Questions asking for recommendations of textbooks on some subject. It can be helpful to indicate whether the request is for self-study, for use in a course one teaches, for use accompanying a course one takes etc., and to give some additional details on the context. Typically, additional tags are ...

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83
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32answers
50k views

Best Algebraic Geometry text book? (other than Hartshorne)

I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc. One suggestion ...
39
votes
8answers
3k views

Natural transformations as categorical homotopies

Every text book I've ever read about Category Theory gives the definition of natural transformation as a collection of morphisms which make the well known diagrams commute. There is another possible ...
38
votes
7answers
11k views

Is Mac Lane still the best place to learn category theory?

For a student embarking on a study of algebraic topology, requiring a knowledge of basic category theory, with a long-term view toward higher/stable/derived category theory, ... Is Mac Lane still ...
34
votes
1answer
2k views

Wanted: a “Coq for the working mathematician”

Sorry for a possibly off-topic question -- there are four StackExchange subs each of which could be construed as the proper place for this question, and I've just picked the one I'm most familiar ...
27
votes
11answers
8k views

Homological Algebra texts

I would like to hear the communities' ideas on good Homological Algebra textbooks / references. The standard example is of course Weibel (which I'll leave for someone else to describe). As usual, ...
26
votes
11answers
10k views

Algebraic Topology Beyond the Basics:Any Texts Bridging The Gap?

Peter May said famously that algebraic topology is a subject poorly served by it's textbooks. Sadly,I have to agree. Although we have a frieghtcar full of excellent first-year algebraic topology ...
26
votes
6answers
4k views

Book on mathematical “rigorous” String Theory?

I've been looking high and low for a mathematical Book on String Theory. The only Book I could find was "A Mathematical Introduction to String Theory" by Albeverio, Jost, Paycha and Scarlatti. I only ...
25
votes
8answers
3k views

An “advanced beginner's” book on algebraic topology?

It has so happened that I have come this far knowing nothing on the subject of algebraic topology (as in homology theories of topological spaces and their applications). I've decided to finally read ...
24
votes
16answers
6k views

introductory book on spectral sequences

I have studied some basic homological algebra. But I can't send to get started on spectral sequences. I find Weibel and McCleary hard to understand. Are there books or web resources that serve as ...
24
votes
11answers
7k views

Lecture notes on representations of finite groups

Next term I am supposed to teach a course on representation of finite groups. This is a third year course for undegrads. I was thinking to use the book of Grodon James and Martin Liebeck ...
22
votes
1answer
11k views

If I want to study Jacob Lurie's books “Higher Topoi Theory”, “Derived AG”, what prerequisites should I have?

I've been told that it's important to know modern physics, Differential Geometry and Algebraic Topology for understanding higher structures. Is there any other prerequisite for understanding Lurie's ...
22
votes
3answers
3k views

Elementary Number Theory Text from a Categorical Perspective

My question is somewhat similar to this previous question, but from a slightly different perspective. Is there any textbook on elementary number theory that develops the properties of $\mathbb{Z}$ as, ...
20
votes
1answer
1k views

Why not a Roadmap for Homotopy Theory and Spectra?

MO has seen plenty of roadmap questions but oddly enough I haven't seen one for homotopy theory. As an algebraic geometer who's fond of derived categories I would like some guidance on how to build up ...
19
votes
3answers
2k views

Real algebraic geometry vs. algebraic geometry

This question is predicated on my understanding that real algebraic geometry (henceforth RAG) is the version of algebraic geometry (AG) one gets when replacing (esp. algebraically closed) fields with ...
19
votes
5answers
2k views

A toolbox for algebraic topology

This question has a very general part and a rather concrete part. General: When one wants to prove something in algebraic topology (actually in all parts of mathematics) one obviously needs some ...
18
votes
10answers
6k views

Riemannian Geometry Introductory Text

I have studied differential geometry, and am looking for basic introductory texts on Riemmanian geometry. My target is eventually Kähler geometry, but certain topics like geodesics, curvature, ...
17
votes
15answers
14k views

Good Algebraic Number Theory Books

I have just finished a master's degree in Mathematics and want to learn everything possible about algebraic number fields and especially applications to the generalized Pell equation (my thesis ...
17
votes
5answers
4k views

Textbook for Etale Cohomology

What is the best textbook (or book) for studying Etale cohomology?
16
votes
2answers
5k views

Introductory text on Galois representations

Could someone please recommend a good introductory text on Galois representations? In particular, something that might help with reading Serre's "Abelian l-Adic Representations and Elliptic Curves" ...
16
votes
12answers
1k views

Insightful books about elementary mathematics

What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful ...
16
votes
14answers
2k views

Teaching a pedagogy course

At my institution incoming graduate students must take a semester long course on pedagogy taught by current grad students. I may soon be in the position of having to teach this course and I'm looking ...
15
votes
29answers
11k views

Alternative Undergraduate Analysis Texts

Other than the standard baby Rudin, Spivak, and Stein-Shakarchi, are there other alternative and comprehensive analysis texts at the undergraduate level? For example something that has general results ...
15
votes
16answers
2k views

Math books for advanced high school students

I'm working in a program for teaching a group of students selected in a Olympiad competition. The program is aimed to acquaint the students with the diverse aspects of higher mathematics in a way ...
15
votes
3answers
2k views

A reference for geometric class field theory?

The classic reference of this topic is Serre's Algebraic Groups and Class Fields. However, many parts of this book use Weil's language, which I find quite hard to follow. Is there another reference ...
14
votes
19answers
11k views

Good combinatorics textbooks for teaching undergraduates?

Hello, can anyone recommend good combinatorics textbooks for undergraduates? I will be teaching a 10-week course on the subject at Stanford, and I assume that the students will be strong and motivated ...
14
votes
12answers
5k views

A book in topology

I will have to teach a topology course: it starts in point set topology and ends at fundamental group of $S^1$. In the past I have used two different books: Elementary Topology. Textbook in ...
14
votes
5answers
1k views

Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...
14
votes
1answer
4k views

Matsumura: “Commutative Algebra” versus “Commutative Ring Theory”

There are two books by Matsumura on commutative algebra. The earlier one is called Commutative Algebra and is frequently cited in Hartshorne. The more recent version is called Commutative Ring ...
13
votes
13answers
4k views

Looking for an introductory textbook on algebraic geometry for an undergraduate lecture course

I am now supposed to organize a tiny lecture course on algebraic geometry for undergraduate students who have an interest in this subject. I wonder whether there are some basic algebraic geometry ...
13
votes
10answers
3k views

Good book on Riemann surfaces and Galois theory?

I'm supervising an undergraduate project on Galois theory and covering spaces. I want to have him read about the fact that from a branched cover of a Riemann surface you get an extension of its field ...
13
votes
6answers
1k views

Text for Algebraic Number Theory

I have the privilege of teaching an algebraic number theory course next fall, a rare treat for an algebraic topologist, and have been pondering the choice of text. The students will know some ...
12
votes
18answers
17k views

A good book of functional analysis

I'm a student (I've been studying mathematics 4 years at the university) and I like functional analysis and topology, but I only studied 6 credits of functional analysis and 7 in topology (the ...
12
votes
17answers
24k views

Suggestions for a good Measure Theory book

I have taken analysis and have looked at different measures, but I am currently looking at realizing a certain problem in a different light and feel that I need a better background in various measures ...
12
votes
11answers
2k views

Approachable French Masters

It has been my general desire for a few years to acquire the basics in other European languages for the purpose of reading some of the classics in their original language, in a similar vein to this ...
12
votes
19answers
8k views

Textbook recommendations for undergraduate proof-writing class

I am teaching the proof-writing class (for the 3rd time) in the Fall and plan to buck the party line and use a different text than the default Bond and Keane. My parameters are as follows: Logic, ...
12
votes
9answers
3k views

Book recommendation for ergodic theory and/or topological dynamics?

Hello, I'd like to hear your opinion for ergodic theory books which would suit a beginner (with background in measure theory, real analysis and topological groups). I am looking for something well ...
12
votes
4answers
3k views

Textbook or lecture notes in topological K-Theory

I am looking for a good introductory level textbook (or set of lecture notes) on classical topological K-Theory that would be suitable for a one-semester graduate course. Ideally, it would require ...
12
votes
2answers
1k views

Learning path for the proof of the Weil Conjectures

Assume you are an algebraic geometry advanced student who has mastered Hartshorne's book supplemented on the arithmetic side by the introduction of Lorenzini - "An Invitation to Arithmetic Geometry" ...
11
votes
10answers
3k views

Math History books

I'm teaching a course over the summer (it's a sort of make-your-own course for non-majors) and I'm planning on organizing it as a math history course, hitting on major threads through about 1900, and ...
11
votes
16answers
10k views

Undergraduate Differential Geometry Texts

Can anyone suggest any basic undergraduate differential geometry texts on the same level as Manfredo do Carmo's Differential Geometry of Curves and Surfaces other than that particular one? (I know a ...
11
votes
6answers
2k views

Graduate ODE textbook

Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, ...
11
votes
2answers
450 views

Book on the Three body Problem

Hi all, I am looking for a good book about the famous (infamous perhaps?) three body problem - both theoretical and numerical hardless and accomplishments. can you help? Thanks
10
votes
12answers
5k views

undergraduate logic textbook

I am going to teach the standard undergraduate Logic course for math and engineering majors. What are good (bad) text-books and why. I have not taught that course for a while and wonder if there are ...
10
votes
4answers
748 views

Calculus book in the spirit of the 18th century

I have been reading a lovely 18th century book by John Rowe called An Introduction to the Doctrine of Fluxions. It presents calculus completely motivated by geometric questions - finding tangents to ...
10
votes
7answers
3k views

Leibnizian calculus textbook

Where can I find a calculus textbook that emphasizes differentials? Is there such a book that I could realistically require my calculus students to use? I want a textbook that supports me when I tell ...
9
votes
10answers
3k views

The best text to study both incompleteness theorems

Hi! What text on both incompleteness theorems you would recommend for beginner? Specifically, I'm looking for the text with the following properties: 1) The proofs should be finitistic, in Godel's ...
9
votes
4answers
608 views

Introductory reading on the Scholz reflection principle?

The Scholz reflection principle says, among other things, that if $D < 0$ is a negative fundamental discriminant, not $-3$, then the 3-ranks of the class group of $\mathbb{Q}(\sqrt{D})$ is either ...
9
votes
2answers
1k views

Literature on the Springer resolution

Could you suggest me a basic reading list on the Springer resolution? Is there a textbook I can refer to? Or do I need to start with the original paper? Unfortunately googling for "Springer" and ...
9
votes
2answers
564 views

Tor sheaves: what do they tell us about geometry

Hi! I fear that I am up to ask a very vague question, but more than an answer I need a suggestion of references I should look up. I need to know everything about Tor sheaves and what do they tell ...
8
votes
15answers
9k views

Learning Topology

EDIT (Harry): Since this question in its original form was poorly stated (asked about topology rather than graph theory), but we have a list of Topology books in the answers, I guess you should go ...