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I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden).

I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on if I want to study beyond Masters. I'm not there yet as I'm on a four year course and had a gap year between Years 3 and 4.

Please recommend for Algebra, Linear Algebra and Categories - Analysis, Set Theory, Measure theory (an area I have seen too little books dedicated for).

E.g. Spivak is very good for self learning basic real analysis, but Rudin really cuts to the heart.

Adam
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    I think Rudin is great for readers already familiar with the subject who want to learn it meticulously, but hard for beginners – Daniel Freedman Dec 29 '11 at 00:03
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    I feel "Elementary Analysis by Ross" is a good book with a lot of examples. You can use this book for self-studying. – Alexander Mar 10 '15 at 07:03

6 Answers6

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EDIT: I now think that this list is long enough that I shall be maintaining it over time--updating it whenever I use a new book/learn a new subject. While every suggestion below should be taken with a grain of salt--I will say that I spend a huge amount of time sifting through books to find the ones that conform best to my (and hopefully your!) learning style.

Here is my two cents (for whatever that's worth). I tried to include all the topics I could imagine you could want to know at this point. I hope I picked the right level of difficult. Feel absolutely free to ask my specific opinion about any book.

Basic Analysis: Rudin--Apostol

Measure Theory: Royden (only if you get the newest fourth edition)--Folland

General Algebra: D&F--Rotman--Lang--Grillet

Finite Group Theory: Isaacs-- Kurzweil

General Group Theory: Robinson--Rotman

Ring Theory: T.Y. Lam-- times two

Commutative Algebra: Eisenbud--A&M--Reid

Homological Algebra: Weibel--Rotman--Vermani

Category Theory: Mac Lane--Adamek et. al--Berrick et. al--Awodey--Mitchell

Linear Algebra: Roman--Hoffman and Kunze--Golan

Field Theory: Morandi--Roman

Complex Analysis: Ahlfors--Cartan--Freitag

Riemann Surfaces: Varolin(great first read, can be a little sloppy though)--Freitag(overall great book for a second course in complex analysis!)--Forster(a little more old school, and with a slightly more algebraic bend then a differential geometric one)--Donaldson

SCV: Gunning et. al--Ebeling

Point-set Topology: Munkres--Steen et. al--Kelley

Differential Topology: Pollack et. al--Milnor--Lee

Algebraic Topology: Bredon--May-- Bott and Tu (great, great book)--Rotman--Massey--Tom Dieck

Differential Geometry: Do Carmo--Spivak--Jost--Lee

Representation Theory of Finite Groups: Serre--Steinberg--Liebeck--Isaacs

General Representation Theory: Fulton and Harris--Humphreys--Hall

Representation Theory of Compact Groups: Tom Dieck et. al--Sepanski

(Linear) Algebraic Groups: Springer--Humphreys

"Elementary" Number Theory: Niven et. al--Ireland et. al

Algebraic Number Theory: Ash--Lorenzini--Neukirch--Marcus--Washington

Fourier Analysis--Katznelson

Modular Forms: Diamond and Shurman--Stein

Local Fields:

  1. Lorenz and Levy--Read chapters 23,24,25. This is by far my favorite quick reference, as well as "learning text" for the basics of local fields one needs to break into other topics (e.g. class field theory).
  2. Serre--This is the classic book. It is definitely low on the readability side, especially notationally. It also has a tendency to consider things in more generality than is needed at a first go. This isn't bad, but is not good if you're trying to "brush up" or quickly learn local fields for another subject.
  3. Fesenko et. al--A balance between 1. and 2. Definitely more readable than 2., but more comprehensive than 1. If you are wondering whether or not so-and-so needs Henselian, this is the place I'd check.
  4. Iwasawa--A great place to learn the bare-bones of what one might need to learn class field theory. I am referencing, in particular, the first three chapters. If you are dead-set on JUST learning what you need to, this is a pretty good reference, but if you're likely to wonder about why so-and-so theorem is true, or get a broader understanding of the basics of local fields, I recommend 1.

Class Field Theory:

  1. Lorenz and Levy--Read chapters 28-32, second only to Iwasawa, but with a different flavor (cohomological vs. formal group laws)
  2. Tate and Artin--The classic book. A little less readable then any of the alternatives here.
  3. Childress--Focused mostly on the global theory opposed to the local. Actually deduces local at the end as a result of global. Thus, very old school.
  4. Iwasawa (read the rest of it!)
  5. Milne--Where I first started learning it. Very good, but definitely roughly hewn. A lot of details are left out, and he sometimes forgets to tell you where you are going.

Metric Groups: Markley

Algebraic Geometry: Reid--Shafarevich--Hartshorne--Griffiths and Harris--Mumford

Alex Youcis
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  • @Mathemagician1234 Thank you! I would very much like that, I am always on the prowl for a good, new book. – Alex Youcis Dec 29 '11 at 01:29
  • I had to edit and add more that came to mind. I may add more in the future. – Alex Youcis Dec 29 '11 at 01:43
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    I'm tempted to give you a -1 just for mentioning Hartshorne. :) –  Dec 29 '11 at 01:52
  • @SteveD Haha, I understand the sentiment. I do like Hartshorne in small doses though. It's really not so bad if you are fairly up-to-snuff on your algebra--but I agree, I think for a first time around Mumford or Shafarevich is the way to go. – Alex Youcis Dec 29 '11 at 01:53
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    Let's agree to disagree then. I worked through the first 4 chapters of Hartshorne's book, and I had no trouble with the algebra - it was just the way the book is. Anyway, I think the perfect book for anyone getting into algebraic geometry is Ravi Vakil's lecture notes, surely soon to be published as a book. They are available here: http://math.stanford.edu/~vakil/216blog/ –  Dec 29 '11 at 02:08
  • Great list, Alex! A suggestion: Hatcher for Algebraic Topology! – Bruno Joyal Dec 29 '11 at 02:26
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    @Bruno Hey man. I never really liked Hatcher to be honest--don't shoot me. – Alex Youcis Dec 29 '11 at 02:34
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    Dear Alex, Mumford's name has only two m's, not three. Regards, – Matt E Dec 29 '11 at 02:39
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    @Alex,Bruno: I'm still trying to get around to liking Hatcher myself.Hatcher really does a lot of handwaving for my tastes,although a lot of sections of the text are indeed wonderful. I was really disappointed by Bredon because it's trying to do something really important-namely,create a modern algebraic topology course with an emphasis on manifolds-and it falls flat on it's face.It introduces many concepts without clearly defining them and this really makes the book mystifying to follow at times.My favorites-in addition to Rotman and May-are the translations of 2 awesome books by Prasolov. – Mathemagician1234 Dec 29 '11 at 04:30
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    @SteveD I'm working through Vakil's notes right now, and I love them! – Alex Becker Dec 29 '11 at 04:33
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    Not a single probability or statistics book? –  Mar 11 '13 at 18:57
  • Thanks for such a great answer. Please consider adding books on Operator theory, Non-commutative geometry & Atiyah-Singer Index Theorem. – user90041 Nov 04 '13 at 13:14
  • I like Huppert's character theory book a lot, too. It's comparable to Isaacs: a little more module-y, more example oriented, no Schur multipliers or exceptional characters, great sections on Frobenius groups and induced characters. – Alexander Gruber Jan 28 '14 at 06:46
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    Nothing in combinatorics, graph theory, association schemes, design theory, discrete geometry, ...? – Anurag Apr 21 '14 at 00:16
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    @AlexYoucis how long did it take you to study all those books? – seeker Jun 04 '15 at 11:09
  • I need a book in projective geometry, it's missing on your list, can someone add one to make it more complete? – Santropedro Oct 31 '16 at 23:18
  • Hi Alex - could I ask for some more detail on Ahlfors vs. Freitag vs. say Narasimhan (I'm not sure if you've looked at it or spoken to people about it, but have seen it recommended in another thread on this site) or even something like Rudin. My background/goals: I covered the first few chapters of Conway as a first book several years ago, but there are gaps in my knowledge of the introudctory material. I'm aiming to do a detailed review, then go significantly further, perhaps towards topics that "every mathematician should know" (Great Picard and so on or anything you might suggest) and/or... – Jonathan Rayner May 30 '18 at 13:34
  • ... lay some foundation for analytic number theory/devote time to modular forms/other topics necessary for a basic intro to the Langlands program. I have about a year for this. I see that Freitag goes in this direction, but I wonder if it's not really a bonus for that book, because perhaps I would switch to a book specifically on that topic rather than read the last few chapters of Freitag, what do you think? I have found the exercises in Ahlfors to be extremely illuminating, but the first few chapters don't seem rigorous enough for me, is that just because it's introductory? – Jonathan Rayner May 30 '18 at 13:41
  • ... I haven't had a chance to look at Narasimhan yet. I'd like a challenging text, but also want to me sure that in hindsight from someone more knowledgable than me that it's challenging because it's worth it and not challenging simply because a better alternative exists. – Jonathan Rayner May 30 '18 at 13:44
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    This list is way too long to be a list of books that every undergraduate must read. – Richard Sullivan Oct 31 '19 at 23:37
  • Minor point: please use Royden 4th Edition printed after 2017. The earlier 4th edition has quite a few serious mistakes in Problem sets unfortunately. – Smart Yao Jul 09 '21 at 04:25
  • Nothing on set theory? I can think of some authors for topology too –  Sep 08 '21 at 20:33
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Here are a few of the books I've found especially rewarding:

  • Linear Algebra - Friedberg, Insel, Spence

    An excellent introduction to finite dimensional linear algebra. In fact, for most undergraduate stuff you won't need anything else.

  • Principles of Mathematical Analysis - Rudin

    Perfect intermediate textbook between typical undergraduate books and typical graduate books. The exposition to "Rudin-style books" is almost as valuable as the actual content.

  • Mathematical Analysis - Apostol

    Covers a few more topics than Rudin and is a little more explicit.

  • Complex Analysis - Stein, Shakarchi

    A little difficult as a first introduction but very good.

  • Partial Differential Equations - Rauch

    This book is geared at graduate students, but accessible to undergraduates with a strong background in my opinion. Its a great first introduction to "serious PDE" (i.e. not your typical cookbook course). Another good one is Introduction to Partial Differential Equations - Renardy & Rogers. This second one also does not assume any familiarity with Lebesgue integration, so it might be better for undergraduates.

  • Introduction to the Theory of Groups - Rotman

    This is a great second text in group theory after an elementary exposition (say Contemporary Abstract Algebra - Gallian).

  • Real and Complex Analysis - Rudin

    For those wishing to go further in analysis this is absolutely necessary as far as I'm concerned.

Community
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    +1 for Friedberg, Insel and Spence. This book straddles the line between a rigorous presentation of linear algebra and an applied approach better then any of the others.It's only major defect is a very confusing presentation of the Jordan form,for which Charles Curtis' book does a much better job. I'd have given more points,but I disagree with some of your other picks.This' ll be clearer when I post my list. – Mathemagician1234 Dec 29 '11 at 02:32
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    In a similar vein, "Linear Algebra Done Right" - Axler –  Dec 29 '11 at 02:37
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This may not be relevant to you, but for others who are still in high school or first and second year university the following book by Chartrand, Polimeni, and Zhang, is an incredible introduction to proofs and various areas of mathematics

"Mathematical Proofs: A Transition to Advanced Mathematics" by Gary Chartrand, Albert D. Polimeni, and Ping Zhang.

There is an entire chapter devoted to each of the following:

  • Communicating Mathematics
  • Naive Set Theory
  • Logic
  • Direct Proof
  • Proof by Contrapositive
  • Existence and Proof by Contradiction
  • Mathematical Induction (and Strong Induction)
  • Equivalence Relations (Equivalence Classes, Congruence Modulo n, Modular arithmetic)
  • Functions (Bijective, Inverse, Permutations)
  • Set Theory (up to Schroder-Bernstein Theorem and the Continuum Hypothesis)
  • Number Theory
  • Calculus (Limits, Infinite Series, Continuity, Differentiability)
  • Group Theory (up to Isomorphic Groups)

With Three Additional Chapters online covering:

  • Ring Theory
  • Linear Algebra
  • Topology

All with in depth worked out solutions to each chapter. I think every serious first-year mathematics student should work through this entire book thoroughly doing as many questions as possible!

Samuel Reid
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  • Ordinary Differential Equations by V.I.Arnold or by J. Hale * Combinatorics - Enumerative Combinatorics vol.1&2 by R. Stanley
    • – hkju Apr 17 '12 at 14:56
  • Exactly what I'm looking for. Do you recommend me to spend my money on that book? I mean, is really worthy? Because I need a book full of demonstrations (relations, functions, set theory, etc) Thanks. – Sofía Contreras May 09 '18 at 00:04
  • I know this is late, but thank you very much for posting this! That book is EXACTLY what I need! – zerosofthezeta Jun 25 '19 at 17:58
  • Old post, but I teach an intro to proofs course from this book. There's not time to get past number theory, but it's a very patient, detailed and well-organized account that's perfect for "general audiences". Very solid option. – Joshua P. Swanson Jan 27 '21 at 07:30