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.


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.


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

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

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

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

Quantum Stuff

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

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