# Mathematics Book List

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:

• That you should buy it.
• That I necessarily stand by the words I have written forevermore after I write them.
• That I know what I am talking about.

I maintain a list of books which I own on LibraryThing.

## Culture

### 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.

### 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.

### Topology

Geometric topology.

### 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.

### 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.

## Combinatorics

• 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.
• 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.

## Geometric group theory

• 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

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.

• 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).

## 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.

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).