Berkeley Book List: Mathematics

Vaughan Jones, professor, Mathematics A Course in Arithmetic, Jean-Pierre Serre, Springer Verlag, 1996 Characteristic Classes, John Milnor and James Stasheff, Princeton University Press, 1974 Algebra, Serge Lang, Addison-Wesley, 1992 A Course of Modern Analysis, E.T. Whittaker and G.N. Watson, Cambridge University Press, 1997 Lie Groups and Lie Algebras: Chapters 4-6 (Elements of Mathematics), Nicolas Bourbaki, Springer Verlag, 2002 Concrete Mathematics, A Foundation for Computer Science, Ron Graham, Donald Knuth and Oren Patashnik, Addison-Wesley, 1994 Non-Commutative Geometry, Alain Connes, Academic Press, 1994 Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, Michael Reed and Barry Simon, Academic Press, 1980 The Analysis of Linear Partial Differential Operators 1, Lars Hormander Knots and Links, Dale Rolfsen, American Mathematical Society, 2004 Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo, Prentice Hall, 1976

The books in my office are mostly classified by subject but there is a special section, which resides next to my armchair, which I call my "good books" section. It contains about 30 books. I love them all and it has been a challenge to pare down the list to the 10-well sorry, 11, I give below, in no particular order. These are books the mathematician reads so will not be accessible to those without a certain amount of undergraduate mathematical training.

Jean-Pierre Serre. A course in Arithmetic. Serre is a masterful expositor, besides being a great mathematician (the first winner of the new "Abel Prize"). Any book written by Serre is a masterpiece. This one can double as a party trick-just leave it lying around in the living room and watch the face of the non-mathematician as he picks it up to browse and is at quadratic reciprocity already on page 4.

John Milnor and James Stasheff. Characteristic Classes. A must-read for anyone who wants to understand the geometry and topology of the second half of the 20th century. As a bonus it contains an accessible treatment of some gnarly basic problems in singular cohomology, and a beautiful section on connections and Chern classes.

Serge Lang. Algebra. An advanced undergraduate text in algebra which was particularly influential for me as it provided my first contact with the "modern" way of talking mathematics, with its exact sequences and commuting diagrams. After this book you can certainly fool people into thinking you know what you mean by the terms "natural" and "functorial" and even "canonical".

E.T. Whittaker and G.N. Watson. A Course of Modern Analysis. A classic-this book has it all. Functions analytic, elliptic, Besselian, with tons of tripos questions. Massive information density and a line of first resort for tricky problems in analysis.

Nicolas Bourbaki. Groupes et algebres de Lie, IV,V and VI. Nicolas Bourbaki is a consortium of French mathematicians who set out to put all of mathematics together in a coherent series of books. This is reputed to be his finest. Here you will find Coxeter groups, root systems and all kinds of attendant mathematics presented with alarming elegance and attention to detail.

Ron Graham, Donald Knuth and Oren Patashnik. Concrete Mathematics. A very user-friendly grab-bag of results of a combinatorial nature. A highly accessible treatment of hypergeometric functions and lots of special numbers and tricks. And don't miss all those wonderful quips in the margin.

Alain Connes. Non-Commutative Geometry. Non-commutative geometry is very much the rage these days from string theory to number theory. Every mathematician will find something stimulating and/or upsetting in this book. Parts of it are not an easy read but the overall impression of glimpses of a grand new world is worth the effort.

Michael Reed and Barry Simon. Functional Analysis. The first of a series devoted to mathematical physics, this book serves as a perfect stand-alone course in functional analysis. You come out of this book with an attitude. Brilliant exercises and end-of-chapter notes.

Lars Hormander. The Analysis of Linear Partial Differential Operators I. I can't say I read this one very often but there it stands. A slightly frightening monument of knowledge and technique all of which is no doubt indispensable for the analysis of differential equations.

Dale Rolfsen. Knots and links. A rough diamond which somehow manages to teach, with words and pictures, the conceptual contortions required to get started in three dimensional topology.

Manfredo P. Do Carmo. Differential Geometry of Curves and Surfaces. A delightful undergraduate account of differential geometry in low dimensions. Multivariable calculus in action. Learn about the osculating plane of a curve and the intrinsic and mean curvature of surfaces according to Gauss.

About Vaughan Jones
Professor Vaughan Jones of the Department of Mathematics is a 1990 Fields Medalist, the mathematics equivalent of a Nobel prize winner. A member of the National Academy of Sciences, he is best known for his contributions to knot

theory, which have had major implications for biologists researching DNA as well as for quantum physicists. He gained international acclaim in 1984 when he discovered the Jones polynomials, which provide a list of numbers that help characterize a given knot. Knots have fascinated mathematicians for centuries, and Jones' discovery proved to be the best method to date for determining whether two complex knots that look very different are really the same. The Jones polynomials have proven useful in analyzing the twists and turns of DNA as well as the intricate mathematics of statistical mechanics, and have provided insight into the "string" theories of quantum mechanics.

A native of New Zealand, Jones received his BSc from the University of Auckland and his Docteur ès Sciences from the Ecole de Mathematiques in Geneva, Switzerland. He has been on the mathematics faculty since 1985. For more, see Jones' faculty website.