Last Updated: November 24, 2019
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This section of the best math books list is a collection of specialized and advanced topics.
- Foundations Books
- Combinatorics Books
- Category Theory Books
- Algebraic Topology Books
- Homological Algebra Books
- Commutative Algebra Books
- Algebraic Geometry Books
- Number Theory Books
- Group Theory and Representation Theory Books
- Riemannian Geometry Books
- Functional Analysis Books
- Harmonic Analysis Books
- Partial Differential Equations Books.
Best Foundations Book List
- Jech, Set Theory (Third Millennium Edition).
This is the classic reference for set theory at the advanced level. This edition (the third “millenium” edition) was a huge overhaul from earlier editions, and brought the text completely up-to-date with the latest research developments in set theory. Obviously I have not read this cover-to-cover (it contains all of the set theory in the universe), but I’ve glanced through sections. It’s inspiring, and even has exercises, so presumably you could learn set theory from it? I’m not a set theorist. If you are interested in the research frontier of set theory this is the book you must own. Also, if you read the whole thing, send me an e-mail. You’d have my admiration.
- Kanamori, The Higher Infinite.
Another set theory book that is a must-have for aspiring set theorists. This volume combines earlier surveys that Kanamori wrote and adds a lot of new material to bring the text up to the research frontier (as of the 1980s). Slightly more accessible than Jech (from my experience in perusing them both). The two books are really complementary though since Kanamori goes into much more detail on large cardinals research. I think this and Jech are the two books every set theorist owns a copy of. I’d like to find the time to read them some day.
- Hodges, A Shorter Model Theory.
The “Shorter” in the title is a reference to Hodge’s earlier book, Model Theory, a much meatier reference book. Maybe go there if you need something more substantial, but I found this “shorter” version had everything I needed. I’m only really familiar with two introductory (graduate) books on Model theory. This one, and Model Theory: An Introduction by David Marker, which I did not like.
- Troelstra & Schwichtenberg, Basic Proof Theory.
This is the only standard textbook on Proof theory that I am aware of. I’m not sure you have any other choice if you want to learn proof theory from a book.
Best Combinatorics Book List
- Stanley, Enumerative Combinatorics, Volumes 1 and Enumerative Combinatorics, Volume 2.
This two volume set is the bible of counting. It brings together a huge number of disparate techniques in combinatorics. The volumes are noted for the challenging exercises. Stanley has has a rating system for the exercises, each one receiving a difficulty rating from 1 to 5. Difficulty 1 exercises are (supposed to be) routine, while difficulty 5 exercises are open problems in combinatorics. So, there should be something to suit everyone. I actually own a copy of both volumes, and have been steadily working on them, and can’t recommend them highly enough. If you want to go into something in combinatorics or that makes use of combinatorial tools, then these books are great. Also note, there is a second edition of Volume 1, but there will not be a second edition of Volume 2.
- Flajolet & Sedgewick, Analytic Combinatorics.
I cannot tell if this is a computer science book for mathematicians or a mathematics book for computer scientists. Flajolet & Sedgewick were really computer scientists, and Sedgewick wrote well known books on algorithms (I think?). That said, I think the mathematics in this book is very nice, but I’m especially interested in generating functions and combinatorics. You might want to think of this as the grown up version of Wilf’s excellent generatingfunctionology. It really goes much deeper in the complex analytic aspects of generating functions (as the title suggests) and would make a great second course in generating functions.
Best Category Theory Book List
- Awodey, Category Theory.
This is the most accessible introduction to category theory I know of. I found Awodey’s category theory much easier to get started with than MacLane (see below), and I think it will appeal to a broader audience. Of course, it still covers all of the standard category theory material you should know (categories, limits and colimits, functors, Yoneda’s lemma, etc.). There is also a brief introduction to the lambda calculus for those in computer science. I think Awodey is the place to go if you want to get a basic working knowledge of categories as quickly as possible, with as little overhead as possible.
- MacLane, Categories for the Working Mathematician.
This the classic text/reference on category theory written by one of the primary creators of the theory. If you are going into something heavily categorical then you should probably read it. I found it difficult to get into and switched over to Awodey. Maybe I didn’t have sufficient background in algebraic topology and homological algebra? I think if you are interested in learning enough to category theory to understand applications in other fields you’re better off reading Awodey and then back-filling as necessary. If you’re really into Category Theory then you should read MacLane.
- Spivak, Category Theory for the Sciences.
It seems like everyone wants to use category theory these days. Why do scientists need category theory? If you’re a scientist and convinced that you do, then the mathematics in this book gets the mathematician stamp of approval.
- Walters, Categories and Computer Science.
Unlike “the sciences” I understand why computer scientists want to (and should) learn category theory. This is really a comprehensive introduction to category theory at the undergraduate level, completely motivated by examples coming from computer science. I think this would make a great textbook for mathematically inclined undergraduate CS majors. The abstract category theory might be heavy going, but the concrete computer science applications are good such as the application of distributive categories to abstract data types.
Best Algebraic Topology Book List
- Hatcher, Algebraic Topology.
This is the standard introductory graduate textbook for algebraic topology. This is the book I used to learn algebraic topology, and it’s just ok. There are five chapters (numbered 0 through 4). Chapter 0 is an introduction to some background material on CW complexes, chapters 1 and 4 are on homotopy, and chapters 2 and 3 are on homology. The prerequisites are not hugely daunting, but I think you want a solid course in point-set topology (at the level of Munkres or Willard on the best topology book list) as well as a semester or so of graduate abstract algebra. To be honest, I’ve found this book a little bit frustrating to try to learn from. It is too informal for my taste (particularly the introductory chapter). A lot of Hatcher’s proofs use visual arguments that simply don’t work for me. Maybe this is the sort of thing that works for algebraic topologists, but in the end I find it unsatisfying. I think if I had to do it over again, I would use Bredon (see below) to try to learn the basics from. (Luckily, before I started Hatcher, I had read through Lee’s Introduction to Topological Manifolds, reviewed here, which contained a large chunk of algebraic topology, and having some familiarity helped me navigate Hatcher a bit better.)
- Bredon, Geometry and Topology.
Despite the title, this is mostly an algebraic topology book. I think the first chapter is a basic introduction/review of general topology, but then the book moves quickly onto differential topology (I think only two chapters) and algebraic topology. The prerequisites are roughly the same as Hatcher: a solid first course in topology, and some graduate level abstract algebra. I’m not sure why “geometry” is in the title, since I don’t recall much geometry happening outside of a few examples. I would choose this book over Hatcher for learning algebraic topology for the first time.
- Rotman, An Introduction to Algebraic Topology.
As far as I know, this is the other standard graduate introduction to algebraic topology (along with Hatcher and Bredon). I’m not as familiar with it as the other two, but plenty of people like it. If you read Rotman’s abstract algebra book and like his style, then this is more of the same. I glanced through it, but found it a bit dull. I’d go with Bredon instead.
- Bott and Tu, Differential Forms in Algebraic Topology.
This book is awesome! Everyone who has picked up this book and read any part of it loves it. It’s amazingly clear and well motivated. I don’t think it can serve as an introduction to anything, per se, since you should probably know some differential geometry and some algebraic topology before you start. (Well, I’m not sure you strictly need any algebraic topology, but this doesn’t cover the standard algebraic topology material, so you probably need to read an introductory text on it at some point anyway.) For the differential geometry prerequisites, Loring Tu wrote Introduction to Manifolds (which I reviewed here) with the specific aim of providing the prerequisite material for this book. Anyway, everyone should read this book. It’s really one of the best advanced math textbooks on any subject. After going through parts of it I felt it brought a lot of ideas into focus that I hadn’t understood in topology and geometry before. I just can’t recommend this book highly enough.
- May, A Concise Course in Algebraic Topology, and the sequel, More Concise Algebraic Topology.
Leaving the friendly introductions behind we arrive at Peter May’s two book sequence (the second co-authored with Kate Ponto). After you’ve learned algebraic topology, you can read the first of these to get a concise (as the title implies) introduction to the core concepts from the most sophisticated and abstract viewpoint. The books are extremely well-written and May has taught from them for many years, so they actually are presented as a course that one can learn from. I’ve looked at the second book, but it’s above my head (for the moment). It’s aim is to bridge the gap to modern areas of research, so if you’ve read a couple of algebraic topology books and still want more, this might be for you.
- Spanier, Algebraic Topology.
I think any list of algebraic topology books is obligated to mention Spanier’s opus. I’m not sure this is human-readable, but it contains a huge amount of material stated in the most general and abstract setting. I can’t imagine using it to learn from, but it could work as a reference. Attempt reading it at your own risk.
Best Homological Algebra Book List
- Weibel, An Introduction to Homological Algebra.
This is the standard book on homological algebra, covering the standard material: derived functors, Tor and Ext, projective dimensions and spectral sequences. It isn’t very exciting, but I don’t know of a better alternative, and you’ll have to learn homological algebra at some point.
- Rotman, An Introduction to Homological Algebra.
Hey, another Rotman book. If you like Rotman’s style, you’re in luck. This is really the only alternative to Weibel that I’m at all familiar with. It’s somewhat easier going than Weibel, but I feel like it’s a bit less complete.
Best Commutative Algebra Book List
- Matsumura, Commutative Ring Theory.
This is my favorite introductory commutative algebra book. It’s complete and approximately self-contained (in the sense that it does not assume you know any commutative algebra, but does assume you know abstract algebra at the graduate level). This book covers a lot of material, and I think effectively serves the dual purpose of providing an introduction to commutative algebra for its own sake, and as a tool in algebraic geometry. There are also plenty of exercises at the end of each chapter, and even some hints and solutions in the back of the book. Maybe one day someone smarter than me will volunteer to add all of the solutions to Matsumura to the blog.
- Atiyah & MacDonald, Introduction to Commutative Algebra.
This is a tiny book on commutative algebra. Every time I look at it to find something, it’s not there. Plenty of other people really like it though, and it does provide an absolutely streamlined path to acquiring the commutative algebra you might need in other areas. Certainly a good place to start if you don’t want to tackle Matsumura since Atiyah & MacDonald is shorter, and somewhat easier.
- Eisenbud, Commutative Algebra: with a view Toward Algebraic Geometry.
In terms of pages, this book is the anti-Atiyah & MacDonald. It’s absolutely huge and might cover everything in the universe if you can only find it. I got the feeling that the total number of appendix pages might actually exceed the total number of non-appendix pages, but I haven’t counted. One of the explicit aims of this textbook was to provide all of the commutative algebra background one needs to read Hartshorne’s Algebraic Geometry (see below). You’d have to ask a proper algebraic geometer if it succeeds or not, but I find it hard to believe there is any commutative algebra result not contained in Eisenbud somewhere. Seriously, it is a really great book, I just find it overwhelming to try to find anything in. The idea of reading it cover to cover frightens me just a bit.
- Zariski & Samuel, Commutative Algebra I, and Commutative Algebra II.
A two volume set from two masters of the subject. Designed to make commutative algebra accessible to anyone, it’s a good alternative to Atiyah & MacDonald or Matsumura. Much more complete than either. It provides extremely detailed proofs of many results, and also provides several different proofs in some cases in an effort to highlight different ways of thinking about the subject.
Best Algebraic Geometry Book List
Algebraic Geometry is a sprawling subject. I’ve asked friends for some recommendations. The list is focused on broader introductions to the subject, neglecting even more specialized topics like Intersection Theory, Invariant Theory, Hodge Theory, etc.
- Shafarevich, Basic Algebraic Geometry 1, and Basic Algebraic Geometry 2.
This two volume set is quite classical in orientation, but probably the most complete introduction to the subject from the ground up, building up to schemes and complex geometry. This set is a good place to start before jumping in to more abstract treatments below.
- Gortz, Algebraic Geometry: Part 1: Schemes. With Examples and Exercises.
This is a quite recent text, and Part 2 does not exist yet. (Update 2019. Part 2 still does not exist.) This is a really lovely introduction to schemes, and incredibly complete. It is an update and improvement on Mumford’s classic Red Book of Varieties and Schemes. As the title indicates, this also includes plenty of exercises. This is really the best place to go for learning schemes now, and is a great complement to Hartshorne (see below). Hopefully more volumes in this series will come out in the near future.
- Liu, Algebraic Geometry and Arithmetic Curves.
The algebraic geometry book for number theorists. This does give a fairly complete presentation of algebraic geometry, but is focused on providing the tools to study arithmetic geometry, and number theoretic questions. Includes the background and preparation to understand Mordell’s conjecture, Faltings’ theorem and Fermat-Wiles.
- Griffiths & Harris, Principles of Algebraic Geometry.
This book is a sprawling, beautiful monster of a text. The edition I have is around 900 pages of some of the densest mathematical writing I’ve encountered. It is an amazing book though, and if you are interested in complex algebraic geometry then this is the canonical reference. This is really targeted at geometers rather than algebraists, so if you prefer less commutative and homological algebra in your algebraic geometry then this will suit. It contains no exercises, I’d say it is more of a reference than a textbook. An absolute must for complex geometers.
- Hartshorne, Algebraic Geometry.
The classic algebraic geometry textbook, this seems to elicit strong opinions from people. The exercises are legendary in their difficulty, and every aspiring algebraic geometry student is advised to work through as many of the exercises in Hartshorne as possible. Definitely not appropriate for a first treatment of the subject, and plenty of people think it should be avoided entirely. However, it remains the classic textbook in the subject, and you should get it for the exercises if nothing else.
- Vakil, The Rising Sea: Foundations of Algebraic Geometry.
This a free set of notes (people call them “notes,” but at 800 pages I’d say it’s a book) from Ravi Vakil at Stanford. I haven’t read them, but they are very highly regarded, especially those approaching algebraic geometry from a heavily algebraic background. The title refers to Grothendieck’s metaphor for how he solved problems,
The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it.. yet it finally surrounds the resistant substance.
Grothendieck Récoltes et semailles
Best Number Theory Book List
This list is heavily skewed toward algebraic number theory. The only analytic number theory book I’m really familiar with at this level is Davenport’s classic Multiplicative Number Theory (last book on the list). I’ll try to get some friends to suggest analytic number theory books to fill this out at some point.
- Weil, Basic Number Theory.
Weil was one of the greatest mathematicians of the 20th century. One of his most celebrated achievements was his proof in the 1940s of the Riemann hypothesis for zeta-functions of curves over finite fields. So, if you believe in “reading the masters” then this book should be on your shelf. This book is one of the absolute classics of number theory, and students seriously considering going into number theory should certainly own a copy. Be warned though, despite the title, this book is in no way “basic.” I’ve looked at it from time to time, but don’t have anywhere near the background necessary to understand anything.
- Silverman, The Arithmetic of Elliptic Curves.
Hey, I just went to a talk by Silverman at my university and he was a really nice guy! Not only that, but his books on elliptic curves (the other one is reviewed below) are the (deservedly) standard references for elliptic curves. This book is really a study of the arithmetic properties of elliptic curves from a modern viewpoint, utilizing algebraic number theory and algebraic geometry (as opposed to older formulations based on the `Lefschetz principle’). For aspiring number theorists, this is a must read, and the modern approach it takes leads naturally to the higher dimensional analog of elliptic curves: abelian varieties. The geometry and arithmetic of abelian varieties are central to a great deal of current research in number theory. The new second edition also includes a new chapter on algorithmic aspects of elliptic curves, including applications to elliptic curve cryptography. There are probably easier treatments of algorithmic elliptic curves, but if you want all of the hardcore mathematics you’ll get it here.
- Silverman, Advanced Topics in the Arithmetic of Elliptic Curves.
In case the first 500 pages of elliptic curves wasn’t enough, Silverman wrote a second 500 page book to keep you busy! I haven’t read this one at all, but along with the first book above these two form the standard reference in the subject. If you are interested in number theory then it’s worth looking into. (Maybe I’ll get around to reading it one day.) Although the first book of the series is now in its second edition, this one is still on the first edition, and I don’t think there is any plan to release a new edition.
- Frohlic & Taylor, Algebraic Number Theory.
I think this is the best place to go for an introduction to algebraic number theory at the graduate level. Sadly, I lent my copy to a friend who took it with her to grad school. She keeps promising to give it back, but I suspect I’ll never see it again. Maybe I’ll buy a new copy for myself for christmas, since this is a book I really want to spend more time with. I think this book strikes a nice balance between proving abstract (sometimes unmotivated) theory and examples. You’ll certainly spend plenty of time proving properties of Dedekind domains in the first half of the book, but the back half of the book provides a lot more context. I just found this an extremely readable book for the parts that I’ve gone through, and really found the style amenable to actually learning things. (As opposed to some books that make you feel like someone smacked you in the head with a brick when you try to learn things from them.)
- Neukirch, Algebraic Number Theory.
I would say this is the main alternative to Frohlic and Taylor for algebraic number theory. Less readable, and less introductory, but probably more comprehensive. For serious number theorists, this might be the better choice. I’m not a serious number theorist (or an unserious number theorist for that matter), so I haven’t read it.
- Davenport, Multiplicative Number Theory.
This is an absolute classic in the field, and one of the most important references for multiplicative number theory (and maybe the only one in the field I’m at all familiar with). We used it as a supplement in my analytic number theory class (though we didn’t use it as much as I would have liked). It’s an impressively well written text, and contains the most clearly written proof and discussion of the prime number theorem I know of. Even if you aren’t interested in number theory, or analytic number theory, you owe it to yourself to at least work through some sections of this book. It gives a really nice overview and historical context of some of the crowning achievements of 19th century mathematics, and a sense of the work of Riemann, Dirichlet, and others. This is a really nice book, even if (like me) you aren’t especially interested in multiplicative number theory.
Best Group Theory and Representation Theory Book List
This section has a lot in it, and I had to cut out some books to try to keep it manageable. It’s very likely I’ll be going into research in areas related to groups and representation theory. (Maybe geometric group theory, but I don’t actually know yet.) This is more in depth than any other section on this list, and I’m actively working on some of these books and am using the rest frequently for reference. I’ll try not to let myself get carried away, and will try to keep the reviews to a finite word count. This list below generally moves from introductory to more specialized as you go along. The last book or two are really specialized (but great!).
- Isaacs, Finite Group Theory.
This is my choice for a second course in group theory (after a standard graduate abstract algebra course). This book is demanding, and is intended for graduate students with a strong algebra background, and with a view toward going into research in a related area. Really, this text is preparation to start reading research papers in group theory, and I think it does a great job toward that goal. I should note, however, that this is strictly a group theory book; no representation theory. Still, it’s a book that is well worth having if you want to learn about groups beyond the standard material you’d find in a general algebra text.
- Rotman, Introduction to the Theory of Groups.
The other canonical textbook choice for a second course in group theory. I don’t enjoy Rotman’s writing as much as Isaacs, and I found this book a bit too plodding for my taste. It’s still an excellent book, and if you like Rotman (and plenty of people do), then you’ll be fine with this. The biggest irritation I’ve had with this book (I haven’t read it systematically, but have tried to look things up in it) is that there are just a lot of errors. I’ve seen this mentioned in other reviews, and have to agree. Even dipping into it for reference, I’ve noticed mistakes. This seems especially odd since the book is currently in its fourth edition. That point aside, it’s a nice book, and slightly gentler than Isaacs.
- Aschbacher, Finite Group Theory.
Written by one of the most prominent mathematicians in group theory, and one of the key figures in the classification of finite simple groups, Asbacher’s book provides a streamlined account of the background one needs to start reading research papers in the field. This is an interesting book in that it seems to try to walk the line between textbook and reference source. Although I do think it could be used as a text, I think it succeeds more as a reference. Still, Asbacher is one of the masters of the subject, and it’s probably worth reading his account of the results in finite group theory.
- Gorenstein, Finite Groups.
Another book written by one of the most prominent mathematicians involved in the classification of finite simple groups. The first edition of this book was the canonical reference for everything in the world known about groups up until 1968. The first edition was published as some of the major classification results of the 1950s and 1960s were giving way to the full blown classification program which culminated with the complete classification of finite simple groups. This second edition was published in 1980, somewhat before the complete classification was finished (modulo some minor fixes, the classification was finished in 1981), but far enough along that it was clear the complete classification would be accomplished. Anyway, this is the classic reference in the field, and although a lot has happened with finite groups since 1980, this is still an excellent place to learn the foundations of the subject. (In fact, sometimes these classic treatments during the formative years of a subject can be more enlightening since a lot of the insights and thinking has not yet been compressed and refined, and so somehow gives more insight and motivation.)
- Carter, Simple Groups of Lie Type.
This the standard introduction to the subject. It starts out with introductions to the classical simple groups, the Weyl groups, simple lie algebras, and the Chevalley groups. Since the work of Chevalley in the mid-1950’s, expanded in the ensuing decades by many others, Lie theory has provided a unified way of describing the families of simple groups of Lie type. Carter’s book describes the properties of these Lie families in a coherent framework. Obviously, this is a book for serious group theorists, and as such, requires a strong background in algebra, and probably some additional coursework specifically in group theory. Carter does spend the first few chapters building up the background of root systems, reflection groups, and simple Lie algebras over the complex field. This is required reading for people going into Lie theory (or, it is required reading for me at this very moment).
- Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters.
I’ve never been able to read this book since it is virtually impossible to get a hold of. The last time I saw a used copy show up on amazon it was outrageously expensive. The library at my university has a copy, but a faculty member has had it checked out permanently, and I’m not brave enough to put in a recall request. Anyway, if you can ever get a hold of a copy, you should. I’ve seen it referenced in papers as “The Bible.” Obviously, this is a highly specialized group theory text, but if you are interested in this area it is one of the classic/best references.
- Tao, Expansion in Finite Simple Groups of Lie Type.
Hey, I just got a hold of a copy of this. I’m really interested in seeing what Tao has to say on this, and I’ve been meaning to learn about expander graphs. I’ll update this when I’ve read more, but from browsing through, it looks great! I really like Terry Tao’s writing style, so your milage may vary. I think it is definitely worth checking out though, and if Tao is writing about it, then it’s probably interesting to a wider audience than just me.
- Procesi, Lie Groups: An Approach Through Invariants and Representations.
An amazing book! Absolutely the best choice for an introduction to Lie groups and their representations. I really love Procesi’s writing style: it’s very engaging and clear. I think this is a great book, but it is probably best for people with a little knowledge of algebraic geometry, and who have leanings in that direction. I really like this book, and the algebraic geometry prerequisite is not all that steep. If you’re motivated you could learn what you needed along the way. I think this one is worth the effort.
- Etingof, Introduction to Representation Theory.
The best pure introduction to representation theory. Etingof’s goal is to present a unified approach to the study of associative algebras, group representation theories, and lie algebras. The book has a lot of nice exercises and examples, many of which develop extensions of the material in the main text. If you just want a quick and accessible introduction to representation theory, I think this is a good book.
- Fulton & Harris, Representation Theory: A First Course.
A standard reference in the subject, Fulton & Harris is completely example driven. Really, almost the entire book is a parade of example after example, each building up slightly in complexity and abstraction. If you like your textbooks to contain concrete examples, then this is for you. I tried to read it, but found the example after example style too monotonous, but lots of people really love it, and if you read all the way through it you’ll learn a ton. It’s a book that is not ideal for me personally, but I definitely recommend to people since many people really like the example-driven style.
- Lydon & Schupp, Combinatorial Group Theory.
Okay, this book and the next one are super specialized in an area I find fascinating. Lydon & Schupp is the classic reference in combinatorial group theory, which was the precursor to the subject of “geoemtric” group theory launched by Gromov and his hyperbolic groups. I think particularly interesting are the prominence in the latter half of the text on the connections of combinatorial group theory with topology and logic.
- Davis, The Geometry and Topology of Coxeter Groups.
Another highly specialized text recommendation in my continuing effort to convince everyone to follow me into geometric group theory. This one is still sitting on my shelf waiting for me to learn some more geometry before making a sustained effort at reading it.
Best Group Riemannian Geometry Book List
I’m not a differential geometer at all. I took a poll around the department for the best books on differential geometry and Riemannian geometry and this is what I came up with. I have read parts of both Lee and do Carmo. If you are reading this and are really into differential geometry send me an email of some books you like and I’ll include them.
- Do Carmo, Riemannian Manifolds.
I think Do Carmo’s text is the most popular in this area and is a follow up (of sorts) to his undergraduate book on curves and surfaces. The bits of it I’ve read seem good.
- Lee, Riemannian Manifolds: An Introduction to Curvature.
The final book in Lee’s three book series (after Introduction to Topological Manifolds and Introduction to Smooth Manifolds). I’ve read the first two, and really like Lee’s books overall (even if Smooth Manifolds gets a bit wordy). If I felt compelled to seriously study some Riemannian geometry this is probably where I’d start. (Update 2019. Lee has released a second edition of this book and seems to have changed the title to Introduction to Riemannian Manifolds. It’s the one I linked to at the top.)
- Jost, Riemannian Geometry and Geometric Analysis. A final popular choice. I haven’t read it at all. If you have feel free to write a review and send it to me!
Best Functional Analysis Book List
All the books on this list are classics. I’m not aware of any more modern treatments that are superior to these. If you write one, let me know and I’ll include it.
- Rudin, Functional Analysis.
Hey, another Walter Rudin book. If you read baby Rudin and big Rudin and are still standing, why not try to complete the trifecta? This is probably still the best book on functional analysis, and if you like Rudin’s style then it definitely is. If you don’t like Rudin’s style, I’m not sure there is a great alternative (Conway, maybe?).
- Dunford & Schwartz.
- Linear Operators, General Theory (Part 1)
- Linear Operators: Spectral Theory (Part 2)
- Linear Operators: Special Operators (Part 3)
Written in the 1950’s, this was the standard book on Linear operators for the second half of the 20th century. The exercises are legendary (supposedly one of them became the basis of Langland’s thesis). I don’t know that it is readable by humans today, but there is probably no better reference around. If you managed to read through the first volume, you would know an impressive amount.
- Kadison & Ringrose.
- Fundamentals of the Theory of Operator Algebras, Volume I: Elementary Theory
- Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory
- Fundamentals of the Theory of Operator Algebras, Volume III
- Fundamentals of the Theory of Operator Algebras, Volume IV
The first two volumes are a more modern treatment of operator algebras than Dunford & Schwarz, while volumes III & IV provide complete solutions to all of the exercises from volumes I & II. Of course, I love exercises and solutions as a way to learn math, so I fully endorse these. Dunford & Schwartz is still the classic, but I think Kadison & Ringrose is more palatable today.
- Kreyszig, Introductory Functional Analysis with Applications.
Definitely the functional analysis book for applied mathematicians. If you find a quick plausibility argument more satisfying than an actual proof, then you’ll like this book. All of the space saved by not proving results can then be spent on applications. Obviously, not suitable for most pure mathematicians, this book is extremely popular amongst application oriented folks. Take a look at it if you want to know how functional analysis gets used in the real world!
Best Harmonic Analysis Book List
Harmonic analysis books are interesting in that there is a real divergence between the classical theory and the modern theory. I included three classical books on this list (Hewitt & Ross, Rudin, and Katznelson) and one much more modern treatment (Stein) at the end.
- Katznelson, An Introduction to Harmonic Analysis.
Classic text on harmonic analysis written in 1968, but still a solid way to pick up the basics of the subjects: Fourier series and transforms, H^p-spaces, Paley-Wiener, and spectral theory. It might be a bit dated now, but it’s still an excellently written introduction to the subject.
- Hewitt & Ross.
Another classic textbook for harmonic analysis, this ones goes much deeper than Katznelson and is legendary for its problems. I haven’t read it, but it would be fun to try to blog all the problems to this one. (Fun in the sense that I would fail miserably and everyone could laugh at me.)
- Rudin, Fourier Analysis on Groups.
Well, another Walter Rudin book. If you like Rudin then I don’t see why you wouldn’t like this.
- Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals.
Lastly, a distinctly more modern textbook on harmonic analysis. Stein has a couple of other books in the same vane general direction (Introduction to Fourier Analysis on Euclidean Spaces and another whose title I can’t remember at the moment). Really aimed at introducing recent developments in harmonic analysis: Hardy spaces, oscillatory integrals and Fourier integral operators, connections with the Heisenberg groups, and pseudo-differential operators. Definitely the only place I know of to go for the modern viewpoint on harmonic analysis.
Best Partial Differential Equations Book List
- Partial Differential Equations I: Basic Theory
- Partial Differential Equations II: Qualitative Studies of Linear Equations
- Partial Differential Equations III: Nonlinear Equations.
Taylor’s three volume series contains basically everything in the entire universe. They are definitely the most sophisticated, comprehensive, and thorough books available on PDEs and are suitable for both pure and applied mathematicians. If I had the time and ability this is the set of mathematics books I would most like to sit down and read from cover to cover. This is a really monumental work, giving the abstract view of PDEs on Riemannian and Lorentzian manifolds, but then drilling down and doing computations on coordinates when it is necessary or enlightening. If you’re serious about this material then Taylor is the gold standard, but it is definitely for specialists.
- Evans, Partial Differential Equations.
This is the standard choice for graduate-level partial differential equations courses. Much easier going than Taylor, it spends a lot of time talking. That said, most people like it, and it does a good job of introducing PDEs at the advanced abstract level. If you’re not a specialist, but want a solid introduction then this is the way to go.
Two more options for proof theory are Negri & von Plato, “Structural Proof Theory”; and Mancosu, Galvin, & Zach, “An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs”.