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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 Grothendieck 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. How to sufficiently motivate organization of proofs in math books

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    $\begingroup$ Great question. I look forward to seeing the responses. $\endgroup$ Dec 6 '09 at 13:19

34 Answers 34

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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.

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    $\begingroup$ Seconded. In fact, books by Harris tend to be good for this (and Fulton, though not quite as much in my experience) $\endgroup$ Dec 6 '09 at 4:59
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    $\begingroup$ They themselves make the case for an example-driven book, saying in effect they don't remember how the abstract proofs go until they run through the cases $sl_2$ and $sl_3$ first. $\endgroup$
    – Todd Trimble
    Jun 25 '14 at 11:39
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Visual Complex Analysis, by Tristan Needham.

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

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    $\begingroup$ Thanks for reminding me this book existed. i've been meaning to take a look at it. $\endgroup$ Dec 9 '09 at 2:02
  • $\begingroup$ And see also this answer to an MSE question, which includes a bit more detail on the Needham text. $\endgroup$ Apr 12 '15 at 0:29
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For algebraic geometry, you'll be wanting Joe Harris's "Algebraic Geometry: a First Course"

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"Differential Topology", Guillemin-Pollack, 1974.

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  • $\begingroup$ The book "Measure theory and probability" by Guillemin and Adams is also very good. $\endgroup$ Dec 6 '09 at 18:08
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    $\begingroup$ Guillemin is able to make absolutely any topic in mathematicsl both accessible and interesting. $\endgroup$
    – Deane Yang
    Dec 7 '09 at 0:59
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    $\begingroup$ Now we're talking.Even with all the great introductions to differential manifolds out there right now,this is still one of the best. $\endgroup$ Mar 14 '10 at 8:14
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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.

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    $\begingroup$ I think Koblitz' is actually called "Introduction to elliptic curves and modular forms". Maybe you wanted to refer to "Invitation to the mathematics of Fermat-Wiles" by Yves Hellegouarch? $\endgroup$
    – Jose Brox
    Dec 6 '09 at 17:03
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    $\begingroup$ Thanks, corrected now. I didn't read Hellegouach's book, but people described it as excellent. $\endgroup$ Dec 7 '09 at 9:48
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    $\begingroup$ +1 on account of Cox's magnificent book! $\endgroup$
    – stankewicz
    Mar 14 '10 at 12:57
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    $\begingroup$ Koblitz's book is very good, but its biggest problem is that it isn't a very good introduction to modular forms - in relation to what is actually being researched (today). $\endgroup$ Mar 14 '10 at 17:28
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    $\begingroup$ @Dror: Care to elaborate? $\endgroup$ Feb 1 '11 at 2:27
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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).

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  • $\begingroup$ +1 for one of my favorite textbooks-but this is really an undergraduate textbook. $\endgroup$ Jul 14 '12 at 5:01
  • $\begingroup$ @Andrew Yepp, of course! it does not go very far and you still need some other topology books, if you really want to get some deeper insight. But for a start, I really like it :) $\endgroup$ Jul 16 '12 at 12:21
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Three-dimensional geometry and topology: Volume 1 by William Thurston

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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

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    $\begingroup$ Although I think Whitney's paper "On the topology of differentiable manifolds," Lecture notes in Topology, University of Michigan Press, 1940 is a far more direct motivation -- elegant and to the point. It doesn't dwell on formalities as much as Milnor and Stasheff, but IMO this is a good thing. $\endgroup$ Dec 10 '09 at 21:31
  • $\begingroup$ utterly any pro mathematician is going to face modern formal maths :) as high (or worst) as many book on this subject: topology $\endgroup$
    – janmarqz
    Dec 11 '09 at 4:55
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    $\begingroup$ Anyone who doesn't read Milnor and Stasheff will be the very much poorer for it. In fact,anyone who's thinking about writing an advanced mathematics textbook should read it for inspiration. $\endgroup$ Mar 14 '10 at 8:11
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    $\begingroup$ While I certainly like Milnor-Stasheff a lot, I cannot say that it has a wealth of examples. If I remember correctly, it is not even proven that $S^4$ has no almost complex structure! $\endgroup$ Feb 21 '15 at 20:26
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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.

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    $\begingroup$ I didn't realize that Sesame Street taught topology. $\endgroup$ Dec 6 '09 at 4:57
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    $\begingroup$ Love the subtitle! $\endgroup$ Dec 6 '09 at 12:41
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    $\begingroup$ You're right. Nice notes and written with great sense of humour. $\endgroup$ Dec 6 '09 at 17:22
  • $\begingroup$ estan buenisimas las notas! la primera linea que lei: "el ADN (Asociacion Nacional de Dislexicos)?" esta genial!! $\endgroup$ Dec 8 '09 at 5:01
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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.

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IMHO, algebraic geometry books by Joe Harris are very naturally written: every definition or theorem comes up when you are ready or even expect it, and if you do not understand something, it is probably written on the next page.

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

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

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  • $\begingroup$ There is also Silverman and Tate's "Rational Points on Elliptic Curves" which is even more explicit and introduces the subject to undergraduate students. $\endgroup$ Apr 14 '15 at 20:37
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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.

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Algebraic curves and Riemann surfaces by Rick Miranda

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

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

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    $\begingroup$ I've always felt that this book, due to the concrete examples, is one of the best books on differential geometry. $\endgroup$
    – Deane Yang
    Dec 7 '09 at 0:58
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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).

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

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  • $\begingroup$ +1 for an outstanding geometrically flavored textbook that compliments more "analytic" textbooks on the subject like Conway or Greene/Krantz. $\endgroup$ Jul 14 '12 at 5:02
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Terras, Harmonic analysis on symmetric spaces I, II.

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

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  • $\begingroup$ I agree, but does anyone know why it's out of print? $\endgroup$ Jan 5 '10 at 2:56
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Model Theory: An Introduction by David Marker. A chief strength of this introduction (to what is after all a fairly abstract field) is the working through of explicit examples from all over mathematics, e.g., definability and interpretability of structures, applications of back-and-forth arguments, and so on.

Another good "model theory" book is van den Dries, Tame topology and o-minimal structures. Here it's not so much that many examples are given as it is that a few well-chosen examples are constantly at hand to illustrate the development. I put "model theory" in quotes because in fact the author keeps the model-theoretic jargon to a minimum throughout (even though there is no doubt that model theory informs the development); this makes the book very accessible to a wide audience.

These examples suggest to me that some of my category-theoretic brothers and sisters could take a page from some of these model theory books. It has to be understood that many examples in category theory are not of single objects but rather of categories and mini-theories (so one may have to to readjust one's expectations here), but even so, I find some articles in category theory mildly exasperating when they lack examples even of this type.

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O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov, Elementary Topology Problem Textbook has lots of examples.

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No probability book yet, so let me add a classic.

William Feller, An Introduction to Probability Theory and Its Applications, vol I, II. Full of examples, well motivated.

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A first course in Algebraic topology, again Fulton

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    $\begingroup$ Boy,is this book overrated,I bought it and thought I was robbed.You want a topology book that follows a historical development,read Stillwell's classic supplemented with McCleary's beautiful little book.Trust me,I just saved you about 60 bucks........ $\endgroup$ Mar 14 '10 at 8:13
  • $\begingroup$ I agree with Andrew L here. I too was disappointed at how little stuff was covered in this book. $\endgroup$
    – Anonymous
    Mar 14 '10 at 16:03
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"Explorations in Monte Carlo Methods" by Shonkwiler and Mendivil. Everything is well-motivated by examples. However, it is an undergraduate book.

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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.

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Foliations 1 by Alberto Candel and Lawrence Conlon

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Not really an abstract field, and probably well know to you, but it is worth mentioning:

Partial Differential Equations, by Lawrence C. Evans (first edition in 1998).

In Part I of the book, many fundamental notions are introduced by studying the Transport, Laplace's, Heat, and Wave Equations.

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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.

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Elliptic Curves: Function Theory, Geometry, Arithmetic by Henry McKean and Victor Moll

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