Mathcamp 2001

- Stewart,
**From Here to Infinity: A Guide to Today's Mathematics**.*A good view of modern math, very accessible.* - Hofstadter,
**Metamagical Themas: Questing for the Essence of Mind and Pattern**.*Collection of articles treating subjects from recursion to game theory.* - Conway and Guy,
**The Book of Numbers** - Martin Gardner,
**Any of his math books**.*They're all great.* - Ball and Coxeter,
**Mathematical Recreations and Essays** - Ian Stewart,
**Game, Set and Math: Enigmas and Conundrums** - Dunham,
**Journey Through Genius**.*A history of important theorems and their motivations. Accessible and well-written.* - Dunham,
**The Mathematical Universe**.*An A to Z on all sorts of topics in math. Chapter M, entitled "Mathematical Personality" is particularly amusing.* - Davis and Hirsh,
**The Mathematical Experience**.*One of the best and most accessible books on the philosophy of mathematics -- deals with such questions as "What constitutes a proof?" It's a lot less straightforward than you might imagine!* - Stillwell,
**Mathematics and its History** - Hardy,
**A Mathematician's Apology** - Stewart,
**Does God Play Dice?***On non-linear dynamics (more infamously known as chaos theory) and other things. Very accessible, no equations, doesn't really tell you how to compute these things.* - Méro, Laszlo,
**Moral Calculations**.*See below under Game Theory* - David Wells,
**The Penguin Dictionary of Curious and Interesting Numbers**and other books by the same author. - Polya,
**How to Solve it** - Stephen Hawking,
**A Brief History of Time**.*A very famous popular book on cosmology* - Richard Feynman,
**Surely You're Joking, Mr Feynman**, and other books by him.*This is a collection of autobiographical stories, and it really great fun and inspiring.* - Weisstein,
**The CRC Concise Encyclopedia of Mathematics**.*This is a fabulous handbook of all sorts of mathematical terms and suchlike, and is not too exorbitantly priced. This used to be available online, but has been removed due to legalities. There are bootleg copies available in various places if you can find them.* - Daniel Solow,
**How to Read and Do Proofs** - Escher,
**Visions of Symmetry: Notebooks, Periodic drawings and Related Work of M.C.Escher** - Gian-Carlo Rota,
**Indiscrete Thoughts**.*Stories about mathematicians* - Körner,
**The Pleasures of Counting**.*This is a very readable book introducing one to some very interesting branches of applied mathematics. It has a strong historical flavour which also discusses the mathematics of such things as the Eureka machine, competitive processes in biology and algorithms. It also contains information on the mathematics of bombing U-boats during World War II, which would have formed the basis of the Operational Research class.*

- Brualdi,
**Introductory Combinatorics**.*An elegant presentation of introductory combinatorics* - Graham, Knuth and Patashnik,
**Concrete Mathematics**.*An awesome book, full of everything you ever wanted to know about how to add and a lot more besides. Although it is quite advanced in places, it is a fabulous book. It also covers a significant amount of combinatorial material. The "How to Add" class was based on Chapter 2 of this book, and recurrence relations are also covered. And if you thought Fermat wrote hot marginalia... Plus, a review from Meep* - Wilf,
**generatingfunctionology**.*Cool book, but harder than Concrete Mathematics.Available online from http://www.cis.upenn.edu/~wilf/* - Koh and Chen,
**Principles and Techniques in Combinatorics**.*A harder textbook, but with interesting material on many branches of combinatorics including design theory, generating functions, coding theory and graph theory.* - Cameron,
**Combinatorics**.*Covers a lot of interesting material in many different areas of combinatorics ranging from moderate to quite advanced. The book is arranged in roughly increasing order of difficulty.* - Stanley,
**Enumerative Combinatorics I and II**.*Again, an advanced text*

- Scarne,
**Scarne's New Complete Guide to Gambling**.*Even though it was written in 1974, the principles are still the same. Don't try Martingale betting!* - Mosteller,
**Fifty Challenging Problems in Probability**.*A Dover book (cheap! cheap!) with archetypal problems. A review from Meep.* - Weaver,
**Lady Luck: the theory of probability** - Feller,
**An Introduction to Probability Theory and Its Applications**.*A more advanced text, but a classic in the field and full of useful information.*

- Bollobás,
**Graph Theory; Modern Graph Theory**.*Both of these books are quite advanced, but they present a lot of information very nicely. There are a lot of good exercises, ranging from elementary to highly challenging* - West,
**Introduction to Graph Theory** - Godsil and Royle,
**Algebraic Graph Theory**.*A graduate level textbook -- not many examples, but lots of cool theory.* **Schaum's Outline on Graph Theory**- Herbert Wilf,
**Algorithms and Complexity**.*Reasonably advanced, but covers the material we looked at in week four of the Discrete Track and much more besides. Available online from http://www.cis.upenn.edu/~wilf/*

- Méro, Laszlo
**Moral Calculations: Game Theory, Logic and Human Frailty**.*Terrific book, but do NOT, under any circumstances, attempt the dollar auction. A review from Meep* - Stahl,
**A Gentle Introduction to Game Theory**.*Shows you how to calculate most things of interest for simple games. A review by Meep.*

- Knuth,
**Surreal Numbers**.*A mathematical novelette* - Berlekamp, Conway, Guy,
**Winning Ways for your Mathematical Plays**.*The first edition was two volumes, but is now being reprinted in four volumes. Only the new volume 1 is currently available.*

- Howie,
**Automata and Languages**.*It might be difficult to read because of its reliance on the notation and terminology of abstract algebra.* - Sipser,
**Introduction to the Theory of Computation**.*Covers all of the CS track material and much, much more; medium level* - Lewis and Papadimitriou,
**Elements of the Theory of Computation**.*Covers automata theory, formal languages and complexity, and has lots of examples*

- Cormen, Leiserson and Rivest,
**Introduction to Algorithms**.*An awesome book!* - Knuth,
**The Art of Computer Programming (3 vols)**.*The bible of Computer Science. But it's really advanced!*

- Schneier,
**Applied Cryptography: protocols, algorithms and source code in C**.*A big thick book, but easy to read with little difficult mathematics. Lots of references, and a very solid description of the basics of the subject.* - Buchmann,
**Introduction to Cryptography**

- Niven, Montgomery and Zuckerman,
**An Introduction to the Theory of Numbers**.*Great book* - Davenport,
**The Higher Arithmetic** - Dudley,
**Elementary Number Theory** - Ore,
**Number Theory and Its History** - Silverman,
**A Friendly Introduction to Number Theory**.*A gentle book which walks through some very interesting math, ending up with a discussion of elliptic curves and some major theorems relating to them.* - Baker,
**A Concise Introduction to the Theory of Numbers**

- Hardy and Wright,
**An Introduction to the Theory of Numbers** - Borevich and Shafarevich,
**Number Theory** - Ireland and Rosen,
**A Classical Introduction to Modern Number Theory**.*Awesome book, highly recommended.* - Apostol,
**Introduction to Analytic Number Theory**.*The Riemann Zeta Function and other such delights.*

- Khinchin,
**Continued Fractions**.*Elementary*. - Olds,
**Continued Fractions**.*Even more elementary.* - Gouvęa,
**p-adic numbers**.*Where*1+3+3^{2}+...=-1/2*really is true.*

- Cox,
**Primes of the form x**.^{2}+ny^{2}*Starts easy, gets very hard.* - Silverman and Tate,
**Rational Points on Elliptic Curves**.*An undergraduate level introduction to elliptic curves with lots of explicit calculations.* - Marcus,
**Number Fields** - Stewart and Tall,
**Algebraic Number Theory**.*A gentle undergraduate level introduction to the subject; assumes basic familiarity with ring theory.* - Serre,
**A Course in Arithmetic**.*Awesome, but very hard.*

- NOVA's
**"The Proof"**(video) - Singh,
**Fermat's Last Theorem/Fermat's Enigma**.*A non-mathematical history of the problem leading up to Andrew Wiles' dramatic announcement of a proof.* - van der Poorten,
**Notes on Fermat's Last Theorem** - Ribenboim,
**Fermat's Last Theorem for Amateurs**

- Guy,
**Unsolved Problems in Number Theory** - Shanks,
**Solved and Unsolved Problems in Number Theory**

- Gallian,
**Contemporary Abstract Algebra** - Herstein,
**Topics in Algebra** - Fraleigh,
**A First Course in Abstract Algebra**.*A gentle introduction to groups, rings, and finally Galois Theory* - Dummit and Foote,
**Algebra** - Gallian,
**Comtemporary Abstract Algebra** - Isaacs,
**Algebra: A Graduate Course**.*Much more advanced.*

- Friedberg, Insel, Spence,
**Linear Algebra**.*This is a great book, very thorough, comprehensive and clear!* - Peter Lax,
**Linear Algebra** - Strang,
**Linear Algebra**.*For applications/computations* - Gelfand,
**Linear Algebra** - Bretscher,
**Linear Algebra** - Axler,
**Linear Algebra Done Right** **Schaum's Outlines for Linear Algebra**.*For reference; lots and lots of solved problems*- Hoffmann and Kunze,
**Linear Algebra** - Lang,
**Introduction to Linear Algebra**

- Armstrong.
**Groups and Symmetry** - Rotman, Joseph.
**An Introduction to the Theory of Groups**.*A graduate level introduction, much faster pace than several others listed here.*

- Hartley and Hawkes,
**Rings, Modules and Linear Algebra**.*Although this little book is out of print, it is worth tracking down in a library. It covers the ED => PID => UFD material we studied, and goes on to discuss modules over PIDs, concluding with some beautiful classical structure theorems such as the Jordan Canonical Form for endomorphisms of a vector space.* - Atiyah and Macdonald,
**An Introduction to Commutative Algebra**.*This is for scary people! It's brilliant but very hard.*

- Stewart,
**Galois Theory**.*Very gentle pace.* - Rotman,
**Galois Theory**.*Short and thorough.* - Artin,
**Galois Theory**

- James and Liebeck,
**Representations and characters of groups**.*A very gentle introduction to a beautiful area of math.*

- Humphreys,
**Introduction to Lie algebras and Representation Theory** - Carter, Segal, MacDonald, Taylor,
**Lectures on Lie groups and Lie algebras**

- Prasalov,
**Intuitive Topology**.*Great book, a lot of fun. Based on his lectures for high school students in Moscow 57th.* - Armstrong,
**Basic Topology** - Munkres,
**Topology, a first course**.*Well written, lots of examples.* - Steen and Seebach,
**Counterexamples in Topology**.*Just in case you THINK you know what open sets looks like.* - Weeks,
**The Shape of Space: how to visualise surfaces and three-dimensional manifolds** - Francis,
**A Topological Picturebook**

- Massey,
**A Basic Course in Algebraic Topology** - Hatcher,
**Algebraic Topology**at http://www.math.cornell.edu/~hatcher/#ATI - Rotman,
**An Introduction to Algebraic Topology**.*Introductory only in the sense that most graduate textbooks are introductory -- but thorough.* - Guillemin and Pollack,
**Differential Topology**

- Livingston,
**Knot Theory** - Adams,
**The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots**.*It has a section of jokes at the back, which every math book should have.*

- Edgar,
**Measure, Topology and Fractals** - Alligood, Sauer, Yorke,
**Chaos**.*These are both quite hard books, which would be appropriate for people who followed Noah's class.*

- Downing,
**Calculus the Easy Way**.*Contains good stories.* - Spivak, Michael.
**Calculus**.*This is an awesome book to learn calculus from. It's one of the best rigorous introductions to theoretical calculus out there. Look for jokes in the index.* - Apostol, Tom.
**Mathematical Analysis: A modern approach to advanced calculus**.*Covers a lot of material in a formal way, introducing the basics of analysis, differentiation, Riemann-Stiltjes integration and Lebesgue integration* - Bartle and Sherbert,
**Introduction to Real Analysis**

- Rauch,
**Partial Differential Equations**.*You need real analysis for this book; it's quite hard* - John,
**Partial Differential Equations**.*Not good as a first book on the subject, but is concise and useful* - Dym and McKean,
**Fourier Series and Integrals**.*Again, real analysis is needed* - Körner,
**Fourier Analysis**.*Great book, but again real analysis is needed. There's also a companion "Exercises in Fourier Analysis".*

- Spivak,
**Calculus on Manifolds** - Bredon,
**Topology and Geometry**.*Nice book, but a graduate level text.*

- Ahlfors,
**Complex Analysis** - Churchill, Brown and Vehey,
**Functions of a Complex Variable** - Nehari,
**Conformal Mapping** - Conway, John B.,
**Functions of One Complex Variable**.*A different John Conway. There's a proof in here with the punchline 1/72.* - Beardon,
**Complex Analysis: the Argument Principle in Analysis and Topology**.*Out of print, but worth tracking down.*

- Gelbaum and Olmsted,
**Counterexamples in analysis**.*Wild and woolly functions -- find out that you have no idea of what a continuous function looks like.* - Conway, John B.,
**A Course in Functional Analysis**.*A terribly named book -- this is at least THREE courses in functional analysis! For those of you who complained about the lack of ***** classes.*

- Coxeter, H.S.M.
**Geometry**. - Coxeter, H.S.M.
**Geometry Revisited**.*Out of print for some reason -- but it's worth tracking down.* - Honsberger.
**Episodes in Nineteenth and Twentieth Century Geometry**. - Vossen, Hilbert-Cohn.
**Geometry of Imagination**. - Greenberg, Marvin.
**Euclidean and non-Euclidean Geometry**. - Trudeau.
**The Non-Euclidean Revolution**.*This book makes you confront your prejudices about lines and points. A good, though tough intro.*

- Coxeter, H.S.M.
**The Real Projective Plane**.*A great, thorough introduction.* - Hughes and Piper.
**Projective Planes**.*A graduate text, so enter at your own risk.* - Samuel.
**Projective Geometry**.

- Thurston,
**Hyperbolic Geometry and Low-dimensional Topology**

- Hahn,
**Complex Numbers and Geometry**

- Silverman and Tate,
**Rational Points on Elliptic Curves**.*A fabulous book on the boundaries between algebraic geometry and number theory* - Bix,
**Conics and Cubics** - Reid,
**Undergraduate Algebraic Geometry** - Kirwan,
**Complex Algebraic Curves**.*Uses a manifolds and topology approach.* - Cox et al,
**Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra**.*An introduction to algebraic geometry via computational algorithms and Gröbner bases*

- Cofman,
**What to Solve? Problems and Suggestions for Young Mathematicians** - P'olya,
**How to Solve it** - Larson,
**Problem-Solving Through Problems** - Gilbert, Larson and Mark Krusemeyer,
**The Wohascum County Problem Book**.*That's our Mark!* - Shklarsky et al,
**The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics** - Yaglom and Yaglom,
**Challenging Mathematical Problems with Elementary Solutions, vol. 1** - Holton,
**Let's solve some math problems** - Gardiner,
**The Mathematical Olympiad Handbook: An Introduction to Problem Solving Based on the First 32 British Mathematical Olympiads 1965--1996**.*Nice book, providing a brief introduction with some methods and giving both hints and full solutions for all of the problems. He's been involved in Olympiad training and mathematics education in England for many years.*

- Garnier and Taylor,
**100% Mathematical Proof**.*Good intro to logic in math proofs and simple proof techniques. If you're shaky on how to prove things rigorously through logic, this is a good place to start.* - Solow,
**How to Read and Do Proofs** - P'olya,
**Mathematics and Plausible Reasoning**.*A somewhat more difficult book on logic and proofs*

- Hofstadter,
**Gödel, Escher, Bach: An Eternal Golden Braid**.*Strange loops, strange isomorphisms (in a loose sense of the word) between music, biology, logic, computers, and other similar stuff. Won the Pulitzer Prize. A review from Meep* - Johnstone,
**Notes on Logic and Set Theory**.*Quite a terse introduction to logic, ordinal and cardinal arithmetic, but mostly well-written* - Cameron,
**Sets, Logic and Categories**.*Similar material but less terse and more intuitively presented*

- Smullyan,
**any of his books**.*If you're really into insane and sane liars and truthtellers, he's your man.* **Games Magazine**(Or**Games World of Puzzles**), Paint-by-numbers.*The cutest logic puzzles one ever did see. Interestingly, the two-color puzzles are easier than the one-color ones (once you get in the hang of things).*

- Dudley,
**Mathematical Cranks**.*Some of this is very funny indeed.* - Dudley,
**200% of Nothing**.*The very unfunny way math is misrepresented in real life.* - Huff,
**How to Lie with Statistics**.*An old book, but much of what is described inside still goes on with reporting of statistics. A review from Meep*

- Martin Gardner,
**Gotcha!***Easy to read and contains descriptions of many of the important paradoxes.* - Barwise and Etchemendy,
**The Liar: An Essay on Truth and Circularity**.*Much more advanced book dealing with Russell's paradox.* - Stalker, ed.
**Grue!**.*Collection of articles on Hempel's paradox -- a paradox of scientific (as opposed to mathematical) induction. Rather difficult.*

- Donald Knuth,
**The T**_{E}Xbook*The definitive guide to the T*_{E}X system, written by its author. This describes Plain T_{E}X, not L^{A}T_{E}X, but is the best book for learning how to write complex macros. - Leslie Lamport,
**L**.^{A}T_{E}X: A Documentation Preparation System User's Guide and Reference Manual*Written by the author of the L*^{A}T_{E}X system (which is a set of macros which lives above T_{E}X) - Kopka and Daly,
**A Guide to L**.^{A}T_{E}X*Another well-written introductory book.* - Michel Goossens et al,
**The L**.^{A}T_{E}X Companion, The L^{A}T_{E}X Graphics Companion, The L^{A}T_{E}X Web Companion*Three books describing add-on packages available for L*^{A}T_{E}X for different uses. **The TUG (T**http://www.tug.org/_{E}X Users' Group) website,*This is also a useful source of information. It contains links to CTAN (the Comprehensive T*_{E}X Archive Network), a repository of hundreds of L^{A}T_{E}X -related pieces of software and complete T_{E}X distributions.

- Kernighan and Pike,
**Programming in the UNIX Environment**.*They are among the group who created UNIX. While it's an old book and doesn't have any of the up-to-date snazzy commands or anything about the X Windows system, it teaches the philosophy of UNIX and how to make the most out of it better than any other book I've seen without getting lost in myriad details. It gives the basis necessary for going onto anything else. The second part of the book talks about programming in C, which may or may not be of interest.* - Welsh,
**Running Linux***A nice book on installing Linux and using it. (I was co-poobah of the computer club with Matt. He really knows his Linux -- Meep)* - McCarty,
**Learning Debian GNU/Linux**.*Debian is Julian's favourite distribution of Linux; it is a volunteer effort and he is one of the volunteers. This book is available online at http://www.oreilly.com/catalog/debian/chapter/, more information on Debian is available from their website, http://www.debian.org/*

- Fritsch & Fritsch,
**The Four-Color Theorem**.*Quite an advanced book; it expends a significant amount of effort developing the problem from a topological perspective, finally moving into graph theory and describing the nature of the actual proof.*

- Taylor,
**Math and Politics** - Saari,
**Chaotic Elections! A mathematician looks at voting**

- Abbott, Edwin and Burger, Dionys,
**Flatland and Sphereland**.*Flatland is a classic; Sphereland is for those who thought Flatland was not wild enough.* - Gamow,
**Mr Thompkins in Paperback**.*See what happens when you have quantum effects on pool tables or the speed of light is 65mph (so no more need for state troopers)* - Herbert,
**Quantum Reality: Beyond the New Physics**.*Extremely easy to read for a book on metaphysics* - Feynman,
**The Character of Physical Law**.*For the man on the street. All of Richard Feynman's books are great.* - Rucker,
**The 4th Dimension**.*I can see it! I can really see it!*

- Lang,
**Challenges**.*A mathematician reads the newspapers and gets into politics. A disturbing book by a very interesting person. (He was a visiting speaker at MC2K, and had a great time with Sanjoy Mahajan.)*

**Scientific American**.*Ian Stewart used to write the math column there, but now there's a monthly puzzle and sometimes articles on developments in math in the magazine.***American Mathematical Monthly**.*Journal published by the Mathematical Association of America. Has problems and articles on undergraduate-level math.***Mathematics Magazine**.*Also by the MAA.***Math Horizons****Crux Mathematicorum with Mathematical Mayhem**.*Dave sometimes writes for this.***Quantum**.*High-school level math magazine. Neat puzzles and problems.*

*A CD with all of the CoRT material is available; an order form can be found on http://www.edwdebono.com/. Edward de Bono has also written dozens of books on the nature of thinking, the skill of thinking and other related subjects ranging from very original to somewhat repetitive; in particular,***Six Thinking Hats**presents a very powerful technique for running meetings and discussions highly effectively (assuming that all of the participants are willing to follow the system).- Tony Buzan
*has written several very good books on mind skills. In particular,***Use Your Head**,**Use Your Memory**and**The Mind Map Book**contain amazingly powerful ideas. (The former appears to have a different title in North America:**Make the Most of Your Mind**.)

Meep, last updated Aug 2001