以前在quora 上看到的一些数学的书单 记一下怕忘了：
General stuff:
Mathematical Circles: Russian Experience by Dmitri Fomin, Sergey Genkin and Ilia Itenberg
The Art and Craft of Problem Solving by Paul Zeitz
Mathematics: A Very Short Introduction by Timothy Gowers
Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov and Lavrentev
What is Mathematics? by Richard Courant
Letters to a Young Mathematician by Ian Stewart
Analysis:
Calculus by Spivak
Principles of Mathematical Analysis by Rudin
Calculus on Manifolds by Spivak
Introduction to Topology and Modern Analysis by Simmons
Algebra:
Linear Algebra Done Right by Sheldon Axler
Finite Dimensional Vector Space by Halmos
Algebra by Martin Isaacs
Algebra by Artin
Elements of Algebraic Topology by Munkres
Combinatorics:
A Course in Combinatorics by Van Lint and Wilson
Combinatorics: Topics, Techniques, Algorithms by Cameron
Number Theory:
A Classical Introduction to Modern Number Theory by Ireland and Rosen
Which book in geometry would you recommend to an undergraduate student in pure mathematics for self studying?
Spivak is pretty popular for differential geometry. It certainly has the best cover, but honestly I don't know it well. Some folks say it's too verbose and cumbersome.
My favorite book is Warner's "Foundations of differnetiable manifolds and Lie groups". I hear that Lee's "Introduction to smooth manifolds" is more gentle.
In a different direction I would recommend digging into hyperbolic geometry. Iversen is a good start, and there's a great short summary here.
Finally, if you get over your disliking of topology, there's a huge and beautiful area at the intersection of topology and geometry. Try Guillemin and Pollack, my favorite.
One would have to know what level you are starting with. Assuming you have a solid grasp on linear algebra (if not, read P. Halmos):
The first several chapters of Mathematical analysis by Zorich.
Ordinary Differential Equations by V. Arnol’d. (Pick a latest edition.)
Numerical methods: see if you can get the book by Thomas and Raviart (I haven’t seen an English rendition yet) and Marchuk’s Numerical mathematics.
Absolutely must get acquainted with basic physics:
Transport Phenomena by B. Bird
some chapters from The Feynman Lectures on Physics
a look into J. Murray’s Mathematical biology
some problems from Hannah and Hillier’s Applied mechanics.
Further study will depend on the specific applications you would like to work on. At any rate, the variational approach to optimal control and elasticity theory (start with a look at Variational Calculus by F. Wan) is a must.