Below is a categorized list of textbooks that I have read through completely with attached comments. If you have any trouble getting ahold of any of these sources feel free to reach me via my contact page here.
- Textbook – Dummit & Foote’s Abstract Algebra. This is one of the standard textbooks for both graduate and undergraduate abstract algebra. It is often verbose, and so distracts from the intuition behind the theory. However, it makes clear all of the details of the machinery it presents. It also contains a large collection of exercises; a decent collection of which were solved in a project entitled Project Crazy Project which seems to now have stalled. An archive for the project can be found here.
- Textbook – Axler’s Linear Algebra, Done Right. The best abstract linear algebra text I have come across. It is worthwhile, but not necessary, to have read a computation-based linear algebra text first to familiarize one’s self with the definitions and ideas. Most of the exercises are quite simple, and are worthwhile doing for someone with less mathematical maturity. It does not treat linear algebra in the most generality as Axler works over and only (but this will suffice for most things).
- Textbook – Humphreys’ Introduction to Lie Algebras and Representation Theory. A standard reference for Lie algebras and representation theory. The first three chapters are quite easy to follow, but the fourth and fifth are more terse. The text can serve as a stand alone introduction to root systems if that is what the reader is looking for. A strong background in linear algebra is necessary, and it will help if the reader can translate between the language of linear algebra and abstract algebra quickly.
- Textbook – Munkres’s Topology. A standard textbook with part I serving as the text for a one semester point-set topology course. It is very worthwhile to use this text for learning point-set topology. Part II of the text discusses algebraic topology. The text uses non-standard topologies quite often in many examples and exercises which are usually not dealt with in other fields.
- Textbook – Tu’s An Introduction to Manifolds. A phenomenal textbook for introducing smooth manifolds. Chapter serves as an overview for the remainder of the text and introduces the theory in by connecting it to vector calculus. Chapters 2, 3, 4, and 6 on manifolds, the tangent space, differential forms, and integration respectively are incredibly well-written and focus on introducing the general theory as much as how to perform associated computations. Chapter 5 on Lie groups and Lie algebras is very light and a deeper text should be consulted for further interest. It is best, but not necessary, to read Chapter 7 on de Rham Theory with some familiarity in homological algebra (see Dummit & Foote Chapter 17).
- Textbook – Rudin’s Principles of Mathematical Analysis. A staple for real analysis. While quite light on the intuition behind the arguments presented most proofs are clean and quick. So, it is worthwile to mull over the proofs and see why they work. Many of the exercises are quite long and tricky, but Rudin does have a solution manual.
- Textbook – Abbott’s Understanding Analysis. The only downside of this textbook is that you might fall prey to the idea that every mathematics textbook is written this well. The text coves the fundamentals of a first year undergraduate corse in analysis, but there is so much to be gained beyond the standard definitions and theorems. A bountiful amount of intuition is given in every chapter (especially about uniform continuity, Taylor series, and the Cantor set), and there are many historical remarks for the reader who likes the backstory of mathematical analysis. Many of the exercises are worthwhile doing albeit not exceptionally difficult, and some sections of the text are more D-I-Y type where the reader is encouraged to do most of the work. The last chapter is dedicated to additional topics and provides good motivation for spaces.
- Textbook – Silverman & Tate’s Rational Points on Elliptic Curves. This is an introductory textbook to elliptic curves. Not all statements contain proofs and many proofs are not complete argument. However, this text is quite good at introducting the general setting of the theory and very little backgound is needed.