This list was formerly on my "NCEA notes" GitHub page. I have updated it and moved it here (Nov. 2021).
This list is an effort to make a list similar and supplementary to the Chicago undergrad bibliography,
but with a less useful organisational structure. Library call numbers listed in brackets after review; (*) denotes a book that the library does not own.
Now that the University of Auckland moved the mathematics research collection off-site in the latest set of library closures and service reductions, I see no reason
to include library call numbers any more.
It is less a list of reviews, and more a reminder to me about which books I have enjoyed reading and which books I would like to read more of. I mark my favourite books with a green star, thus.
Disclaimers. The presence of a book on this list does not mean:
I maintain a list of books which I own on LibraryThing.
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.
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:
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)
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.
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. (*)
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)
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).
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.
The first few books here, broadly speaking, are complex calculus books; the latter books are complex analysis books.
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.
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.
See also the entry on Escher under culture above.
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).
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.
Rings and group representations.
Polytopes. See also the books by Stanley and Miller/Sturmfels in the commutative algebra section above.
Arithmetic flavour (e.g. buildings, \(p\)-adic fields).
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.
Geometric manifolds. See also the section on geometric group theory above.
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.
Toric varieties and combinatorial algebraic geometry.
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.
Abelian varieties; Elliptic curves; Arithmetic geometry.
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.
Now it's the challenger's turn to reply to to this verbal bombardment:
Neatly each phrase he dissects. with intelligence subtle and keen;
Harmless around him the adjectives tumble, as he ducks for cover
And squeaks, 'It depends what you mean.'
-- Aristophanes, Frogs (trans. Barrett, p.166).