Mathematics Book List

Culture

• Paul Halmos, I Want to be a Mathematician. This is the autobiography of one of the most important mathematical writers of the 20th century, it is not only interesting as a biography but also as a source of advice on writing mathematics, teaching, and other things. Particularly interesting are his discussions of graduate student life in the US before the second world war, and the stories behind the two books I mention below, Naive Set Theory and Finite-Dimensional Vector Spaces. His discussion of the writing process is particularly enlightening: it reminds me somewhat of another great writer, Vladimir Nabokov - for instance, both wrote on index cards (this is discussed on p.394 of Halmos (Springer softcover reprint 1985) and on p.169 of Brian Boyd's Vladimir Nabokov: The American Years (Vintage softcover 1993)).
• Michael Harris, Mathematics Without Apologies: Portrait of a Problematic Vocation. A more modern non-autobiography, I quite enjoyed it. This book is allegedly addressed to the layperson, which is an opportunity to impose upon you one of my opinions: there are two kinds of mathematical biographies, namely those by laypeople writing about mathematicians and those by mathematicians writing about mathematicians. The first kind is invariably readable only by laypeople, and the second kind invariably only by mathematicians (regardless of the target audience).

  Let me be more precise: when a layperson (an abuse of terminology which I will use to mean 'non-mathematician' for convenience) writes a mathematical biography (two good examples here: Paul Hoffman's The Man Who Loved Only Numbers (about Paul Erdős) and Simon Singh's Fermat's Last Theorem which I would place in the 'biography' super-genre despite being a biography of something inanimate) a full two-thirds of the book is spent on discussing mathematical culture through listing off the same few historical events (e.g. stating the Bridges of Konigberg problem, or giving a precis of a biography of Ramanujan). On the other hand, mathematicians do not do this - and even when they try, as in Harris' book here, the level of discourse is far beyond the average reader with only minimal mathematical training: not so much the mathematical content (here, the basics of number theory) but the speed at which it is expounded which requires a certain amount of mathematical maturity. That is, there is a reason that the non-mathematician spends 2/3 of the book on this stuff when writing for the layperson, rather than the three sections of this 10-chapter book.

  This was recently discussed in XKCD:2501. Another famous example is the rejected obituary of Grothendieck written by Tate and Mumford: for their side of the story see Mumford's blog post Can one explain schemes to biologists. I also like the subsequent post Can one explain schemes to hipsters? on the neverendingbooks blog, and the post The two cultures of mathematics and biology on Lior Pachter's blog.

• Paul Lockhart, A Mathematician's Lament: A complaint about the U.S. education system that is somewhat relevant to NZ.
• Underwood Dudley, A Budget of Trisections and Mathematical Cranks. Both very funny, quite quick to read! In Trisections, Dudley goes through a selection of attempts to trisect the angle using only a straightedge and compass (proved impossible by Wantzel in 1837). Cranks is a short collection of random stories about mathematical cranks. At this point I am reminded that there is (was?) a book in the General Library by a relativity denier whose first ten or so pages were covered in pencil from some conscientous physics student's efforts at giving detailed refutations - after ten pages, of course, they had clearly had enough. (516.204 D84 and 510 D84)
• Stephen Krantz, Mathematical Apocrypha and More Mathematical Apocrypha. Collections of short mathematical biographies and urban legends (possibly even less reliable than Bell).
• Carl E. Linderholm, Mathematics Made Difficult. This book needs no introduction and no recommendation, it is a classic.
• Norman E. Steenrod; Paul R. Halmos; Menahem M Schiffer; and Jean A. Dieudonné, How to Write Mathematics. This is a collection of four essays, the theme being mathematical writing (planning structure, style of writing, etc.). Well worth the read.
• G. H. Hardy, A Mathematician's Apology. This is a classic book, but should be read with the understanding that many of the philosophical ideas put forward by Hardy are old-fashioned at best and damaging at worst. A very good discussion of this from a modern viewpoint is chapter 10 of Harris' Mathematics Without Apologies which I mentioned above.
Another very interesting read is the blog post G. H. Hardy and Mrs Ellis on Izabella Laba's blog.
• Diana Davis (ed.), Illustrating Mathematics. Very nice book of pictures of various geometric objects which have been made "in the real world" (e.g. sculptures or line drawings).
• Mark Ronan, Symmetry and The Monster. An introduction to the classification of the finite simple groups, very good bathroom reading. A bit of "mathematics for the nonmathematician" is included but it doesn't make the book too hard to read (i.e. it is easy to skip).
• M.C. Escher, The Magic of M.C. Escher. Very nice edition of the works of Escher. In this vein see the book below by Grünbaum and Sheppard on tilings of the plane. Also highly recommended is this video of H.S.M. Coxeter discussing the work of Escher (part of the documentary The Fantastic world of Escher) and this video on Coxeter, Gardner, and Escher.
• Lewis Carroll, Alice's Adventures in Wonderland and Through the Looking-Glass. While on the subject of nice editions of mathematical art, I particularly recommend the edition of Lewis Carroll's two novels published in the Penguin Classics series.
• Douglas Hofstadter, Gödel, Escher, Bach. Perhaps it is because I am not a philosopher or computer scientist, but I found it unreadable.
• Michael Gowers (ed.), Princeton Companion. A very nice, locally readable, encyclopaedia of mathematics written for mathematicians. Keep meaning to buy a copy...

Children's books (first and second year undergraduate)

Single-variable calculus

• James Stewart, Calculus (and all other isomorphic books, e.g. Anton's calculus book). If you're an engineering student, this book is probably fine; it's full of routine problems. If you're a mathematics student, I wouldn't bother buying a copy (there are plenty in the library) - for the actual mathematics, see the lecture notes.
• Silvanus Thompson, Calculus Made Easy. This is the book I originally learned calculus from in year 11. It is a little old fashioned, but the intuition in here is very clear and the author has a sense of humour! This is perhaps the book I would recommend to the motivated Y12 student or the interested Y13 student if they want some reading material. It covers various integration tricks (or 'dodges', as Thompson calls them), as well as some geometry (lengths of curves, and curvature) and gives reasonable intuitive justifications for the rules.

  Thompson is unsuitable for even first year university, because it is far from rigorous (it doesn't mention limits, although the recent editions have a foreword and initial chapters by Martin Gardner which do cover them to some extent) and is too informal to really be a good introduction to mathematical thinking.

  Indeed, my favourite introductory grown-up calculus book is, as you can probably guess, the One True Calculus Book:

• Michael Spivak, Calculus. Do I really need to say any more? Indeed, this book is more of an introductory real analysis book than a 'standard' calculus book. I don't believe any university in NZ actually uses it as a course text in first year (or at all), but that is no excuse for students not to use it (there always seems to be at least one copy out of the general library despite not being an official text).

  Highlights include a very readable motivation for completeness of the reals and 𝜀-𝛿 proofs; most exercises are also very interesting. I would avoid the complex analysis chapters at the end, but beyond that there are no real faults with this book. (The University of Toronto is one university that uses this book for their flagship first-year course, which has a reputation for being a trial-by-fire for new mathematics students.) One final bonus: it is cheap (well, not really, but cheaper than the shiny calculus books that the book shop sells) and concise (again, it is a doorstop, but much more concise than the texts that try to include absolutely everything from biology examples to Stoke's theorem). (Some comments) (515 S76)

Multi-variable calculus

• Peter Baxandall & Hans Liebeck, Vector Calculus. This book is very geometric, contains proofs in the usual "basic calculus" style, requires a minimal level of linear algebra, and covers Gauss' and Stokes' theorems after Green's theorem (useful for physics - and in fact it uses electromagnetics and fluid flow as motivational examples throughout). It even ends up stating the real Stokes' theorem, the one involving differential forms:- however, it does so in around twenty pages right at the end, only does it in three dimensions, and doesn't really motivate it that well. Oh well, at least it's there...! (It really is a very good book though, apart from that. Also, a Dover book - so very cheap.)
• Michael Spivak, Calculus on Manifolds or James Munkres, Analysis on Manifolds. Vector calculus with rigor, leading to differential forms. Spivak's Calculus doesn't include multi-variable calculus; the "standard" sequel to Spivak in this regard is Calculus on Manifolds by the same author. However, I find this concise introduction a little too concise (and that it introduces technicalities in all the wrong places). Munkres' Analysis on Manifolds is an expanded and updated book which is (anecdotally) based on Spivak; but I have never liked it that much either.
• Lynn Loomis and Shlomo Sternberg, Advanced Calculus. This book started off as notes for the legendary Math 55 at Harvard (a first year course!); it starts off with a recap of linear algebra (really one of the best `second courses in linear algebra' I've looked at), and ends up around 500 pages later with a very nice approach to differential forms. This is my favourite calculus text of all time, but it is probably a little too much for students in first year! The preface recommends Spivak's Calculus as prep, but I think it likely that a little more maturity is required that that for reading this book.

  Highlights include a rigorous (and clear) treatment of infinitesimal functions and differentials, and a final chapter on applications to theoretical physics. The exposition tries to explain a great deal of the 'philosophy of doing mathematics', which I quite like - see, for example, the chapter on uniformity and compactness.

Linear algebra

• David Poole, Linear Algebra: A modern introduction. This was the "recommended" textbook for the linear algebra portion of the standard UoA undergraduate series. It's fine, I guess. Lots of routine exercises. Very chatty. Lots of random applications that aren't covered in lectures; nice to take out of the library to flick through. Very geometric.
• Stewart Venit & Wayne Bishop, Elementary Linear Algebra. A shorter book, same content essentially. Includes section on linear programming (missing from Poole but vaguely interesting). Spends a very long time on computations.
• Gilbert Strang, Linear Algebra and its Applications. An incredibly computational book: I use this if I need an algorithm/computation notes. It's probably very good for compsci students, there's a lot of optimisation and efficiency stuff in here (my edition gives all the computer code in FORTRAN and some flavour of BASIC, but it's an old edition). Probably not recommended for mathematics students though.
• Paul Halmos, Finite-Dimensional Vector Spaces. My favourite linear algebra book! Like every other Paul Halmos book, it is incredibly well-written; just take care not to get bogged down by the sections on multi-linear algebra and determinants. The exercises are actually quite similar to those that you should be solving for anyway, just take note of his preface (some exercises are introduced before the material that makes them easy). Like Naive Set Theory below, the theorems themselves are the real exercises.

Logic, "proofs", and naive set theory

• Kevin Houston, How to Think Like a Mathematician. I'm philosophically opposed to a book that tries to teach "proofs" without teaching mathematics, but if I had to pick a book to do it it would be this one. It's very slow, very friendly, and incredibly fluffy. I photocopy some parts for high school students occasionally.
• Daniel Velleman, How to Prove It. Again, I don't like this kind of book in general; this book in particular I can't stand. Many people like it.
• Paul Halmos, Naive Set Theory. This is the set theory book for the casual reader. Not many exercises that are marked as such, but the proofs are often vague enough that you should be filling bits in on your own anyway. There is an exercises book lying around in the library somewhere if you really feel like you need it.
• L.E. Sigler, Exercises in Set Theory. In fact, this is that book of exercises. Might be worth a look but as I remember it's a bit dry and not actually that good of a companion to Halmos.
• Judith Roitman, Introduction to Modern Set Theory. Begins axiomatically, quite a bit harder than Halmos; I would have enjoyed it if I were into logicky semanticky stuff. I liked the style.
• Edmund Landau, Foundations of Analysis. Construction of the complex numbers and their properties from the Peano axioms. This book is a work of art. It makes excellent bathroom reading, despite what the Chicago bibliography would have you believe.

Elementary number theory

• Underwood Dudley, Elementary Number Theory. This is a cute little introduction to number theory; the first few chapters cover the material needed for 255, and the remainder are a mixture of topic chapters (on number representations, prime counting, and so on). As you might expect from this author, the book has a definite sense of humour. There is absolutely no algebra used at all, even if it would simplify any argument. This is good from the persepctive of 255.
• Kenneth Rosen, Elementary number theory and its applications; Joseph Silverman, A Friendly Introduction to Number Theory. Alternative viewpoints. Both have the advantage of being written by practicing number theorists; Silverman is supposed to be good for learning about some basic cryptography (i.e. in my copy of Jones and Jones on p.93 is handwritten in capital letters "CONSULT SILVERMAN FOR CRYPTO" and I guess it is this book rather than Diophantine Geometry!).

Differential equations

• George F. Simmons, Differential Equations with Applications and Historical Notes. I have a feeling that most differential equations books tend to be a list of one unmotivated trick after another; this is another such book, but it is well-written, includes some fairly nice examples, and some pretty geometry. (515.312 S59)
• Wolfgang Walter, Ordinary Differential Equations. A more modern book; quite geometric, quite physical but mathematically rigorous. The exercises are quite difficult.
• Vladimir Arnol'd, Ordinary Differential Equations. This book is a very nice treatment of the geometry of differential equations, especially those forms which one would expect to see in physical situations.

Geometry

• George Birkhoff and Ralf Beatley, Basic Geometry. This is perhaps the best high school geometry book I know of, although much of it is quite dated in style and culturally USA-centric.
• Serge Lang and Gene Murrow, Geometry: A High School Course. Wait, a Lang book for high school students?! (Please allow me to repeat the standard joke: the reason Bourbaki stopped writing new textbooks is that he realised Lang was just one person.) (516 L27)

Higher undergraduate books

Mathematical logic

• A. G. Hamilton, Logic for Mathematicians. This is the text for the logic paper I took in undergraduate, it failed to keep my attention but I think that's a feature of most logic books.
• H.-D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic. I slightly prefer this to Hamilton. Perhaps if I leave it under my pillow I will learn to like logic via osmosis.

Algebra

• M. A. Armstrong, Groups and Symmetry. A very geometric book. It's similar in philosophy to Jänich's topology book below, in that it's less rigorous than one would really like for third year. Essentially a lot of treatment of group actions.
• Nathan Carter, Visual Group Theory. This book is worth finding in the library for the pictures, which simply do not exist in most other books. However, the actual development of the theory is done much better in other books.
• Joseph Gallian, Contemporary Abstract Algebra. Lots of computational exercises. Some "topic" readings at the back (Galois theory, crystal point groups, etc), but not enough mathematics is developed to make them worthwhile (his stated goal is something like showing off algebra as a modern and up-to-date subject, but he doesn't do enough group and field theory to be able to introduce anything beyond a short historical sketch). I can't stand this book: he spends so much time doing computational examples that he hardly gets around to the algebra! Worth buying? No. (Also, it's disgustingly expensive.) The library has copies if you feel you need it.
• Charles Pinter, A Book of Abstract Algebra. Introductory book. Lots of interesting exercises covering both applications and pure mathematics: the proof of the structure theorem for abelian groups is left as a structured set of exercises, for example. Also very cheap.
• Michael Artin, Algebra. This is the best introductory algebra book I know of. Lots of geometry (e.g. he classifies all of the motions of the plane). Focuses heavily on linear algebra. Maybe a little light on the proofs (not that he's not rigorous, but make sure you do the exercises, or you'll miss out on stuff). Highlights include a chapter on Galois theory (first edition only) and a chapter on group representations.
• Ian Stewart, Galois Theory. Once you have an OK grounding in basic facts about groups, rings, and fields then this book is a readable introduction to Galois theory. Famously full of typos. Later editions (3rd and 4th) start by proving everything over \( \mathbb{C} \) before moving to arbitrary fields. Covers all the basic material (field extensions, proof of impossibility of general solutions to the quintic using radicals). Plenty of exercises, computational ones as well.
• Harold M. Edwards, Galois Theory. This is very classical. I have not read it in detail, though I found the first few chapters enlightening. I would like to find the time to read it at some point, but likely as part of a reading group.

Combinatorics and finite geometry

• Miklos Bona, A Walk Through Combinatorics. This is a very good introduction to combinatorics. Cute book, but a little pricey. Version in the library is an older edition without design theory. Perhaps a bit chatty and slow to get started, but does cover a lot of random topics that should be of interest both for computer science and mathematics.
• Bela Bollobas, Modern Graph Theory; Reinhard Diestel, Graph Theory. Very nice graph theory books. I prefer Bollobas, but Diestel is more modern. Overall I do think that these books are in more in the Erdős-esque camp of mathematical exposition. (Of course Bollobas was also a Hungarian prodigy, so that makes some sense I suppose.)
• W.D. Wallis, Combinatorial Designs. This book is a very nice introduction to finite geometries and other combinatorial designs. Very readable, a little old.
• Matthias Beck and Sinai Robins with David Austin, Computing the Continuous Discretely. A very readable introduction to polytope combinatorics for undergraduates. Repackages a lot of the stuff in Stanley's book on combinatorial commutative algebra and in Ziegler's book in a form digestable with minimal algebra. Very nice little UTM.
• Branko Grünbaum and G. C. Sheppard, Tilings & Patterns. Introduction to the combinatorics of symmetries with a very low bar to entry; lots of very nice pictures. I would compare it with Coxeter's book on convex polytopes (see below in the section on polytopes), if you like one then you'll like the other. The open problems are a bit outdated.
• Branko Grünbaum, Configurations of points and lines. It seems to be a trend in geometry that good authors write many many books. This is another book on combinatorial geometry, this time about intersections of lines in the plane.

Number theory

• Gareth A. Jones and J. Mary Jones, Elementary Number Theory. This is a step up from Dudley (see in the elementary number theory section above). There are some very interesting problems in here, but the typesetting really annoys me. Don't let the easy nature of the first few chapters fool you. For chapter 5, our lecturer gave all the proofs via ring theory; if you know some algebra, write down these proofs yourself (they follow from the fact that a cyclic ring can be decomposed into a product of cyclic rings of prime power order; see, for example, Ireland & Rosen) and compare with the ones in here. (512.7 J77)
• Kenneth Ireland and Michael Rosen, A Classical Introduction to Modern Number Theory. This is the classic modern number theory book. It does start a little slowly, but it eventually gets to elliptic curves and even proves the Mordell-Weil theorem over \(\mathbb{Q}\). It is in this section and not the graduate section below because of how self-contained it is: the bare minimum of algebra is needed (basic knowledge of groups, rings, and fields at, say, the level of Pinter).
• Fernando Q. Gouvêa, \(p\)-adic Numbers. This book focuses on the metric structure of the \(p\)-adic numbers; it mainly develops the basic theory for its own intrinsic interest. Unfortunately spends a lot of time trying to avoid doing any commutative algebra, and I found it very slow.

Single-variable analysis

• Kenneth R. Davidson and Allan P. Donsig, Real Analysis and Applications. For an introduction to real analysis for someone who didn't learn calculus from Spivak or a comparable book this might be a better choice than Rudin, I find it incredibly readable, it has a very nice tone, plenty of interesting exercises and examples, and the bits which depend on linear algebra are done a lot better than Rudin (but in less generality). I also like the second half (on applications).

  Let me compare it directly with Rudin: Rudin is shorter, denser, and deeper; D&D is longer, friendlier, more detailed, and covers slightly less material.

  One complaint: there is no explicit construction of the real numbers. (They try to explain a construction via decimal expansions in a very handwavy way; there is also almost an outline of a construction via Cauchy sequences in exercise 2.8.L; I know that Tao does this same construction in the same kind of detail as the rest of D&D, so it might be an idea to look there for this material. Or in Rudin, or Landau.)

  Like Jones & Jones in the number theory section, this is another new Springer book with really annoying typesetting, but to a lesser degree. (*)

• Walter Rudin, Principles of Mathematical Analysis. The first eight chapters are an incredibly clean and terse exposition of metric space theory. People I know have called it dry, but I didn't actually find it that bad - it is a little lacking in motivation, so maybe it would be a good idea to pick up a shiny book like Davidson & Donsig above as well for historical details and more examples. (I personally used a mixture of the two when I learned real analysis.)

  I prefer Loomis & Sternberg for multivariable analysis. (I did try to work through chapter 9, and despite knowing what he was trying to do at every point it's incredibly clunkly and awful. Belive me, I thought the Chicago bibliography would be exaggerating on this point, but it turns out that they were entirely correct.) (Errata and notes)

• A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis. This is not an introductory textbook; it is a high-level geometric (in the Russian sense) view of real analysis (in particular, metric space theory). There is plenty of topology, some nice differential equations, and a lot of very pretty mathematics.

Topology

Point-set topology.

• Klaus Jänich, Topology. An easy introduction to topology which isn't rigorous enough for a third- or fourth-year course (and probably not appropriate for anyone that's actually done any proper analysis, it's very handwavy), but it does have some very nice pictures. I liked it in first year when I wanted to learn more about topology.
• James R. Munkres, Topology. This is the standard topology book, it's very clear with lots of examples and pictures. I didn't like the section on metric spaces, though - use Rudin here, it's a good supplement. Kolmogorov & Fomin is also nice to read alongside.

  Having thought about it for a while, I think it's incredibly likely that this is the textbook I have spent the most time with over the past year:- it was used in the topology course I took in 2018, and I must have spent at least an hour a day with it, doing readings or problems. Unfortunately the algebraic topology part is not so good, it is very short and does not do the subject justice (it is overly technical, I think).

  (Errata)

Geometric topology.

• Jeffrey R. Weeks, The Shape of Space. This is supposed to "fill the gap" between the simple geometric examples of topology (Klein bottles, tori) and the "sophisticated mathematics of upper-level courses". I don't think it does this job very well, but it is a very nice book of geometry that might even be accessible to the layperson. What I would like to see is a book like this that makes explicit the actual links with point-set topology.

  I wrote the previous review in 2018, here is an update for 2021: Of coure what Weeks meant (and I didn't understand at the time) was to link algebraic topology and not point-set topology to the explicit geometry of Thurston etc. I can even name a book which does this with more sophistication (Weeks is really a book for an interested first year student): Bredon's Geometry and Topology which I talk about below.

• David Mumford, Caroline Series, and David Wright, Indra's Pearls: The vision of Felix Klein. Another highly visual book, this time about Schottky groups (the easiest of all the Kleinian groups) and their fractal limit sets. Ends up discussing the Maskit slice (the moduli space of punctured torus groups). A lot of computer experiments too (the website for the book has some more stuff, if you dig through the Wayback Machine there is also an archive of FORTRAN-77 code which took a little work to get going on a modern compiler like gfortran). See also the lecture The Suggestive Power of Pictures (Caroline Series).
• Francis Bonahon, Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots. This is a rigorous geometric topology book for undergraduates: the basic theory of Kleinian groups and hyperbolic surfaces, culminating in a short discussion of 3-geometry. Requires very little technical background, it even develops the needed group theory from scratch.
• Richard Evan Schwartz, Mostly Surfaces. Another surfaces book for undergraduates assuming very very little technical background, this time heading towards Teichmüller theory.
• George K. Francis, A Topological Picturebook. How to draw topological pictures by hand: knot theory, mapping class groups, sphere inversions, etc. I appreciated this book when I was an undergraduate.

Complex calculus and analysis

The first few books here, broadly speaking, are complex calculus books; the latter books are complex analysis books.

• Stephen D. Fisher, Complex Variables. The main problem with this book is that it's trying to be both a rigorous book and a calculus book and failing miserably at both. I don't hate the sections on the geometry of analytic functions (chapter 3), but chapters 1 and 2 attempt to build the basic theory of analytic and holomorphic functions without using phrases like 'uniformly convergent'. The main mechanism used for this is Green's Theorem (for double integrals), and I found it unsatisfying and insufficient. Further, a non-negligible number of the problems seem to be pitched at a higher level of rigour than the proofs in the text (either that, or one must hand-wave away enough topology that they become trivial). There are a minimum of examples.
• E.B. Saff and A.D. Snider, Fundamentals of Complex Analysis for Mathematics, Sci...etc. Basically a calculus book, so if you like Stewart or Anton you'll probably like this. I don't hate it, and I found it useful for complex calculus, if a little dry. The problems are much better than Fisher (in that they fit the style of the book). More examples than Fisher. It's a bit expensive, because it's a calculus book...
• Tristan Needham, Visual Complex Analysis. This book attempts to construct complex analysis purely geometrically. It is quite successful in building intuition, but there are no proofs at all and a number of theorems are only (strongly) hinted at rather than stated - even important ones like Cauchy's theorem. It has the great advantage of doing an awful lot of geometry.

  In 2021 Needham published a second book like this, Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts - I haven't had a chance to look at it yet, but judging by VCA it should be well worth looking at.

• Elias Wegert, Visual Complex Functions. Another visual book, of a different style to Needham. This book develops the theory with proper proofs, and has a much more analytic flavour. The tool of choice here is the 'graph' (in some sense) of the complex function; this is in contrast to Needham's approach via differentiation.
• Joseph L. Taylor, Complex Variables. This is a new and shiny (2011) book that is roughly at the level of a first course. It's OK.
• John Conway, Functions of One Complex Variable I. I found this book to be readable, but very slow.
• Lars Ahlfors, Complex Analysis. This is a little old-fashioned, but very geometric. My main issue is that everything is done by partial differentiation and differential forms; there is also not a lot of emphasis on series representations (analyticity) in the first half of the book.

  I wrote the previous review in 2018, here is an update for 2021: (1) To add to what I said, because of the reliance on differential forms you probably should have had a read of one of the multivariable calculus books like Spivak first. (2) What I said was an issue is now (I think) a good thing: the language of differential forms is a very pleasant way to phrase complex analysis. The simplest manifestation of this is, of course, the statement of the Cauchy-Riemann equations in this language: they just become \( \frac{df}{d\overline{z}} = 0 \). In any case this is now my preferred treatment of geometric complex analysis (but I still prefer Rudin overall, it's just such a nice book). I should of course say that the downside of the differential form language is that it is much less accessible to the beginner, and this is my excuse for not being so positive three years ago - I also remember taking a complex calculus (not analysis) course in 2018 where the lecturer very clearly would much rather have been teaching a course for mathematics majors instead (I was in it by mistake, I misunderstood the course structure) and spent most of the time going off on tangents including one about how the "derivative with respect to \(\overline{z}\)" language (more properly called the Wirtinger derivative --- yes, the same Wilhelm Wirtinger as the knot group presentation). I really disliked the course (the second most boring mathematics course I have ever taken) and so I really disliked this language. Here ends the story.

• Walter Rudin, Real and Complex Analysis. From chapter 10, the best book on complex analysis I have found. It is possible to read the complex analysis half, broadly speaking, without having read the real analysis half; one just needs to replace all the integration theorems with less general theorems involving the Riemann integral.
• Umberto Bottazzini and Jeremy Gray, Hidden harmony — Geometric fantasies. The rise of complex function theory. And now for something completely different: a very nice historical treatise on complex analysis which I have dipped into from time to time.

Classical geometry

• Robin Hartshorne, Geometry: Euclid and Beyond. Hartshorne follows Euclid, putting him on a firm axiomatic foundation. The capstone of the book is a discussion of polyhedra. You should have a little algebra for the last stretch, but the rest should actually be accessible to a high-school student!
• John M. Lee, Axiomatic Geometry. This book is like Hartshorne but in a much more formal style; it's written for future teachers, but it develops Euclidean geometry (and only Euclidean geometry, although there is some hyperbolic stuff at the end) in a synthetic and formal way. I secretly quite like this book, but I will never admit it. This is the book I use to look things up if I feel the urge to know how (for example) the inscribed angle theorem is proved. For some reason I prefer it to Hartshorne's similar book actually.
• Alfred S. Posamentier and Charles S. Salkind, Challenging Problems in Geometry. Heard of Ceva's theorem and want to know more? Want to learn some geometry? Buy this (cheap) book and do all the problems. All are very good. Difficulty from school level upwards.
• Marta Sved, Journey into Geometries. This is a very strange book which motivates non-Euclidean geometry (hyperbolic geometry in particular) with the characters of Alice in Wonderland. It's well worth the read.
• Harold Coxeter, Introduction to Geometry and Geometry Revisited. These books are both full of a lot of nice university geometry with some quite nice links to other subjects: for example, crystallography and biology. Revisited doesn't require any more than basic school geometry, but Introduction is quite sophisticated and difficult to read cover-to-cover as he uses a wide variety of techniques from different fields (group theory, linear algebra, calculus) without any warning. It is very easy to expand a lot of what Coxeter says in half a page into hours of fruitful messing about with a blackboard thinking. (I have used part of Intro. as a basis for a set of NCEA level 3 geometry notes, and managed to expand three pages of Coxeter into around thirty pages of notes.) The discussion of differential geometry in Intro. is not so good, but that is just a small section - Coxeter was not a differential geometer, and he basically does differential geometry in the style of "calculus applied to geometry" (like do Carmo which I discuss below). It is still my favourite broad introduction to geometry.

  See also the entry on Escher under culture above.

• Harold Coxeter, Projective Geometry. A quite axiomatic view of projective geometry; not my favourite book by Coxeter, but still a nice read. More comprehensive than the section of Introduction on projective geometry, but not as comprehensive as the chapters in Berger. Solid book.
• Marcel Berger, Geometry. But this is my all-time favourite geometry book. It is a little long (two volumes), but you can dip in and out of it rather than reading linearly. There are many nice pictures, including of physical models; in addition, almost every geometric theorem I've ever seen in any paper is included here (although often under a different name). There are a lot of very pretty results in both affine and projective geometry. Linear algebra and group theory are used extensively (of course). The only things lacking, really, are the more combinatorial parts of the subject (things like polytope theory); also, the section on elliptic and hyperbolic geometry is a little short. Really this is a book on classical geometry, and you need to look elsewhere for curved geometries --- for instance, in Berger's more recent A Panoramic View of Riemannian Geometry (which is at a higher level, of course).
• Marcel Berger, Geometry Revealed. A more modern book by Berger, this one is focused more on convexity and combinatorial geometry. I have not spent too much time with it (since I have not had access to a paper copy since I was in Toronto - I am less and less comfortable giving money to Springer directly and all the second hand copies cost far too much money) but it looks really good.
• Tom Apostol and Mamikon Mnatsakanian, New Horizons in Geometry. I would describe this as `Archimedean' geometry: calculus applied to problems in classical geometry (isoparimetric problems, lines sweeping out curves, liquids flowing out of containers...). Very very nice book which I enjoyed in first year (and still enjoy now). A bit expensive but overall a well-made book (worth the price).

Algebraic geometry for undergraduates

• Miles Reid, Undergraduate algebraic geometry. I find this book overly informal and chatty to the point of annoyance, but it is a very easy read if you like that kind of book.
• David Cox, John Little, and Donal O'Shea, Ideals, Varieties, and Algorithms. A very popular book. I don't like it so much but it's just a matter of taste. It treats the subject too much as an extension of linear algebra for my liking (which is a fair enough way of doing things), really my issue is that there is very little of the kind of geometry which I like (and very little application of algebraic geometry to geometry at all - which makes sense given the authors, who are interested in problems like solving systems of equations).
• Audun Holme, A Royal Road to Algebraic Geometry. This is a very gentle introduction to the subject that is very well-motivated if you have some experience with projective geometry and classical algebraic geometry (c.f. Kendig's Conics below). Unfortunately the second half seems to be completely unedited and is full of typos and mistakes which make it almost unreadable if you are trying to learn the subject.
• Joe Harris, Algebraic Geometry: A first course. This is my favourite introductory book. I think there is a good range of content and it has the advantage too of doing a lot of geometry (compared to Reid or the book by Cox/Little/O'Shea). It includes a very readable proof of Bezout's theorem (chapter 18), a vast number of interesting examples of varieties contructed in ways other than just taking a random polynomial (he constructs Grassmannians, scrolls, flag manifolds, determinantal varieties, various group varieties,...), and spends some time motivating various constructions rather than just stating them. On the other hand it is a bit light on some proofs but I think that this is a tradeoff that has to be made in writing an introduction to this subject given how abstract it can be. Of course he also assumes everything is algebraically closed, I think the alternative requires a lot of attention to technical detail and is best left to scheme theory.
• Igor Shafarevich, Basic Algebraic Geometry I. A Russian algebraic geometry book, also one of my favourites. It is more comprehensive than Harris (e.g. there is a proof of the Riemann-Roch theorem for curves, a more careful study of intersection numbers, and lots of stuff on differential forms) and I tend to oscillate between which one I prefer. On the other hand, it has the feeling of a book on algebraic geometry written by an analyst (I am not sure why, since Shafarevich was an arithmetic geometer!) and is very very classical, unlike Harris which develops more rigid intuition (I know this is very vague and I apologise) and the result is that I think Harris is a better book for budding pure algebraic geometers while Shafarevich is better for those who want to apply algebraic geometry to other parts of mathematics (beyond number theory, of course, which is more rigid). This kind of explains why I don't like volume 2 (on schemes) so much --- I write about this below in the section on scheme theory.
• Keith Kendig, Conics. This book is everything you always suspected about conic sections and it is amazing. Please read it.
• George Salmon, A Treatise on Conic Sections. If you read Kendig and want to know more, this book (from the 1800s) is very good but a little difficult to read due to its age. Serving recommendation: read and try to make Salmon's statements precise with projective geometry and/or complex algebra.

Differential geometry.

• Manfred do Carmo, Differential Geometry of Curves and Surfaces. Differential geometry, without any more machinery than multivariable calculus. Quite dry, very computational and really very little geometry. My complaints about Fisher's book on complex calculus above apply to this book as well: it is written in the style of a calculus book trying to be rigorous and in the process being bad at both.

Postgraduate texts

Algebra

General algebra.

• Joseph Rotman, Advanced Modern Algebra. This book is supposed to be very good, but something about it annoys me. I'm not too sure what the problem is: maybe it's a little too chatty, and it motivates everything (at least initially) with number theory? I will come back to it at some point, maybe I will like it better.
• Serge Lang, Algebra. I like the ability to open up at a random page and begin to work without needing to read the previous hundred pages; I dislike the confusing text, lack of detail, incredibly imprecise cros-references ("this was proved in chapter 6" with no further details) and the inconsistent level of difficulty. Before Aluffi, though, this was really the only properly comprehensive algebra book.

  Perhaps here is the best place to link to the famous review of a Lang book by Mordell (for completeness, here is Lang's subsequent review of Mordell's book, and an article by Lang in DMV Mitteilungen).

• Paulo Aluffi, Algebra: Chapter 0. This is the book which has replaced Lang as my reference book for basic facts in algebra. Instead of expecting the reader to learn category theory by osmosis (like Lang does), Aluffi introduces it in a very practical way; the exercises are excellent, the book is very well-written. Highly highly recommended. Lots of connections given to various areas of mathematics (e.g. some basic algebraic geometry; some number theory; some topology).

Group theory.

• Joseph J. Rotman, An Introduction to the Theory of Groups. I have always found group theory a little dry, this book is good but it didn't really convince me otherwise, and this is probably my own fault rather than that of the book.
• John D. Dixon, Problems in Group Theory. A problem book in elementary group theory (i.e. the level of Rotman). Again, I am not a group theorist so you might like it more than me.

Category theory

• William Lawvere and Stephen Schanuel, Conceptual Mathematics. The ambiguous title hides that this is a book on category theory, written primarily to appeal to undergraduates with limited experience with the traditional motivating topics like algebraic topology. I remember trying to read it a couple of years ago and being annoyed by the lack of examples (the main theatre of examples used is the study of diagrams themselves); but if you can come up with your own examples then it is likely a nice friendly introduction. For this reason it would be nice to have this book as an alternative, but not as a replacement, for something like Mac Lane.
• Horst Herrlich and George E. Strecker, Category Theory. This book is very well-motivated with lots of examples. More for computer science though.
• Jiří Adámek; Horst Herrlich; and George E Strecker, Abstract and Concrete Categories. This book is also well-motivated but in a different sense. The illustrations are amusing and the book has a sense of humour. Now a Dover book.
• Saunders Mac Lane, Categories for the Working Mathematician. This is the standard introductory text. You really need a good sense of the fields where he draws examples from (algebraic topology mainly), and it is a little dry. I would recommend Aluffi as an introductory text nowadays if you don't have this background.
• Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic. Applications of pure category theory to logic, algebraic topology, and algebraic geometry. Very readable book, surprisingly so given how abstract this stuff can get.
• Denis-Charles Cisinski, Higher Categories and Homotopical Algebra. When you finish Mac Lane/Moerdijk, you can read this: an introduction to \(\infty\)-categories. I must confess that this is the point at which I feel the abstraction beginning to overcome me.
• Jacob Lurie, Higher Topos Theory (and other such books). But wait, there's more. Just as the Chicago bibliography draws the arbitrary line for "difficulty" at Hartshorne ("difficulty level unbounded above..."), I draw the line here. I have come past the point where Hartshorne looks legendarily difficult, and now it is this set of books by Lurie which are the ones that, to me, feel legendarily abstract.

Linear algebra.

• W. H. Greub, Linear Algebra and Multilinear Algebra. Part of me is unsure whether you can really put a section on linear algebra in a list of postgraduate books (and surely such a section would be called "commutative algebra"!), but this book talks about universal properties and such things so I would say it is more at the level of Aluffi's treatment of linear algebra than Poole's (even if the content is similar in the linear algebra part, culminating in such exotic things as unitary maps and quadrics). Actually, looking again now it has some graded spaces and homology so I should stop complaining!

  I want to mention this here because I remember being in the Gerstein library looking at books and I picked this one up and was so utterly confused by the definition of the tensor product via the universal property. (For the longest time I thought that the book I remembered was a book by Lang on linear algebra, but I just looked at a PDF of the third edition of his UTM and he seems not to mention tensor products in there. Greub is my second guess as to what book it is, but perhaps it was an older edition of Lang. Or some other book - I definitely remember it was a Springer book, though.)

  In any case this is a nice reference. How does the Chicago bibliography put it... "One day, you may just have to know fifteen different ways to decompose a linear map into parts with different nice properties. On that day, your choices are Greub and Bourbaki. Greub is easier to carry." I don't think I can improve on that. On the other hand, nowadays I would pick up Aluffi first before I tried Greub - Aluffi is just an all-round good reference for this kind of thing.

• While I am here I might as well mention Steven Roman, Advanced Linear Algebra. Primarily because it has a chapter on the umbral calculus. (Otherwise it is not as comprehensive as Greub.)

Commutative algebra.

• M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra. A little book with everything you will ever need to know about commutative algebra if you are applying it to other subjects, and enough to show the interest of the subject. The exercises are very good and contain many applications, including a basic introduction to scheme theory (which was really new at the time of writing). Errata
• David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. This is my favourite mathematics book of all time. Not an introductory text (see Atiyah-Macdonald for that), but a book about applying commutative algebra to geometry and number theory. Particular highlights include the chapters on filtrations, Hensel's lemma, flat families, and dimension theory. The exercises are also very interesting: they tend to be applications, generalisations, or intuition-building. For instance, the Cayley-Bacharach theorem appears as an exercise, as do many different forms of the Nullstellensatz.
• Richard P. Stanley, Combinatorics and Commutative Algebra; Ezra Miller and Bernd Sturmfels, Combinatorial Commutative Algebra. Applications of combinatorics to commutative algebra, and vice versa. Primarily the combinatorics involved is that of polytopes and more general complexes. As products of the theory you get things like counting magic squares, Ehrhart reciprocity (which counts the interior points of a lattice polytope), and McMullen's \(g\)-theorem (characterising the possible face numbers of simplicial polytopes). Very much like the book of Stanley, it has a lot of nice geometric motivation, the book of Miller and Sturmfels is not so exciting and is less geometric (it is more interested in answering purely algebraic problems and has less motivation).
• M. Scott Osborne, Basic Homological Algebra. Very fast introduction to homological algebra and ext and tor (for an even faster one, see the appendix to Eisenbud). Very little motivation and no applications, but if you need to learn homological algebra by tomorrow this is the book for you. On the other hand this is yet another topic done incredibly well by Aluffi.

Rings and group representations.

• T.Y. Lam, A First Course in Noncommutative Rings and Lectures on Modules and Rings. These books taught me to love noncommutative ring theory. Excellent books, very nicely written. I still prefer commutative algebra, though.
• J.-P. Serre, Representations of Finite Groups. Fast and clean introduction to representation theory. Not my favourite book by Serre. I think the problem is that in his exposition the subject seems quite shallow.
• William Fulton and Joe Harris, Representation Theory: A First Course. I usually like Joe Harris' books, but this one did not click with me. I think it is just my dislike of pure group theory and lack of real knowledge about Lie groups. Perhaps I will come back to it at some point and really enjoy it, I wouldn't be surprised.

  Indeed yes, I came back to it and did enjoy it. This is actually a very very nice introduction to Lie groups as well! (2022)

Number theory

• J.-P. Serre, A Course in Arithmetic. Two halves, the first half is algebraic and studies the classification of quadratic forms over \(\mathbb{Q}\) (the Hasse-Minkowski theorem), in the process he studies very basic local class field theory. Very short and clean. Second half is analytic and deals with arithmetic progressions and modular forms - I haven't looked at those parts in detail.
• Daniel Coray, Notes on Geometry and Arithmetic. An introduction to arithmetic geometry for people who've not seen algebraic geometry before. A very nice introduction to things like the \(p\)-adic numbers, the Hasse principle, and diophantine problems.
• Z.I. Borevich and Igor R. Shafarevich, Number Theory. Yet another first introduction to local class field theory and the Hasse-Minkowski theorem. I am more inclined towards geometry so prefer this to Serre. In the Russian style, if that's your thing.
• Toshitsune Miyake, Modular Forms. This book does very little to motivate modular forms (see this post on MO for that); they are essentially functions on \( \mathbb{H}^2 \) preserved under arithmetic symmetries. This is a fairly thick book of basic results and applications to arithmetic number theory.
• Pierre Guillot, A Gentle Course in Local Class Field Theory. Yet another introduction to local class field theory, this time even more gentle. For such a friendly book there is a lack of exercises, but there is plenty of motivation.
• J.-P. Serre, Local Fields. Serre clearly sat down and thought to himself, "all of these students learning local class field theory have a severe lack of introductory books which are not gentle and friendly"; this is the result. Actually it is not so bad (and really it is not supposed to be an introductory book, so I am being unfair). Anyway, it is another book about local class field theory.
• Jürgen Neukirch, Algebraic Number Theory. Not only an introduction to local class field theory: this book includes the global theory too! Assumes only a little algebra and proves the main local-to-global results. Very geometric, too, which is nice.
• John Voight, Quaternion Algebras. Open access! A nice friendly introduction to quaternion algebras over local and global fields, as well as the associated arithmetic group theory and 3-geometry, assuming even that you know none of those words beforehand. Prerequisites are probably just basic group theory & number theory, but a lot of breadth is covered and so the reader should be prepared. Every postgraduate student in algebra seems to have a copy now. (800 pages, not too dense though, and quite readable - can dip in and out). See also Maclachlan-Reid below in geometric group theory for a harder book on similar subjects.

Combinatorics

General combinatorics.

• Martin Aigner, Combinatorial Theory; Jack Graver and Mark Watkins, Combinatorics with emphasis on the theory of graphs. These books are more algebraic in style than combinatoric, developing a more abstract theory and then applying it to combinatorial problems. Aigner studies mainly enumerative and order-theoretic combinatorics, and Graver/Watkinds studies mainly design-theoretic combinatorics, so together they kind of cover the whole field.
• Richard P. Stanley, Enumerative Combinatorics (2 volumes). A very good reference on counting problems and the related techniques. The exercises are excellent introductions to the applications (to representation theory, number theory, geometry, chess,...), worth reading in and of themselves. I must also mention the following rather odd fact which was noted on the Facebook page Graduate Texts in Memes: there is a Facebook profile under the name Richard Stanley which seems to spend most of the time posting pictures of small children reading this book. (I will not link it here because I feel uncomfortable linking what could be someone's private account.)

Polytopes. See also the books by Stanley and Miller/Sturmfels in the commutative algebra section above.

• H.S.M. Coxeter, Regular Polytopes. Another book by Coxeter that you need to read right now. Very classical introduction to polytopes, various constructions, and their symmetries.
• Günter M. Ziegler, Lectures on Polytopes. A much more modern book on the algebraic theory of polytopes. Lots of nice applications of the theory, and very interesting historical notes. This is the book you want to go along with the books on toric geometry below. (While I am thinking about it, you should also look at the book by Ewald below in that section for another introduction to convex polytopes that I like.) Highly recommended.
• Branko Grünbaum, Convex Polytopes. This is an incredibly comprehensive tome. The latest addition is annotated by Volker Kaibel, Victor Klee, and Günter M. Ziegler, and includes copious historical notes. I should really own a copy, I don't know why I haven't bought it yet.

  An interesting blog post.

• Alexander Barvinko, A Course in Convexity. An alternative introduction to convex polytopes with a different point of view, still worth looking at.

Algebraic topology

• Alan Hatcher, Algebraic Topology. Free online. Many people like this book, but it is not to my taste (I think it is overly verbose). It is incredibly long but doesn't cover much (any?) more material than other books: the extra space is given over to many pictures and a lot of exposition.
• Glen E. Bredon, Topology and Geometry. This is my favourite algebraic topology book, but I don't like the notation that much. More geometric than purely topological; I really like the choices of topics. I think that Munkres' Topology plus some (rigorous) multivariable calculus (probably Spivak or Munkres) is the best preparation for this book.
• John Lee, Introduction to Topological Manifolds. If you like Lee's writing style, this is probably the book for you. It is a little too differential-geometric rather than purely geometric for my liking, but the material on covering spaces is very clear.
• John Stillwell, Classical Topology and Combinatorial Group Theory. Essentially an introduction to homotopy theory from a combinatorial point of view rather than a topological one. If you are more interested in heading towards knot theory and low-dimensional geometry and topology, this might be the book for you. Also talks about related computational results (e.g. the word problem in groups).
• Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology. A very algebraic second course in algebraic topology, a lot of cohomology theory. Particularly algebraic in style. (Best preparation for Hartshorne's chapter III, I think). Particularly nice is the lengthy motivating discussion, which is missing from many other textbooks.
• Robert Ghrist, Elementary Applied Topology. A basic introduction to algebraic topology with applications to graph theory, computation theory, statistics, splines, and other fields. Put this in the category with Needham's Visual Complex Analysis.
• George W. Whitehead, Elements of homotopy theory; Jeffrey Strom, Modern classical homotopy theory. I now mention two harder books, both on homotopy theory. This is another subject which I wish I knew more about; the book by Whitehead is a massive doorstop of a GTM (a quote from the preface promises "a leisurely exposition in which brevity and perhaps elegance are sacrificed"), and the book by Strom is a massive GSM which is more modern but on the other hand seems to leave every second proof as an exercise to the reader (and the preface, I seem to recall, claims that this was done by design). Anyway, homotopy theory is a very interesting subject and I prefer Whitehead despite the age - but of course I am not a specialist.

Differential and Riemannian geometry

• John Lee, Introduction to Smooth Manifolds. 700 pages of pure differential geometry for the most gluttonous graduate student. Essentially an encyclopaedia. Even the pure differential geometers in the office agree: this is not a book you read, it is a book you look things up in if you can't find them elsewhere! Various people recommend Tu's book below, or the first two volumes of Spivak's Comprehensive Introduction, or (of course) anything by Milnor. My opinion is, this book is probably OK for a book to use during a course or some other situation where you have someone who knows the subject well telling you what is and what isn't important, otherwise you will spend a year or two reading it and only then get to doing some interesting modern geometry.
• John Lee, Introduction to Riemannian Manifolds. This one is only about 400 pages long, but just as dense as Smooth. Curvature is of course the main concept here, covered in much detail. The notation annoys me.
• Loring W. Tu, Differential Geometry. A comparably more friendly book on curvature and Riemannian geometry. The organisation annoys me a bit (it feels like slightly edited lecture notes rather than a book - really that should not be a problem and I like plenty of other books which come from lecture notes, I am not too sure what the real problem is here) but I suppose it will appeal to a lot of people. Overall I tend to prefer the books by Lee to the books by Tu (except for Bott and Tu versus Lee's Topological).
• Marcel Berger, A Panoramic View of Riemannian Geometry. I like Berger's writing, this book is very very good; it is not a textbook, it is a guidebook to the theory, like what the book by Holme below on algebraic geometry was supposed to be. Unfortunate that it's published by Springer, I would like to own a quality hardcover but they don't do that kind of thing any more.
• Peter Petersen, Riemannian Geometry. Now into the third edition (but I have only looked at the first). I find this easier to read than Lee's Riemannian book; I would say it is closer to the writing style of Bredon's Geometry and Topology (see algebraic topology above). Unfortunately still uses the Einstein summation convention...

Geometric group theory

General books

• Martin R. Bridson and Andre Haefliger, Metric Spaces of Non-positive Curvature. General theory of geometric group actions in both the archimedean and non-archimedean case. A lot of applicability to hyperbolic geometry, and chapter \( III.\mathcal{G} \) contains a very nice exposition of the covering space theory of orbifolds (I prefer the discussion here to that in Ratcliffe, and it is a complementary approach to that in chapter 13 of Thurston).
• James E. Humphreys, Linear Algebraic Groups. The study of geometric groups (things like \(\mathrm{GL}(n,k)\)) as geometric objects in their own right. Applications also to representation theory.
• William Waterhouse, Introduction to affine group schemes. Introduction to SGA-style algebraic groups; contains a nice introduction to the functorial approach to scheme theory, which is a nice motivation for things like Yoneda's Lemma. He says that "there is no prerequisite beyond a training in algebra including tensor products and Galois theory," but clearly "algebra" here means graduate (i.e. Lang-level) algebra. I think that a decent amount of algebraic geometry is also a prerequisite to really getting what is going on; this is the advantage of Humphreys' book (which essentially develops the theory of varieties in-house, as it were). On the other hand this is a very nice introduction once you have got your head around the first couple of chapters of Hartshorne. Another book I should really own but don't.

Arithmetic flavour (e.g. buildings, \(p\)-adic fields).

• J.-P. Serre, Trees. This is an introduction to the study of actions of free groups. You need to know a bit about the \(p\)-adics: the main goal is Ihara's theorem (every torsion-free subgroup of \(\mathrm{SL}(2,\mathbb{Q}_p)\) is a free group). This book will then lead you to the very basics of the theory of Bruhat-Tits buildings, which I might talk about below (I don't know yet). Essentially the idea is to look at groups which act particularly on trees in such a way as to preserve the combinatorics.
• Peter Abramenko and Kenneth S. Brown, Buildings. Indeed yes I will talk about buildings. Buildings are to geometric groups as schemes are to commutative rings (I suppose). Very attractive book which motivates the subject incredibly well, it makes me wish I had time to learn more of the theory in greater detail.

  While here, I might as well mention the very nice expository paper A (very short) introduction to buildings by Brett Everitt, and the Buildings Gallery (Bram Bekker).

Geometric flavour (Kleinian groups and hyperbolic manifolds). See also the section on low-dimensional geometry below.

• Alan F. Beardon, The Geometry of Discrete Groups. An elementary introduction to Fuchsian groups (discrete subgroups of \(\mathrm{PSL}(2,\mathbb{R}) \)) and their related complex and hyperbolic structures. Minimal topology needed. It doesn't really do enough interesting geometry for me, but it is a good start. The introduction to hyperbolic geometry here is mainly useful as a reference for computational formulae.
• Bernard Maskit, Kleinian Groups. Now this is more like it. Much faster introduction and now more generally about Kleinian groups (discrete subgroups of \(\mathrm{PSL}(2,\mathbb{C}) \)), with lots of geometry to think about. Once you get the hang of the basics you can get away with skipping around a bit to see the highlights: for instance, the various geometric constructions in Chapter VIII. A very classical viewpoint throughout: for a book to read after this, see Matsuzaki-Taniguchi below.
• Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyperbolic manifolds and Kleinian groups. Even more geometric than Maskit, and now post-Thurston (so for instance there is some material on Dehn surgery). There are a lot of very nice pictures - c.f. my complaints about the lack of pictures in Ratcliffe, and compare the pictures of the figure 8 knot tetrahedron gluing in Ratcliffe (pp.448-449) with that in M-T (p.34). This seems to be the only book which presents the quasiconformal deformation theory of Kleinian groups in detail - otherwise you need to try to read the series of papers Quasiconformal Homeomorphisms and Dynamics I-III by Dennis Sullivan (part III coauthored with Curtis McMullen), or the lecture notes Deformation Spaces by Kra (chapter 4 of A Crash Course on Kleinian Groups, Springer LNM 400); and both of these are written for people already in the subject. Unfortunately this book is very hard to come by without spending vast sums of money.
• Michael Kapovich, Hyperbolic Manifolds and Discrete Groups. Another Thurston-esque book which I would put more on the group theory side rather than the geometry side, this book ends with a proof of Thurston's hyperbolisation theory for Haken manifolds via a lot of interesting geometric group theory (moduli spaces of groups, measured laminations, the Bass-Serre theory of trees,...).
• Colin Maclachlan and Alan W. Reid, The Arithmetic of Hyperbolic 3-manifolds. Of a slightly different flavour, this book is about Kleinian groups which happen to be arithmetic. Very dense, quite hard. You should already know local class field theory and some hyperbolic 3-geometry (e.g. as in Ratcliffe). See also Voight's book above in number theory.

Low-dimensional and hyperbolic geometry

Geometric manifolds. See also the section on geometric group theory above.

Of course the canonical reference is
• William Thurston, Three-Dimensional Geometry and Topology (volume 1) and Geometry and topology of three-manifolds (aka the Princeton lecture notes, downloadable here). These are hard books, but very rewarding. The book is an edited version of the first few chapters of the lecture notes, covering the basic theory of geometric manifolds. The real work starts in the second half of the lecture notes, containing the material on Dehn surgery, rigidity, etc.

  The lecture notes themselves, historically only available in rough forms (either typewritten or in the LaTeXed form linked above with drawings scanned from the paper versions), will be published in book form by the AMS in mid-2022 as part of their four-volume series of Thurston's complete works.

  See also the book In the Tradition of Thurston edited by Ken'ichi Ohshika and Athanase Papadopoulos.

• John G. Ratcliffe, Foundations of Hyperbolic Manifolds. Contains many of the ideas from Thurston's book and the `differential geometry' parts of his notes rewritten with full proofs. My comments on Lee's Smooth apply to this book, which is another doorstop. Not for the faint of heart, incredibly technical with not enough pictures (e.g. the definitions of cycles and sequences of faces in section 6.8, which are given entirely algebraically with no indication of the geometric meaning). That said, if you want to understand Thurston's ideas you will need a source other than his notes sometimes, and this is a very complete reference. Excellent historical notes and bibliography.
• Albert Marden, Hyperbolic Manifolds (first edition was Outer Circles). Much more friendly than Ratcliffe and more detailed than Thurston's book; very nice introduction to the big theorems (until recently only conjectures) of 3-manifold geometry - the ending lamination conjecture, the density conjecture, tameness, etc. - and quasiconformal deformation spaces. Later chapters often tend more towards trying to explain the ideas rather than giving proofs, but everything is heavily referenced. Overall perhaps the first hollistic introduction to Kleinian groups and 3-manifold geometry accessible to a beginning graduate student. Noticably large number of typos which is disappointing given the price. Highly recommended, but unfortunately very expensive.
• Jennifer Schultens, Introduction to 3-Manifolds. An even more friendly introduction, requires only minimal algebraic topology and differential geometry. Nice little book.
• R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry. Nice friendly reference for the "big theorems" of hyperbolic geometry (Mostow rigidity and the Margulis lemma), convergence, and the Jorgensen-Thurston theory of Dehn surgery.
• John Hempel, 3-Manifolds. Pre-Thurston 3-manifold theory. It is of historical interest (e.g. to see a lot of concepts that you normally only see in books on surfaces; and to see a book on 3-manifolds which does not use the word "hyperbolic"), and also a good reference for some of the topology since it is ritten at a more elementary level than some post-Thurston books (e.g. provides rigorous foundation for things like cutting manifolds along 2-sided embedded surfaces).

Knots.

• W.B. Raymond Lickorish, An Introduction to Knot Theory. Essentially an introduction to knot invariants like the Alexander and Jones polynomials. This is one of the genre of knot theory books which spends half of the book developing basic algebraic topology, doing only a little knot theory at the end.
• Richard H. Crowell and Ralph H. Fox, Introduction to Knot Theory. Another introductory book but this time more focused on combinatorial group theory.
• Masanori Morishita, Knots and Primes: An introduction to arithmetic topology. In that vein, here is a book in the latter category. Decidedly not an introductory book, this talks about analogies between number theory (you need some class field theory) and knot theory; so you need to be an expert in both fields to read it.
• Peter Cromwell, Knots and Links. An intermediate level text which doesn't spend 50 pages on defining the fundamental group; this one actually proves the equivalence "two bridge" \(\iff\) "rational". Unfortunately, there is not much geometric topology.
• Jessica Purcell, Hyperbolic Knot Theory. Modern book (2021) which is an introduction to knot theory tending towards geometry rather than towards knot invariants or topological theory. Lots of nice pictures, complements Marden's Hyperbolic Manifolds (see above) very nicely. Technically minimal prerequisites beyond knowing some basic algebraic topology & some familiarity with manifolds (in the broadest possible sense). Highly recommended introduction to knot theory and the parts of Thurston's hyperbolic programme not found in many `easy' books.

Complex analysis and Teichmüller theory

• Otto Forster, Lectures on Riemann surfaces. This is a very streamlined account of Riemann surfaces, from definitions to the Riemann-Roch theorem in a very short space of time. Quite algebraic which is an advantage to me but might be a disadvantage to an analyst, primarily interested in algebro-geometric problems.
• Hershel M. Farkas and Irwin Kra, Riemann surfaces. A more analytic account which heads more in the direction of Kleinian groups and moduli theory. Very readable account of the uniformisation theory.
• Yōichi Imayoshi and Masahiko Taniguchi, An introduction to Teichmüller spaces. Elementary introduction to moduli spaces and mapping class groups in the complex context, more analytic than geometric; contains a very well-motivated introduction to the theory of quasiconformal mappings which is an advantage for going on to study moduli spaces of Kleinian groups, for instance as described in the later book co-authored by Matsuzaki and Taniguchi. For a geometric viewpoint on Riemann surface moduli spaces see Thurston's notes in the section on geometric topology, or Farb and Margalit's book below.
• Benson Farb and Dan Margalit, A Primer on Mapping Class Groups. A slow friendly introduction from a more geometric point of view, incredibly clear and readable. One advantage that might not be immediately clear is that the theory is developed for surfaces in such a way that it can be adapted for marked surfaces (i.e. surfaces with points which may either be deleted or ramified). Also, this is the only book I know of which actually motivates clearly the point of quadratic differentials (see section 11.3).
• Kari Astala, Tadeusz Iwaniec, and Gaven Martin, Elliptic PDEs and Qausiconformal Mappings in the Plane. I know nothing about PDEs, but the sections on quasiconformal mappings, holomorphic motions, etc. can be read without that background. Very useful reference particularly because of the large bibliography and lucid writing. Unfortunately it has a tendancy to reference entire books without page or section numbers (e.g. on p.296, "That two different elements share a fixed-point in a Kleinian group has algebraic implications [48]", where [48] is the 300 page book by Beardon on discrete groups -- admittedly it is fairly easy in this case to find the relevant section, but there are more annoying examples I can't find right now).
• Benson Farb, Richard Hain, Eduard Looijenga (eds.), Moduli Spaces of Riemann Surfaces. The lecture notes by Minsky and Hamenstädt are the best short introductions (less than 80 pages all together) to mapping class groups and Teichmüller theory which I know of.

Algebraic geometry

Scheme theory.

• Robin Hartshorne, Algebraic Geometry. This is the traditional introduction to scheme theory, and the first real textbook on the subject after Grothendieck invented it. It is not really an introduction to algebraic geometry: you need a very good background in varieties (i.e. you should really know everything in Chapter I from something like Harris or Shafarevich vol. 1 before you try properly to read this book). Chapters II and III are highly technical, and to see any geometry you need to do all the exercises. The final two chapters (on curves and surfaces respectively) are more geometric and rely only a little on the details of chapter III, so I would recommend skipping the bulk of III and going back to it once you understand the applications. Also as I say above in the algebraic topology section, Hartshorne assumes that you have already seen cohomology before in some other context (e.g. the de Rham cohomology) and so you need to fill in the geometric motivation yourself --- again, it was the first real book on the subject and the primary audience was people already in classical algebraic geometry who wanted to learn this new stuff that Grothendieck had just come up with. The book is highly deficient in arithmetic examples, see Eisenbud and Harris (for instance) for some number theoretic applications.
• Igor Shafarevich, Basic Algebraic Geometry (vol. 2). Primarily about schemes over \(\mathbb{C}\), I really like volume 1 but this is not so much my thing.
• David Eisenbud and Joe Harris, The Geometry of Schemes. Very good set of examples, just ignore chapter I (it is there basically to fix notation and I think it is wrong to try to read it to learn the definitions and motivation) and go straight to chapter II and maybe IV. But I think it is wrong to try to talk about schemes before learning the motivating constructions from classical geometry, they are not very interesting or useful without doing a lot of stuff first since the advantages are very high-level and the barriers are very concrete (i.e. abstraction in three different directions at once) - E-H has the advantage of actually motivating scheme-theoretic constructions like blowups and comparing to what is possible with just varieties, which Hartshorne does not do much of. There are many omissions: for instance, there is no intersection theory, no cohomology, and mainly curves. It does include a brief discussion of the functor of points approach to scheme theory in chapter 6, but I preferred the treatment in Waterhouse's book above.
• Ravi Vakil, The Rising Sea: Foundations Of Algebraic Geometry. Downloadable here. The more modern, friendly introduction to scheme theory. Even though it is friendly, it is still (and this should be in big red letters) a massive mistake to try to read these without a good knowledge of the basic theory. You cannot learn scheme theory without first internalising the theories which it is supposed to be a generalisation of! (You probably can, in the sense of memorising all the theorems and proofs, but you will miss the geometry - and the whole point of scheme theory is to be able to do geometry in a more general setting.) I would recommend this to the beginner before Hartshorne, certainly. I hope that one day the notes will be published in book form.
• David Mumford, The Red Book of Varieties and Schemes. Another classic, and includes what might be the first pictures of schemes. See also this nice blog post.
• Yuri I. Manin, Introduction to the Theory of Schemes. A newer book, essentially an introduction to schemes but with a more Russian style. This has the advantage of many examples (including some more arithmetic ones) but not so useful as a reference as it is not so comprehensive.

Toric varieties and combinatorial algebraic geometry.

• David A. Cox, John B. Little, and Henry K. Schenck. Toric Varieties. This is a massive book which is accessible to the beginner (it might be possible to use this when teaching a second course in algebraic geometry after something like Harris or Shafarevich vol 1) and also includes research-level material. A lot of the foundations are skipped, but this has the advantage that you don't need to wade through a bunch of linear group theory first. It also has exercises.

  As always I tend not to prefer books of this type (I am yet to fully understand what the problem is in general, but my feeling is that books tend to be larger when they are unfocused and lack a clear direction — this explains why there are some large books which I do like, for example Aluffi's algebra book). Anyway, I prefer Fulton below.

• Günter Ewald. Combinatorial Convexity and Algebraic Geometry. A less comprehensive introduction, this book is half on convex polytopes and then half on the associated algebraic geometry. (I suppose at this point I should try to advertise the subject, or at least try to explain what it is about: the philosophy is that, whenever you have a polytope in a lattice, the lattice points inside the polytope define the ring of functions on a variety that reflects the combinatorial properties of the polytope. In fact it turns out that many many varieties arise in this way and so the theory is very applicable. It is also nice in the sense that toric varieties have very regular behaviour compared to arbitrary varieties.) I would recommend this book to people interested only in the combinatorics of polytopes as well, since that half of the book is very easy to read and has a different perspective to something like Ziegler (see the section on polytopes).
• William Fulton. Introduction to Toric Varieties. The classical introduction to toric varieties, much faster than Cox/Little/Schenck and also much shorter (~100 pages vs ~800 pages). Some nice pictures, too, but leaves even more of the background material assumed.
• Tadao Oda. Lectures on Torus Embeddings and Applications and Convex Bodies and Algebraic Geometry : an introduction to the theory of toric varieties. Oda was one of the original practitioners of the subject, so these are mainly of historical interest.

Abelian varieties; Elliptic curves; Arithmetic geometry.

• Christina Birkenhake and Herbert Lange, Complex Abelian Varieties. An abelian variety is a connected projective algebraic variety that admits an algebraic group structure (i.e. you can define a multiplication of points on the variety satisfying the usual group axioms, with the caveat that the multiplication and the function sending each point to its inverse have to be morphisms). In particular abelian varieties over \( \mathbb{C} \) happen to be precisely the complex torii (i.e. \(\mathbb{C}^n / \Lambda \) for some lattice \(\Lambda\) together with a bilinear form on \(\Lambda\) satisfying two simple relations). Of course a simple example is an elliptic curve. In any case the theory has parallels to the theory of toric varieties. This is a very nice book but you should have some maturity in the theory of algebraic groups, for instance by looking at the book by Cox/Little/Schenck on toric varieties or the book by Humphreys on linear algebraic groups (in the section on geometric group theory above). Some knowledge of Riemannian geometry is also an advantage (really this subject lies in the intersection between algebraic geometry and Riemann geometry).
• Joseph Silverman, Rational Points on Elliptic Curves (co-authored with John Tate); The Arithmetic of Elliptic Curves; Advanced Topics in the Arithmetic of Elliptic Curves; and Diophantine Geometry (co-authored with Marc Hindry). I am not an arithmetic geometer, but these books almost make me want to be. Essentially they form a series.

  RPoEC is a UTM so by rights should live in the undergraduate algebraic geometry section above, but it does require readers to be comfortable with projective geometry and basic algebra (basic Galois theory). The highlight for me is the discussion of Fermat's Last Theorem in section 6.6 which goes into much more detail than most other elementary books about the proof method.

  AoEC requires basic algebraic geometry and for you to know what a local field is (i.e. you should know about the \(p\)-adics).

  ATAoEC is too hard for me (it really needs you to know much more class field theory than I am comfortable with).

  DG is somewhat parallel to the two elliptic curves GTMs, doing much of the same thing but with abelian varieties in more generality.

• Carel Faber, Gerard van der Geer, and Frans Oort (eds.), Moduli of Abelian Varieties. This is here primarily for the paper Toroidal Resolutions for Some Matrix Singularities by Faltings which I once tried to read in the algebraic geometry reading group. Consider it as motivation to understand crystalline cohomology, but it was definitely too hard for that reading group.

Miscellaneous.

• David Cox (and five others), Applications of Polynomial Systems. Applied algebraic geometry, applied to the "real world" not so much to mathematics: for example splines, rigidity (for this see also Erik Demaine and Joseph O'Rourke's book Geometric Folding Algorithms), and dynamics. Some very nice mathematics here, but more computational than I am really interested in.
• Robin Hartshorne, Deformation Theory. Moduli spaces of varieties from a very modern viewpoint. You need to understand chapter III of Hartshorne's Algebraic Geometry to read this, but the mathematics is lovely. As with AG the exercises are really the heart of the book. Anyway if you have learned the abstract theory this is as good a place as any to start doing some real geometry with it.
• Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real Algebraic Geometry. Basic introduction, just some knowledge of scheme theory needed. Applications are included but not very much geometry (the subject has a much more logical flavour).
• Frank Sottile, Real Solutions to Equations from Geometry. This is a lot of real algebraic geometry, but actually applied to geometry (primarily enumerative geometry). Another very nice applied-style book with some very interesting content, and less computational.
• David Cox, John Little, Donal O'Shea, Using Algebraic Geometry. Another applied algebraic geometry book, with topics similar to a mixture of the ones in the books above (but collected in one place): integer programming, polytopes, splines, etc. It also includes coding theory, which (I think) is not in any of the other books I mentioned. This is less geometric and more algebraic (i.e. the kinds of problems studied are more on the end of solving equation systems rather than geometric questions)
• Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry. Really, this is a book that would fit more in the section on complex analysis above. Basically this is the generalisation of geometric theorems from the case of Riemann surfaces to more general algebraic varieties over \(\mathbb{C}\). It might be a good place to look if you are reading Hartshorne and don't really understand the geometry behind the intersection theory and cohomology (these are not found in Eisenbud-Harris).

Dynamical systems

• Alan F. Beardon, Iterations of Rational Functions. Slow introduction to complex dynamics, one of the few GTMs not available on SpringerLink even as a PDF.
• S. Morosawa, Y. Nishimura, M. Taniguchi, and T. Ueda, Holomorphic Dynamics. More advanced characteristic 0 dynamics than Beardon which is not afraid to use more powerful tools (and prefixes like 'quasi').
• Joe Silverman, Arithmetic of Dynamical Systems. A very nice introduction to dynamics over our favourite mathematical objects (local fields, of course). You need only minimal arithmetic geometry for this, and really it is even a nice introduction to dynamics over \(\mathbb{C}\). I like Silverman's writing style anyway, so this is my preferred complex dynamics book.