The following list is mostly intended for undergraduates. The recommendations here generally emphasize clarity of exposition over technical detail.
Algebra and Number Theory
- Ash and Gross. Fearless Symmetry: Exposing the Hidden Patterns of Numbers.
- Carter. Visual Group Theory.
- Conway. On Quaternions and Octonions.
- Cox. Primes of the Form
: Fermat, Class Field Theory, and Complex Multiplication.
- Kuga. Galois’ Dream: Group Theory and Differential Equations
- Silverman. Rational Points on Elliptic Curves.
- Stillwell. Naive Lie Theory.
Geometry and Topology
- Adams. The Knot Book.
- Mumford, Series, and Wright. Indra’s Pearls: The Vision of Felix Klein.
- Needham. Visual Complex Analysis.
- Reid, Szendroi. Geometry and Topology.
- Richeson. Euler’s Gem: The Polyhedron Formula and the Birth of Topology.
- Stillwell. The Four Pillars of Geometry.
- Stillwell. Geometry of Surfaces.
- Toth. Glimpses of Algebra and Geometry.
- Weeks. The Shape of Space.
Other
- Aaronson. Quantum Computing Since Democritus.
- Doxiadis, Papadimitriou. Logicomix: An Epic Search for Truth.
- Gowers (ed). The Princeton Companion to Mathematics.
- Lawvere, Schanuel. Conceptual Mathematics.
- Mahajan. Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving.
- Parikh. The Unreal Life of Oscar Zariski.
- Sipser. Introduction to the Theory of Computation.
- Stillwell. Yearning for the Impossible: The Surprising Truths of Mathematics.
- Wilf. generatingfunctionology.
Annoying Precision 
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Is the Euler’s Gem link an easter egg?
How many of these did you read cover to cover?
I read Galoia’s dream, when I was a junior high school student.
Nice list. By the way how do you display math formulas in your blog? I found no button for mathjax plugin on my control panel (and it seems wordpress.com has banned the users freely using plugins)
WordPress has some LaTeX support by default.
It would be great if you can refine the list!
Regarding graph theory: I’ve really enjoyed Pearls in Graph Theory (see link if need be: http://www.amazon.com/Pearls-Graph-Theory-Comprehensive-Introduction/dp/0486432327 ). Sadly, both authors passed away.
Sir this is an excellent list. I have about 9 of these and they are all at about my level of comfort. Ash and Gross’s Elliptic Tales is a great companion to Fearless Symmetry with a focus on BSD.
How about some unorthodox means of instruction like the short video series ‘The Catsters’ by Cheng & co; I couldn’t get a handle on Category Theory until I saw them.
How about Sets for Mathematicians or Conceptual Mathematics?
I haven’t read Sets for Mathematicians but I’m happy to add Conceptual Mathematics to the list. Thanks for the suggestions!
I notice you recommend a lot of “soft” books, yet reference “harder” books in your blog posts. What has worked better for you?
I mostly don’t learn from books either way, but part of the motivation behind this list is to counter a tendency towards a “harder = better” mindset when it comes to mathematical resources.
It takes a lot of courage to go against that mindset. I find that the softer books discuss the “why” that I yearn for in the harder books, at difficult spots.
Out of curiosity, what’s your main source of learning/knowledge if not books? I’m guessing that right now you’re probably at the point where most of your knowledge comes from attending lectures and conferences, talking to other mathematicians, and reading papers. But I’m more interested in how you got to this point, especially during your undergrad years, since your pace of learning would be impossible to maintain solely on undergrad math courses. If not primarily books, what sources did you use (and would recommend) to quickly build up a solid foundation of mathematical knowledge pre-grad school?
Thanks!
Some combination of papers, Wikipedia, the nLab, math blogs, occasionally books, MathOverflow, and blogging.
good recommendations, thanks c:
Am I blind or is there no Serre?!
You are not blind and there is no Serre!
Have you paid for, and read all these booiks?
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