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How to Become a Pure Mathematician (or Statistician):
   a List of Undergraduate and Basic Graduate Textbooks and Lecture Notes
怎樣研習數學﹝或統計學﹞:
   本科與基礎研究課程參考書目

Print friendly version: here [.pdf]

Last Revised: 8/1/2009

Stage 1:
Elementary Stuff, Intro. Discrete Mathematics, Intro. Algebra, Intro. Calculus
Stage 2:
Linear Algebra, Intro. Higher Algebra, Calculus,
Complex Variables, Differential Equation, Probability & Statistics
Stage 3:
Intro. Analysis, Abstract Algebra, Intro. Number Theory, Intro. Topology, Differential Geometry,
Optional: Math. Modelling, Stat. Inference, Probability & Stochastic Processes, Stat. Computing
Stage 4:
Foundations and Discrete Mathematics, Analysis, Algebra, Number Theory, Geometry, Topology,
Optional: Further CalculusMathematical Physics, Probability, Statistics, Biostatistics 

It's like if you want to be a good pianist,
 you have to do a lot of scales and a lot of practice,
 and a lot of that is kind of boring, it's work.
 But you need to do that before you can really be very expressive and really play beautiful music.
 You have to go through that phase of practice and drill.
- Terry Tao

About this article:

  • What for?
    OK, there is a way to become a good theoretical physicist. Here is a guide to study pure mathematics, or even more. This list is written for those who want to learn mathematics but have no idea how to start. Yup, a list for beginners. I don't claim that this list makes you a good pure mathematician, since I belong to the complement of good pure mathematician. I make no attempt to define what pure mathematics is, but hopefully it will be clear as you proceed. I also highlighted several books that you would really like to keep in your own library. You probably like to read those books again and again in your life. Free material excluded. Note the highlighted list does NOT indicate those books are good for beginners. I shall try to keep this list up to date whenever I exist.
  • Assumed knowledge.
    I assumed you have high school mathematics background (i.e. basic trigonometry, Euclidean geometry, etc). The aim of this page is to introduce what different branches of mathematics are; and recommended a few notes or texts.
    Scientists in other fields and engineers may skip first or second stages and begin at later stages, according to their prior knowledge.
  • Time.
    It takes approximately one year for each stage (except for stage 4, I list more material in each field for more advanced studies), for a full time student. Part time students may double the time. But its better for anyone to understand most parts of stage n before proceeding to n+1, for some integer n in {1, 2, 3, 4}. If you decided to attend a class, don't expect the professor can teach, it always happen, especially in higher level courses. What's the order of courses to study within a stage doesn't really matter, usually. One doesn't need to read every listed book within a subject to master the subject. I listed more than enough so that you can scout around to find one that you feel comfortable with. Some people like to consult a few books, beware of the symbols from different books in such cases.
    Moreover, it often happens that you couldn't solve a problem within an hour. It's not surprise to spend a week or more to tackle one problem. Things may come to your head suddenly. Shouting eureka is the high point of a mathematician.
  • "Axiom of choice".
    My selection will not be bounded by any publication press, author's nationality or religion. It relies on two factors: well written or cheap. These two factors are not mutually exclusive. I treat "free" as an element in "cheap". Note the price factor may by irrelevant, sometimes I get a HK $2xx book and Amazon says its US $1xx (~HK $7xx)...... with the only difference is, perhaps, I got the international edition. Moreover, some Chinese press in mainland China published photocopied of English text with a relatively cheap price.
    Bear in mind that, just because one is a good mathematician doesn't imply he's a good author or educator. Perhaps Terry Tao is an exceptional case. To study science, reading the classics (the Elements, Dialogo sopra i due massimi sistemi del mondo, the Principia, Disquisitiones Arithmeticae, Principia Mathematica, etc) is optional. While for literature or philosophy, I wonder if any well educated student has never study Shakespeare or Plato.
    To view [.pdf] get Adobe Reader, to view [.ps] download this and this, or visit this page, to view [.djvu] get this. Get WinRAR for [.rar] files.
  • Comments.
    Links to Amazon for most of the listed book are included, so that you have an easy access to other users' comments. Note that the comments are sometimes quite extreme: for the same book, one rated it with 5 star (with dozens of people supporting) and at the same time another rated 1 star (with dozens of people supporting again), especially for introductory discrete mathematics, probability and statistics books. It seemed to me that lots of people study these subjects because they need to, they want to apply mathematics. Large proportion of these readers are lack of mathematical maturity. If they can't pass the exam, you know... In contrast, most pure mathematics students study because they like the subject and enjoy it. So, ask yourself, why do you study?
  • Other resources.
    Although I'm not into reading books online, I should remind you that MIT's open recourse, the Archimedeans and Wikibooks provide another great sources of materials. These are excluded in the following list. The list below aimed to recommend books or (usually) printable notes. Google books allow you to preview sections from a book. Schaum's Outlines series are cheap, but I seldom include them, you may search the relevant if you like.
  • Me.
    A product of School of Mathematics and Statistics, UNSW, Sydney, Australia. I've taken all undergraduate core pure mathematics and statistics courses there, with all pure mathematics courses in higher level (perhaps equivalent to Honors Courses in the US which focus on theory), whenever they exist. Also, I was Terry Tao's teacher's student, Michael Artin's student's student, Gottfried Leibniz's student Jacob Bernoulli's student Johann Bernoulli's student Leonhard Euler's student Joseph Lagrange's student Simeon Poisson's student's student's student's student's student's student's student's student's student, Max Planck's student's student's student's student and Thomas Kuhn's student's student. Just feel like I'm an idiot. 
  • Disclaimer.
    I'm not responsible for any external link.
  • Comments/suggestion for a book, etc. Either make comment in the blog http://hbpms.blogspot.com/ or email me, contact can be found in the main page.
Stage 1

For many reasons, high school mathematics is taught in an informal way, we'll fix things up here. The first stage contains the basic material that is required for most scientists and engineers, discrete mathematics is particularly meant for computer scientists. 

Elementary Stuff:
Here is a list that helps you to refresh and enrich your high school mathematics.

Introductory Discrete Mathematics:
You will study logic and set theory in an introductory discrete mathematics. The concepts you learnt here will play a key role in your later studies and improve your thinking skill. You will also study proofs, functions (injective, surjective, bijective inverse), introductory graph theory and number theory, Euclid's algorithm, discrete probability (counting, nCr, nPr) etc in this course.

Only for Proof, Logic and Set Theory:

In general:

Introductory Algebra:
Complex number, polynomials, matrix, system of equation, Gaussian elimination, vector space, linear transformation, etc. T.M. Aposotol's Calculus Vol. 1 in the calculus sections also introduce topics that is relevant to this area. 

A few Americans (exchanged to Sydney) in my third year abstract algebra class told us they couldn't follow because they haven't learnt much complex number. In order to have fun with linear algebra and complex variables in stage 2, it's better to learn complex number now.

Introductory Calculus:
As you may have seen, calculus is crucial. It is not only important in advanced mathematics but also can be applied to many other fields. If you learn first year calculus in a college, your suggested text is probably J. Stewart's Calculus. This is a standard modern text, it is colorful and quite detailed. If you prefer it, that's fine. This book is thick, you would like to leave a few section, say vector calculus, double, triple, line and surface integrals, to the next stage. I would suggest a few non-standard texts and notes.

The first thing we need to fix up is the definition of limit, learn the epsilon and delta proof. Other key concepts are the mean value theorem, fundamental theorem of calculus, logarithm and exponential functions, inverse trigonometric functions, hyperbolic functions, basic differential equations, limit of sequences, indeterminate form, infinite series, basic idea of several variables.

 

Stage 2

Mathematics program in the University are constructed in the way that courses are offered in favor of other schools (Physics, Chemistry, Biology, Engineering, Economic, etc). For example, Physics students have to take linear algebra, several variable calculus, mathematical analysis and differential equation. As a consequence, textbooks are written and catalog like the following way. Several authors focus on the application and often do calculation without justifying, beware of it.

Linear Algebra:
Vectors, vector spaces, linear transformations, multilinear map, inner product spaces, norms, orthogonality, Gram-Schmidt algorithm, QR-factorisation, least square, Householder algorithm, normal matrices, Jordan canonical forms, Cayley-Hamilton theorem, minimal and characteristic polynomials, direct sum decompositions, generalised eigenspaces, functions of matrices, exponentials of matrices, etc will be studied in this course. Such material can be applied to linear programming, computer graphics, fractals, and many areas in natural sciences and social sciences. Introductory Higher Algebra:
I've taken this course entitled "Finite Mathematics", a course for computer scientists, software engineers and pure mathematicians (optional but useful). Assuming you have learnt the very basic of number theory in the discrete mathematics section, you are ready to get a taste of higher algebra here. Things like prime numbers, tests for primality, Fundamental Theorem of Arithmetic, Fermat's little theorem, Gauss' lemma, Euler's theorem, Chinese remainder theorem and their applications (coding, RSA etc.) are concerned in this section. This is a bridge that connects first stage basic algebra, discrete mathematics and the third stage abstract algebra. Calculus (Introductory Real Analysis, Several Variables Calculus, Vector Calculus, etc):
Learn the very basic concept of real analysis, like open set, close set, boundary point, closure, limit point, bounded set, connected set, compact set, Bolzano-Weierstrass theorem. To get yourself ready for the stage 3 analysis, make sure you understand those stuff. Next, the Lagrange multiplier, inverse and implicit function theorems can also be studied. For the several variables section, gradients, double, triple and surface integrals, cylindrical coordinates, Green's theorem and divergence theorem are the key things. Also learn basic Fourier series. Complex Variables (Introductory Complex Analysis):
Another beautiful branch of mathematics. While you're in high school, you may wonder what's the point of introducing imaginary number i, does it really exists? Is it really useful? In an introductory complex analysis course, you will see the beauty of this construction. Gauss, Riemann, Weierstrass and Cauchy are key figures in this area. Key concepts to learn are: analytic functions, Cauchy-Riemann equations, contour integral, Cauchy-Goursat theorem, residues and poles. Differential Equation:
Isaac Newton originally studies behaviour of dynamical systems using differential equations. Mathematical models are often described with differential equations and they are widely used in many fields, including physics, chemistry, biology, economics etc. Mathematical methods used in solving differential equation also play an important in advanced studies. Key topics to learn are: Wronskians, series solutions, reduction of order, variation of parameters, Frobenius normal form, Bessel's equation, Legendre's equation, two points boundary problem, Fredholm alternative, Green's function, complete orthogonal system, Sturm-Liouville problems (inhomogeneous, singular), basic partial differential equation, elliptic eigen-problems, heat equation and wave equation. Probability and Statistics:
Well, being a well educated mathematician, you should have basic knowledge of statistics. Things to learn: basic probability theory (independent events, conditional probability, Bayes' Theorem), random variables (r.v.), expectation, convergence of r.v., maximum likelihood estimator, basic hypothesis testing, p-values are the basic. If you want to go a bit further, check out linear regression, linear model, residual, categorical predictors, logistic regression, ANOVA (analysis of variance) etc.

Probability:

Statistics:

Second course in Statistics:

 

Stage 3

At this stage, you begin to learn modern Pure Mathematics. Yup, this is the beginning. Focus on the proof. Don't expect you can solve a problem by plugging numbers into formulae. It often happens in advanced text that the author skips some steps in a proof or calculation, while elementary text gives detail explanations. But as G.F. Simmons has said, '[t]he serious student will train himself to look for gaps in proofs, and should regard them as tacit invitations to do a little thinking on his own.' Moreover, '[i]t is a basic principle in the study of mathematics, and one too seldom emphasized, that a proof is not really understood until the stage is reached at which one can grasp it as a whole and see it as a single idea. In achieving this end, much more is necessary than merely following the individual steps in the reasoning. This is only the beginning. A proof should be chewed, swallowed, and digested, and this process of assimilation should not be abandoned until it yields a full comprehension of the overall pattern of though.'

Also get maturity and learn how to write mathematics, by reading good books (?). I don't think Gauss or Galois can get full mark in mathematics assignments these days, for 1) something obvious to them may not be obvious to everyone else, and 2) they don't bother to explain to others. The point is, every steps in a proof should be logically related and everyone can go easily from the previous step to the next. This is not the case for non-pure mathematics. If you insist, I can show you a copy of full mark statistics assignment consists work of mere calculation and computation. (Of course not done by me!)

I had tried to list more books so that you can compare and choose one or two that fit you. Despite that most of following courses are 'pure', they can be applied to other graduated level science subjects. In the analysis class, Ian Doust even showed us an article in the Econometrica (Hildenbrad W. and Metrens J.F. Upper Hemi-continuity of the Equilibrium-set Correspondence for Pure Exchange Economies, Vol. 40, No. 1) talks about liminf, limsup, measure, weak topology and stuffs like that. If one insists that the pure stuff is useless, one is just ignorant, I don't bother to argue with those people anymore.

Introductory Analysis:
At this stage you should have learnt limits, continuity and convergence, over the reals. These are central concepts of calculus in both one and several variables. These can be generalised. You will study the idea of several spaces (metric space, topological space, Banach, Hilbert), compactness, connectedness, linear operators, elementary Lebesgue theory and measure theory, etc.

Contents of analysis courses varies almost everywhere (a.e.). As you may see, the term analysis appear in the first and second year calculus to forth year analysis. Kolmogorov and Fomin's text fits this stage well, they cover most of the above topics. Apostol, Marsden and Hoffman, and Simmons make good references. Several books entitled analysis are indeed advanced calculus (stage two calculus) or not aimed for mathematicians. Stay away from Mathematical analysis for business and finance or the like.

Only for metric spaces and topological spaces (also consult the list of Topology below):

Also deals with Lebesgue theory, Banach spaces and Hilbert spaces, etc:

Abstract Algebra:
One of the beautiful branches of mathematics. You will see how we study symmetry. Topics to be covered are basic group, ring and field theory.

Only discuss Groups:

Rings and Fields:

In general:

Introductory Number Theory:
Gauss once said "mathematics is the queen of sciences and number theory the queen of mathematics." G..H. Hardy once said in A Mathematician's Apology that, "If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and less mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean." He wrote it on November 1940. It turned out coding theory, where algebra and number theory are applied, played a key role in the World War II.

Introductory Topology:
Topology is an extension of geometry and built on set theory, it's about continuity. There is a joke that topologists are those who cannot distinguish between a doughnut and a coffee or tea cup. Wikipedia has a [.gif] to demonstrate this. You may have studied open sets, open neighbourhood, interior, closure, boundary, basis, continuity, compactness and connectedness etc. in the analysis course, move on to homemorphism, homotopy and fundamental group.

As in abstract algebra and analysis, several selected books or notes cover more than enough. You may leave a few sections to study at the next stage.

Differential Geometry:
You may wonder, geometry takes up a large portion in high school mathematics, why isn't there any geometry course in the first two stages? In fact, geometry is kind of imbedded in stage two calculus (several variables) and linear algebra courses, they are usually assumed and will be used for this course. Here differential calculus is used to study geometry. Key things to study: multilinear algebra, curvature and torsion, Serret-Frenet equation, fundamental theorem of curves, Poincaré Index theorem (plane and surface), exterior calculus, Gauss' theorema egregium, geodesics, Gauss-Bonnet theorem.

Moving frame, due to Élie Cartan, is an approach to geometry of surface. According to my lecturer John Steele, it is "computationally the easiest, notationally the neatest, aesthetically the best, makes the definitions more natural and the proofs of the two major theorems easier. The moving frame method also points the way towards several important ideas in modern differential geometry and theoretical physics. The downside (if there is one) is the reliance on exterior calculus of differential forms." O'Neill, for example, uses this approach and he manages to prove Gauss' theorema egregium in half page, see p.281.

In general:

Tensor Analysis and Manifolds:

Differential Forms:

Mathematical Modelling (optional):
"All models are wrong, some are useful." George Box. Using mathematics to model real world phenomena is useful, but not the main concern of pure mathematician.

Statistical Inference (optional):
Build upon stage 2 statistics, you will learn here the Cramer-Rao bound, uniform minimum variance unbiased estimators, Neyman-Pearson theory, Bayesian inference, basic bootstrap and robustness, introductory non-parametric (including the sign test, Wilcoxon signed rank test, McNemar's chi-square test, Wald-Wolfowitz runs test, Mann-Whitney U test, Kolmogorov-Smirnov two-samples test, Kruskal-Wallis analysis of ranks, Spearman's R and Kendal's Tau), etc. Note a few texts say they are aimed at graduated level, partly because people from other fields only learn this material in graduated schools, as a result the texts have to set up basic probability and statistics (you have done it, stage 2 stuff) for them. Probability and Stochastic (Random) Processes (optional):
Conditional expectation, Poisson process, Markov chains, renewal theory, queueing theory, reliability theory, Brownian motion and stationary processes. This topic is particularly useful for electrical and computer engineers, actuarial study, finance or things like that. See also left material from stage 2 probability. If you want to learn this with measure, see stage 4 probability list.

Statistical Computing (optional):
These days, you can't do statistics without computer, in practice. R is a freeware, S-PLUS uses the same language, so I recommend books of this kind. Also, see UCLA's Statistical Computing page for online resources.

 

Stage 4

Equivalent to the (fourth) honours year in Australia, or the basic graduated year in other places. I shall outline the key topics and give short lists of references, you may scout around to see what makes you interest. I'm far from being an expert, let me know if you have a suggestion! Note that at this stage, the levels or difficulties of books within a certain topic may vary a lot. A few subfields may be highly related, e.g. further reading in Number Theory and algebraic geometry in Geometry. Moreover, further reading should belong to stage 5.

Foundations and Discrete Mathematics:

Foundation:

Logic (Mathematical, Symbolic, Proof Theory, etc.):
According to Wikipedia, Enderton, Hamilton, and Mendelson are meant for undergraduate; while Andrews, Barwise (ed.) and Shoenfield are meant for graduate.

Set Theory:

Graph Theory:

Combinatorics:

Cryptography, Coding and Information Theory (not so pure):
First stage introductory discrete mathematics and basic linear algebra are usual assumed knowledge.

Analysis:
Materials listed in stage 3, which have not been covered there, may be studied here.

Functional Analysis:

Measure Theory:
See also the probability section below.