This list is an effort to make a list similar and supplementary to the Chicago undergrad bibliography, but with a less useful organisational structure based vaguely on the University of Auckland mathematics courses. Library call numbers listed in brackets after review; (*) denotes a book that the library does not own.
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 have just reorganised the list; within each section, the books are roughly ordered by 'difficulty' (whatever that means).
Note. The university course numbers listed below are indicative of the content only; I make no claims about difficulty or relevance to the course as it is usually taught!
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.
- Contents
- Culture
- Elementary texts: stage I and II
- Upper-level undergraduate texts: Stage III
- Postgraduate texts
- Physics books
- Books I want to get around to reading and/or writing about here
- My favourite books
- Paul Halmos, I Want to be a Mathematician. As he explains in the preface, this isn't an autobiography. It has a chapter entitled "how to do everything", so practical life advice is included. Read this book. (510.973 H19Y)
- Michael Harris, Mathematics Without Apologies: Portrait of a Problematic Vocation. A more modern non-autobiography, I quite enjoyed it. (510 H31)
- Paul Lockhart, A Mathematician's Lament: A complaint about the U.S. education system that is somewhat relevant to NZ. (510.71 L81)
- 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. (516.204 D84 and 510 D84)
- Stephen Krantz, Mathematical Apocrypha and More Mathematical Apocrypha. "If it's the quotient rule you wish to know,/It's low-de-high less high-de-low./Then draw the line and down below,/Denominator squared will go." (510 K897m and 510 K897ma)
- Carl E. Linderholm, Mathematics Made Difficult. This book needs no introduction and no recommendation, it is a classic. (510 L74)
- 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. (510.149 H84)
- 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. (510 H26)
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James Stewart, Calculus. This is the "recommended" textbook for every UoA calculus course. 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 (I haven't taken 150, but the notes for 250 and 253 are high-quality). Note that the content of the courses does not necessarily reflect the content of this textbook! (515.15 S84e 2012)
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Howard Anton, Calculus. Isomorphic to Stewart. (515 A63)
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Silvanus Thompson, Calculus Made Easy. This is the book I originally learned calculus from (in year 11, or around 2013-4). 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. (515 THO)
Indeed, my favourite introductory grown-up calculus book is, as you can probably guess, the One True Calculus Book:
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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)
Other books people recommend but I've not read: Apostol, Calculus.
- Peter Baxandall & Hans Liebeck, Vector Calculus. This book covers the same content as 253 but in much greater detail. It is very geometric, contains proofs similar in level and style to those in 253, requires the same level of linear algebra, and covers Gauss' and Stokes' theorems after Green's theorem (useful for physics - and in fact it uses electromag 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.) (151.84 B35v)
My favourite multi-dimensional calculus book, is, in fact, Loomis & Sternberg, Advanced Calculus; see below.
- David Poole, Linear Algebra: A modern introduction. This is the "recommended" textbook for the linear algebra portion of the 150-250-253 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. Like Stewart, the content of the book is almost orthogonal to the way the course is taught! (512.5 P82)
- Stewart Venit & Wayne Bishop: Elementary Linear Algebra. A shorter book, same content essentially. Includes section on linear programming (missing from Poole; this isn't part of any course AFAIK, but is vaguely interesting). Spends a very long time on computations. Not even in the UoA library; don't bother buying a copy. (I had a copy given to me.) I actually use this if I need an algorithm/computation notes. (*)
- Howard Anton: Elementary Linear Algebra. Isomorphic to Venit & Bishop. (512.5 A63)
- Gilbert Strang: Linear Algebra and its Applications. I like this book, it is the the book I would recommend someone use during a stage I course. Lots of intuition, so you can move on to something more rigorous like Halmos for stage II. He does somewhat fetishise the dimension theorem for a linear transformation (he calls it the 'fundamental theorem of linear algebra') but it's not enough to ruin the book. It's probably very good for compsci students, there's a lot of algorithm stuff and 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). (512.5 S89)
- Paul Halmos, Finite-Dimensional Vector Spaces. My favourite linear algebra book! This book covers the same content as 253 but in much greater detail. 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 253 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. (515.5 H19f)
Other books people recommend but I've not read: Curtis, Linear Algebra: An Introductry Approach; Axler, Linear Algebra Done Right; Treil, Linear Algebra Done Wrong.
- 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 college students occasionally. (510 H84)
- 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. Some people like it. (511.3 V43)
- Paul Halmos, Naive Set Theory. This is the set theory book that you should read. 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. (511.322 H19)
- 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. (511.32 S578)
- 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. (511.32 R74)
- 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. (512.7 D84g)
- 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. (512 L253Y)
- 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. (515.352 W23)
- 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. (515.312 A75)
Other books people recommend but I've not read: Blanchard, Devaney and Hall, Differential Equations.
- A. G. Hamilton, Logic for Mathematicians. This is the text for 315, it failed to keep my attention but I think that's a feature of most logic books. (164 H21 1988)
- M. A. Armstrong, Groups and Symmetry. A very geometric book; it is decent if you liked the algebraic parts of 326. 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. I have actually used some parts for scholarship calculus students in the past. (512.2 A73)
- 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. (512.2 C325)
- Joseph Gallian, Contemporary Abstract Algebra. This is the book that 320 is based on. 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. (512 G16)
- Charles Pinter, A Book of Abstract Algebra. Introductory book at the level of 320. 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. (512 P65)
- 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. (512 A79)
- John Dixon, Problems in Group Theory. Pitched at higher level than 320, but I like it and it should be good for 720 and 721 as well, so well worth the cheap price. (512.22 D62)
- Ian Stewart, Galois Theory. Once you have an OK grounding of the material in 320 (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 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. (512.32 S84)
- 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 with other people. (512.32 E26)
Other books people recommend but I've not read: Vinberg, A Course in Algebra.
- Robin Wilson, Introduction to Graph Theory. Elementary book. Easy exercises. (511.6 W75)
- Robert Wilson, Graphs, Colourings, and the Four-Colour Theorem. Proves most of the results for planar graphs and graph colourings that are needed for 326; no exercises. (511.5 W74)
- Miklos Bona, A Walk Through Combinatorics. This is the only book you need for 326; it covers everything (bits with more detail than needed, bits with slightly not enough). 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. (511.6 B697)
- Bela Bollobas, Modern Graph Theory; Reinhard Diestel, Graph Theory. First couple of chapters from these are good for the graph theory part of 326. Nice exercises, but take care that these books are harder than you need. (511.6 B69m and 511.5 D56)
- W.D. Wallis, Combinatorial Designs. This book is the must-have book for the block designs section of 326, in my opinion. I really like the tone, it goes through a lot of different examples and explicit constructions, and it has plenty of literature references. The book is quite good for proofs, and he presents some of the arguments in a more explicit way than the lecture notes. The exercises are fine but nothing special (not very many, often computational). The typesetting isn't that great. (511.6 W21)
- Martin Aigner, Combinatorial Theory; Jack Graver and Mark Watkins, Combinatorics with emphasis on the theory of graphs. These books are far too difficult for 326, but I found the more abstract perspectives in the first few chapters rather enlightening; Aigner, for example, develops the theory of counting entirely via counting bijections and surjections and so forth; and G&W develops graph theory in a much more abstract way first. (511.6 A28Y) and (511.6 G77)
- Alvin E. Roth (ed.), The Shapley value: Essays in honor of Lloyd S. Shapley. This book includes the original paper by Shapley on assigning power values to voting systems, together with a number of essays and papers motivating and extending the paper. (*)
- Gareth A. Jones and J. Mary Jones, Elementary Number Theory. This is a step up from Dudley (see 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 book develops number theory with algebra at about the level of 320. It does start a little slowly (is this a feature of most number theory books?). (512.7 I65)
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Kenneth R. Davidson and Allan P. Donsig, Real Analysis and Applications. For an introduction to real analysis for someone who understands calculus at the level of, say, 150 but not 250+ this might be a better choice than Rudin - it's probably comparable to whatever text is recommended for 332, 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 above, this is another new Springer book with really annoying typesetting, but to a lesser degree. (*)
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Terence Tao, Analysis I. This might be a good alternative to Davidson & Donsig for concrete real analysis (everything is done in one dimension over ℝ); Tao is a very good writer, but I personally found this book to be a little slow. (It might actually work as an alternative to Spivak's Calculus for good first-year students.) The construction of ℝ is done via the equivalence-class-of-Cauchy-sequences method, which is my personal favourite; it is well-motivated and done in detail, and in fact takes a significant part of the book. The reasoning for this is explained in some detail in the preface. (515 T17, but a new edition is available online through the library)
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Terence Tao, Analysis II. This is the sequel (obviously) to Analysis I, and covers analysis on metric spaces through to a construction of Lebesgue integration. I have not read it, but judging from the first volume it will likely be a good alternative to the post-chapter-9 section of baby Rudin. (Available online through the library)
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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 (see below) 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 just material from 253 - 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) (515.8 R91)
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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. (515.8 K81eYs)
Other books people recommend but I've not read: Pugh, Real Mathematical Analysis.
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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. (514.3 J22Y)
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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. (514.3 W39)
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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.
(Errata) (514.3 M96)
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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. (515.84 S76, 515.84 M96)
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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. (515.8 L86)
- 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. (516.35 R35)
- 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). (*)
- Joe Harris, Algebraic Geometry: A first course. This is my favourite introductory book, it is faster and less chatty than the 334/734 notes but it has many more examples and exercises. It includes none of the background commutative algebra, see below for useful references. (516.35 H31)
The first few books here, broadly speaking, are complex calculus books; the latter books are complex analysis books.
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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. (515.93 F53)
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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... (515.9 S12)
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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. (515.9 N37)
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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 mathematical 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. (Available through SpringerLink)
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Joseph L. Taylor, Complex Variables. This is a new and shiny (2011) book that is roughly at the level of 341. (515.9 T23)
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John Conway, Functions of One Complex Variable I. I found this book to be readable, if a little slow. (515.93 C76)
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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. (515.93 A28 1966)
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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. (515.7 R916)
Elementary books.
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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. (516.2 B61)
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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)
Axiomatic geometry books.
- 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! (516 H33)
- 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. (516 L47)
- 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. (*)
Conic sections.
- Kieth Kendig, Conics. This book is everything you always suspected about conic sections and it is amazing. Please read it. (516.352 K33)
- 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. (516.3 S17)
Modern geometry.
- 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. (516.0076 S96)
- 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 blackboardthinking. (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.) (516 C87, 516 C87g) - Marcel Berger, Geometry. But this is my 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). (516 B49)
- Manfred do Carmo, Differential Geometry of Curves and Surfaces. Differential geometry, without any more machinery than multivariable calculus. Quite dry and very computational. This is actually useful for physics (in particular, PHYSICS 201). (516.36 C28Y)
- Michael Spivak, A Comprehensive Introduction to Differential Geometry. More differential geometry than you need in your life. (516.36 S76)
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. (512 R84a)
- Serge Lang, Algebra. I am currently spending a significant amount of time with this book as I relearn a lot of the algebra I haven't used for a couple of years. 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 and the inconsistent leaps of logic. (512 L26a)
Commutative algebra.
- M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra. Contains most (all?) of the commutative algebra needed for 334/734. (512.24 A87i)
- David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. Behold, more commutative algebra (512.24 E36).
Category theory.
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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. (511.3 L42)
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Horst Herrlich and George E. Strecker, Category Theory. This book is very well-motivated with lots of examples. (512.55 H56)
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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. (512.55 A19)
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Saunders Mac Lane, Categories for the Working Mathematician. This is the standard introductory text. (512.55 M16)
- Coxeter, Regular Polytopes.
- O'Rourke, Geometric folding algorithms.
- Something on knot theory? (Cromwell, Knots and Links ?)
- Aluffi, Algebra: Chapter 0
- Artin, Algebra
- Berger, Geometry
- Coxeter, Introduction to Geometry
- Halmos, Naive Set Theory
- Loomis and Sternberg, Advanced Calculus
- Rudin, Principles of Mathematical Analysis
- Wallis, Combinatorial Designs