Abstract/dp/0132413779)) 3. Topology by Munkres (http://www.amazon.com/Topology-2nd-James-Munkres/dp/01318162... (http://www.amazon.com/Topology-2nd-James-Munkres/dp/0131816292))…