Recommended Reading

Sometimes my students ask me for recommendations of texts on certain topics. So here are my answers, in advance (of the next request).

The algorithm used to select these is complex and time-dependent. The style of writing is an important factor, occasionally outweighed by the need to provide a reference.

Combinatorics

• P. Delsarte: An Algebraic Approach to the Association Schemes of Coding Theory (Phillips Research Reports 1973). [on Bill Martin's web site (with permission).]
• J. Matousek: Using the Borsuk-Ulam Theorem (Springer 2003).
• Terence Tao and Van Vu: Additive Combinatorics (Cambridge 2006).
• M. Pollicott and M. Yuri: Dynamical Systems and Ergodic Theory (Cambridge 1998).
• G. M. Ziegler: Lectures on Polytopes (Springer 1995).

By way of explanation, Pollicott and Yuri provide a full proof of Szemeredi's theorem. Learning about dynamical systems and ergodic theory is just icing on the cake.

Algebra

• D. Eisenbud: Commutative Algebra (Springer 1995).
• Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Elena Udovina, Dmitry Vaintrob: Introduction to Representation Theory [arXiv 0901.0827]
• R. Lidl and H. Niederreiter: Finite Fields (Cambridge 1997).
• Dino Lorenzini: An Introduction to Arithmetic Geometry (AMS 1996).

It won't be quick, but if you work through Etinghof and tribe, you'll be really well prepared. (I used to recommend Drozd and Kirichenko: Finite Dimensional Algebras.) Lorenzini treats algebraic number theory and the theory of curves in parallel; I find that this works really well. Eisenbud is the standard introduction to commutative algebra. There is also an abridged version of Lidl and Niederreiter, unfortunately the abridged stuff is what you may need.

Linear Algebra

• D. Farenick: Algebras of Linear Transformations (Springer 2001).
• Peter Lax: Linear Algebra (Wiley 1996).
• V. V. Prasolov: Problems and Theorems in Linear Algebra (AMS 1994).
• T. Kailath: Linear Systems (Prentice-Hall 1980).

Farenick and Lax provide beautifully written introductions. Farenick takes a non-traditional viewpoint, and could serve as a warm-up to a course in non-commutative algebra. Lax offers something more like a traditional approach to linear algebra, but still includes many useful and unusual insights. Prasolov provides a comprehensive overview in a pleasingly concise fashion.

Unfortunately none of these books considers linear algebra over finite fields. Control theory is a subject which provides an extraordinary range of applications of linear algebra, and is unjustifiably disregarded in almost all linear algebra texts. There are numerous books on control theory, but nearly all of these are written by and for engineers. Kailath's treatment is the most accessible I have found.

Permutation Groups

• H. Wielandt: Finite Permutation Groups (Academic Press 1964).
• Peter Cameron: Permutation Groups (Cambridge 1999).
• John D. Dixon and Brian Mortimer: Permutation Groups (Springer 1996).

Peter's book is well written, topical and interesting. (What else would you expect?) Dixon and Mortimer provide a very different but equally valuable treatment. If you want to actually work with automorphism groups of graphs, Wielandt offers a wealth of useful tools which you can either learn here, or be forced to reinvent.

Finite Geometry

• Simeon Ball and Zsuzsa Weiner: Introduction to Finite Geometry. (2007)
• A. Beutelspacher and U. Rosenbaum: Projective Geometry (Cambridge 1998).
• H. Luneburg: Translation Planes (Springer 1980).
• P. Samuel: Projective Geometry (Springer 1988).

Luneburg's book is dense and technical in (most) places, but it covers much more than just translation planes.

Quantum Stuff

• L. D.Faddeev and O. A, Yakubovskii: Lectures on Quantum Mechanics for Mathematics Students (AMS 2009)
• N. David Mermin: Quantum Computer Science (Cambridge 2007).

I recommend Mermin's book very highly - if you read it carefully you will some understanding of quantum physics. There are also books by Kitaev, Shen, Vyalyi and by Kaye, LaFlamme, Mosca. These are good too and, like Mermin's, also weigh less than your laptop.

Other Mathematics

• Michael Barr and Charles Wells: Toposes, Triples and Theories (Springer 1985). [on line]
• Daniel Bump: Lie Groups (Springer 2004).
• A. Hatcher: Algebraic Topology (Cambridge 2002). [on line]
• T. Korner: Fourier Analysis (Cambridge 1989).
• F. William Lawvere and Robert Rosebrugh: Sets for Mathematics (Cambridge 2000).
• Peter Lax: Functional Analysis (Wiley 2002).
• Joseph H. Silverman and John Tate: Rational Points on Elliptic Curves (Springer 1992).

This list is basically books I think are really good but don't fit in any of the other sections.

Programming

• Abelson and Sussman: Structure and Interpretation of Computer Programs, 2nd Edition (MIT Press 1996).
• R. Bird and P. Wadler: Introduction to Functional Programming (Prentice-Hall 1988).
• Jennifer Niederst Robbins: Learning Web Design (O'Reilly 2007).