First some background, I am originally a computer scientist, got a bachelor at university, started working. I had always had a passion for mathematics and I was relatively good at it throughout my courses. I had done things like linear algebra, a bit past inverses of 3×3 matrices, multivariable calculus , a bit of analysis. I have the mathematical maturity to work through proof heavy texts and I prefer books to learn, I also don't mind spending a long time on very large books either (say 1000 pages+) or a lot and lot of questions.
It has however been a very long while, currently I would say I have regressed a bit worse than the average high school student in mathematics. I can say things like I sort of remember how to do inverses of 3×3 matrices and I could do them. I am however far too rusty and want to brush up my mathematics. It has gotten to the point where matrices , trigonometry and standard angle chasing are all the same difficulty to me simply because my foundation is crumbling.
I had a look back at some real analysis and I seem to be able to make progress until I hit any spots where my foundation is weak at which point it slows me down but doesn't stop nor patch any points in my foundation. I believe it is better to restart my mathematical path from the ground up to tackle any weaknesses before I continue with more abstract content.
Is there any book or set of books that suit my need?
I also have access to a very large library so I believe I should be able to get my hands on any book, I can't seem to follow video lectures as well as books though.
I am particularly interested in:
- Combinatorics
- Statistics
- Algebraic Geometry
- Topology
- Logic
- Analysis
Any help is greatly appreciated.
Best Answer
For topology, a nice beginner textbook is Munkres's text Topology while Willard's textbook General Topology is a bit deeper (I prefer the second one, but if I hadn't already taken a class with topology, I think the first one is easier to self-teach from). If you want to delve beyond general topology and touch on algebraic topology, Hatcher has a nice textbook that is freely available online.
For analysis, a couple of beginner-type textbooks are Wade's text An Introduction to Real Analysis and, a bit more difficult but still at the beginner level, Rudin's Principles of Mathematical Analysis. I think some top-notch recommendations for analysis texts are Rudin's Real and Complex Analysis as well as Ahlfors's Complex Analysis. A comprehensive overview of complex analysis can be found in Markushevich's text Theory of Functions of a Complex Variable.