N. Koblitz, p-adic number, p-adic analysis, and zeta-functions. GTM. Perfect in every respect.
N. Koblitz, Other books. It seems that all of them are also excellent, but I am less familiar with them (the previous one I read from cover to cover).
K. Kunen, Set theory: an introduction to independence proofs. This is the best exposition of P. Cohen’s method of proving the independence of continuum-hypothesis (there is no other method). I do not think anymore that this independence is such a big deal as people used to think and many still think. The reason is that I do not attribute to this theorem any philosophical significance, and this is because I know its proof, which I learned from Kunen’s book. But Cohen’s proof is very beautiful and subtle. I learned this from Kunen’s book too. All this beauty and subtlety are missing from popular expositions, even from ones written for mathematicians.
I. Lakatos, Proof and refutations. This is a rather unusual book devoted to the philosophy of mathematics. Definitely not a textbook, but highly recommended. Brilliantly written.
S. Lang, Algebra. The last edition is more than two times longer than the first. A lot of people hate this book as too abstract. They miss the point: the goal of the book is to teach to think in abstract terms. GTM
S. Lang, An introduction to algebraic and abelian functions. GTM
S. Lang, Other books. The collection of Lang’s books is huge and uneven. I will not suggest reading his undergraduate calculus textbooks, but his lectures for high school students are excellent. Many people don’t like Lang’s books without realizing that to a big extend Lang defined the modern style of an advanced mathematics textbooks, and that many books they like are either written in this style, or are just watered down versions of books written in this style (or even of books written by Lang himself).
O. Lehto, Univalent functions and Teichmüller spaces. GTM
G. Mackey, Lectures on mathematical foundations of quantum mechanics.
S. MacLane, Homology. This is a classic written with perfect timing: when a new branch of mathematics (homological algebra) just turned into a mature subject.
S. MacLane, Categories for the working mathematician. GTM
Yu.I. Manin, A course in mathematical logic for mathematicians. It is worthwhile even just to browse this book looking for general remarks. There are a lot of deep insights hidden in it. GTM
Yu.I. Manin. Other books, if you mastered the prerequisites.
W. Massey. Algebraic topology. An introduction. Later versions include homology theory. My recommendation is only for the fundamental groups part. GTM
J.W. Milnor, Morse theory.
J.W. Milnor, Topology from the differential viewpoint.
J.W. Milnor, An introduction to algebraic K-theory.
J.W. Milnor. All other books by Milnor are also exceptionally good with the only possible exception of the book about h-cobordism theorem (this one is really a long research-expository paper).
D. Mumford, Algebraic geometry. Complex projective varieties. One of the best books in mathematics I ever read.
D. Mumford, The red book of varieties and schemes. Probably, the best introduction to schemes.
D. Mumford, Curves and their Jacobians. These lecture notes cannot serve as a textbook, there are no complete proofs, but there is a wealth of insights and ideas; the exposition is masterful. These notes are included into the last Springer edition of The red book of varieties and schemes.
D. Mumford, Lectures on theta-functions I, II, III.
D. Mumford, Other writings. Everything (including research papers) written by Mumford the algebraic geometer is great if one has the required prerequisites. Unfortunately, he left the field and the pure mathematics in general in early 1980ies.
R. Narsimhan, Analysis on real and complex manifolds.
D. Ramakrishnan, R.J. Valenza, Fourier analysis on number fields. GTM
Elmer G. Rees, Notes on geometry. UTM (Springer Undergraduate Texts in Mathematics)
J. Rotman, Homological algebra. The first edition (Academic Press) is shorter and better than the second one (Springer). The first edition is a gem. The second edition contains much more material, which is at the same time a plus and a minus.
W. Rudin, Principles of mathematical analysis. I learned the basics of the mathematical analysis from this book within a month. This month was fairly horrible in almost all other respects.
W. Rudin, Functional analysis.
W. Rudin, Real and complex analysis.
W. Rudin, Fourier analysis on groups.
C. Rourke, B. Sanderson, Introduction to piecewise-linear topology. The book is perfect, but field is out of fashion. The reasons for the latter are not internal to the field; they are the same as in the fashion industry.
J.-P. Serre, Lie algebras.
J.-P. Serre, Lie groups.
J.-P. Serre, A course in arithmetic.
J.-P. Serre, Linear representations of finite groups.
J.-P. Serre, Trees. Perfect.
J.-P. Serre, Everything else, if you mastered the prerequisites.
I.R. Shafarevich, Basic of algebraic geometry, V. 1, 2. The best introduction to the algebraic geometry, but it is too slow if you are planning to be an algebraic geometer.
M.A. Shubin, Pseudo-differential operators and spectral theory.
E. Stein, Singular integrals and differential properties of functions.
E. Stein and Rami Shakarchi, 4 volumes of “Princeton Lectures in Analysis”. I did not read them, but I am sure that they are very good.
J.-P. Tignol, Galois' Theory of Algebraic Equations.
R. Wells, Differential analysis on complex manifolds. Reprinted 2008. GTM
F.W. Warner, Foundations of differentiable manifold and Lie groups. GTM
H-h. Wu, The Equidistribution Theory of Holomorphic Curves. This is a fairly old book and at the same time the last book I read from cover to cover (about two or three years ago). It is brilliant. Don’t be scared by long computations, especially in the last chapter: the author presents them in a way which shows their inner working.
Wu's book completes this list.
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