My Favourite Math Books

• Paul M. Cohn Algebra 1-3 John Wiley and Sons, New York, 1989
  Light, readable, informative- everything a good book should be. Taken alongside Jacobson's Basic Algebra, this book could form the core of the ideal graduate-level Algebra course I think. It's a shame that this book is not better appreciated.

• David Eisenbud Commutative Algebra with a View Toward Algebraic Geometry Springer-Verlag, New York, 1995
  The definitive book on Commutative Algebra. This book is well-organized, well indexed, and somehow manages to contain all the essential information that one wants. How does Eisenbud do it? This book is as essential a reference text to an Algebraic Geometer as a dictionary is to a translator.

• Sergei I. Gelfand and Yuri J. Manin Methods of Homological Algebra Springer-Verlag, 1999
  Derived Categories, Triangulated Categories, all that good stuff, presented just the way you want it. This book is not an easy read- you have to work to understand it, because the exposition is perhaps a bit condensed for my taste, but it is worth every minute! You actually feel your brain expanding when you read this.

• Nathan Jacobson Basic Algebra II W.H. Freedman and Company, San Francisco, 1980
  Classical and brilliant, but perhaps somewhat dry. The right way to read this book, I find, is to read it alongside another book with a lighter exposition, so that the other book gives you the idea, and Jacobson's book allows you to take the next step and dive into Non-Commutative Algebra's murky depths. In this role, Jacobson's Basic Algebra II is unsurpassed. But don't let the word "basic" in the title fool you! "Basic" it ain't.

• Jacobus H. van Lint and Richard M. Wilson A Course in Combinatorics Second Edition, Cambridge University Press, 2001
  A well-presented, readable and easily searchable account of all the basic results that you want to know in Combinatorics. Despite lacking depth, it easily eclipses any other Combinatorics book around today.

• Jean-Louis Loday Cyclic Homology Second Edition, Springer-Verlag, 1998
  The definitive book an Cyclic Homology. Readable and fascinating- but hard. If the exposition was spruced up a bit to motivate all the formalities, this could be my favourite book of all time. A shame really.

• Shigeyuki Morita Geometry of Characteristic Classes Translations of Mathematical Monographs. Iwanami Series in Modern Mathematics. 199. Providence, RI: American Mathematical Society (AMS), 2001
  The ideal book on de-Rahm homotopy and on characteristic classes. Incredible clarity of exposition, and an exciting read even and especially if one already knows about a lot of what's being talked about. This book deserves to become a major classic.

• Jurgen Neukirch Algebraic Number Theory Springer-Verlag, Berlin 1999
  I don't know much about number theory, but this book is magical. Stylistically the best Mathematics book I have ever read.

• Alexander Polishchuk Abelian Varieties, Theta Functions and the Fourier Transform Cambridge Tracts in Mathematics 153, Cambridge University Press, 2003
  A wonderful "second book" on abelian varieties and theta functions, in which the Fourier-Mukai transform plays a leading role. The presentation is stimulating and non-standard, making for a wonderful read. This is not a book to learn from- it is a book to enjoy. And while enjoying it one may find oneself also becoming better informed.

• Douglas C. Ravenel Complex Cobordism and Stable Homotopy Groups of Spheres AMS Chelsea Publishing, 2003
  The juiciest book about Homotopy Theory around. You get a nice, readable, comprehensible account of quite a large slice of modern Homotopy Theory, including some quite important topics which are not covered by any other book, such as BP-Homology and the Adams-Novikov Spectral Sequence. And enjoyable read- a book in which a surprising amount of information is taught in a surprisingly entertaining fashion. And with BP-Homology all the rage among the youth of Japan at the moment, it's a good book for the intrepid explorer of the Orient to be aware of.

• Christophe Reutenauer Free Lie Algebras London Mathematics Society Monographs New Series no. 7, Oxford Science Publications, 1993
  Free Lie Algebras are a useful object to have in one's toolkit, and Reutenauer's book is the only accessible text explaining how to use them. It is very well written and well-organized, and manages to bring the reader from 100-year-old results to the forefront of Free Lie Algebra research circa. 1993 in an entertaining and comprehensible manner. My one complaint would be that such a book ages badly (as the forefront of research is by definition ever-changing), and a Second Edition is badly needed.

• Edwin H. Spanier Algebraic Topology McGraw-Hill Series in Higher Mathematics, 1966
  There are millions of books on Algebraic Topology, and yet Edwin Spanier's 40-year-old exposition has not been improved on to this day. Difficult to read, but an outstanding reference text, which distills basic results in one of the most important fields of Mathematics today in a way that is easy to search and easy to comprehend. This is one of those books that may still be as important in another 40 years as it is today.

• Arild Stubhaug The Mathematician Sophus Lie Springer-Verlag, 2002
  An excellent biography, the kind of book one just can't put down. There's little if any Mathematics in here, but the exposition of the life of one of the most influential Mathematicians of all time is so vivid that one may forgive this book its lack of Mathematical content.

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  Daniel Moskovich
  dmoskovich@[remove-me]gmail.com
  Department of Mathematics
  Danmarks Tekniske Universitet