Question

It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is that a lot of math books lack the motivating examples or only provide very contrived ones stuck in between pages of definitions and theorems. Reading such books becomes a huge chore for me, even in areas in which I am interested. Besides I am certain no mathematical field was invented by someone coming up with a definition out of thin air and proving theorems with it (that is to say I know the good motivating examples are out there).

Can anyone recommend some graduate level books where the presentation is well-motivated with explicit examples. Any area will do, but the more abstract the field is, the better. I am sure there are tons of combinatorics books that match my description, but I am curious about the "heavier" fields. I don't want this to turn into discussion about the merits of this approach to math (i know Grothendick would disapprove), just want to learn the names of some more books to take a look at them.

Please post one book per answer so other people can vote on it alone. I will start:

Fourier Analysis on Finite Groups and Applications by Terras

PS. this is a similar thread, but the main question is different. http://mathoverflow.net/questions/6083/how-to-sufficiently-motivate-organization-of-proofs-in-math-books

Answer 1

Fulton and Harris's "Representation Theory: A First Course". There are three full chapters on representation of $\mathfrak{sl}_2 \mathbb{C}$ and $\mathfrak{sl}_3 \mathbb{C}$ before delving into the general theory.

Answer 2

Visual Complex Analysis, by Tristan Needham.

Really nice to get a thorough geometrical understanding of (one) complex variable.

Answer 3

For algebraic geometry, you'll be wanting Joe Harris's "Algebraic Geometry: a First Course"

Answer 4

Cox' "Primes of the Form x^2+n*y^2", Cohn's "Introduction of the construction of class fields", Koblitz' "Introduction to elliptic curves and modular forms", Waterhouse's "Affine group schemes". I recomend to look for good surveys in Asterisque, Bull. AMS etc., e.g. I found Katz' "Slope filtrations of F-crystals" in Asterisque 63 or Berger's "Encounter with a Geometer I/II" on Gromov's work, Petersen's "Aspects of global Riemannian geometry" good to read.

Answer 5

"Differential Topology", Guillemin-Pollack, 1974.

Answer 6

Robin Hartshorne just came out with a new book titled "Deformation Theory" based on these lecture notes. It is full of examples and exercises (the latter are not in the online notes).

Chapter 1 of the book is also available (with exercises and an improved exposition) on Springer's website.

Answer 7

Characteristic classes by Milnor-Stasheff, 1974. This book from Princeton marks (i think) the synthesis of several years of maturation for the real beginnings of modern topology, the next years that came...

In their 20 chapters, preface, 3 appendices, bibliograph and index, anyone gonna see a jewel master piece of math

Answer 8

The Topology textbook by Jänich (german, I guess there is an english version by now as well) is quite entertaining and has a lot of very nice motivation. Essentially, the book deals most of the time with motivation only, several theorems are only stated but not prove. However, being so well-motivated this does not even matter so much. I regularly suggest this book to students who want to get some overview before they go into the details (for which you may need some other textbooks as well).

Answer 9

J. Silverman's "The Arithmetic of Elliptic Curves" is excellent, and has lots of explicit examples throughout the book.

Answer 10

I learned point-set topology from the lecture notes by Fernando Chamizo available here: Topología (La Topología de segundo no es tan difícil) (yes, they're in Spanish). They also happen to be the most hilarious mathematics lecture notes I have ever come across.

Answer 11

"Riemannian Geometry", Gallot-Hulin-Lafontaine, 1987, plenty of examples and exercises and the motivation: the own one helps...

Answer 12

Peter Petersen's book "Riemannian Geometry" has a whole chapter on examples, most of which are nontrivial ones.

Answer 13

Milne's lecture notes contain many good, standard examples discussed in depth. For example, in Algebraic Number Theory, in the section about Frobenius elements, Milne proves quadratic reciprocity (which IMO is the "correct" proof of quadratic reciprocity).

Answer 14

Three-dimensional geometry and topology: Volume 1 by William Thurston

Answer 15

Trees by J-P Serre. The first half is pretty much all theory, but in the second he looks at the explicit example of $SL_2$.

Answer 16

A first course in Algebraic topology, again Fulton

Answer 17

Terras, Harmonic analysis on symmetric spaces I, II.

It has some very impressive sections with examples and applications from e.g., solar physics.

Answer 18

Kock/Vainsencher's "An invitation to Quantum Cohomology". The friendliest, best motivated and most fun-to-read book I have ever had in my hands!!

Introduces Moduli of Curves, Gromov-Witten invariants and in the end just the rough idea of Quantum Cohomology.

Answer 19

Complex Analysis: Theodore Gamelin's Complex Analysis.Probably the single most user friendly text on the subject there is. Wonderfully written,TONS of examples and covers an enormous breadth of topics.There are lots of good ones on this topic,but for self study,there's probably none better then this one. My one complaint is that Gamelin is sometimes TOO gentle where a proof instead of a picture would be more appropriate. But then the book is designed to be read by a vast audience from freshman to PHD level,so he can be forgiven.

Answer 20

Foliations 1 by Alberto Candel and Lawrence Conlon

Answer 21

Algebraic curves and Riemann surfaces by Rick Miranda

Answer 22

"Explorations in Monte Carlo Methods" by Shonkwiler and Mendivil. Everything is well-motivated by examples. However, it is an undergraduate book.

Answer 23

I can give a couple of dozen examples-but for now,I'll just list my favorite for topology/geometry: The trilogy by John M.Lee is probably the best written,laid out and flat out wonderful introduction to the study of differential and Riemannian manifolds there is for anyone looking to learn it on thier own. I hate to say it,but it's better then Spivak's opus.

Answer 24

Complex Analysis by Raghavan Narasimhan

Answer 25

Complex functions: an algebraic and geometric viewpoint by Gareth A. Jones, David Singerman