I’ve read a few popular maths books and I would highly recommend these ones:

  • Edward Frenkel, Love and Math is an introduction to the Langlands Program at a popular level. It’s extremely well-written and describes lots of very advanced mathematics from across the board, from Galois theory and modular forms to D-branes and mirror symmetry. Definitely a book to read to get (keep?!) you inspired to do maths.
  • Yau and Nadis, The Shape of Inner Space is a book written (in part) by Fields Medal winner Shing-Tung Yau and describes his work on Calabi-Yau manifolds and their relation to string theory and mirror symmetry. An extremely interesting read and an opportunity to see the interactions which occur between high-level mathematics and theoretical physics.

Here are some more technical books which I would recommend to anyone looking to go further:

  • Lawvere and Schanuel, Conceptual Mathematics is where I first learned the basic concepts of category theory. It teaches you how to “think categorically” (in terms of morphisms rather than objects) and presents these ideas through basic examples of categories – sets, graphs and dynamical systems. While it doesn’t get much further than explaining universal mapping properties (I believe the word “functor” is only used once in the whole book!) it certainly helps you understand things like products and exponentials well. It consists of several main technical “articles” interspersed with “dialogue” sections which were recorded from when the authors taught the book originally as a course. These bits keep you rooted in reality and help weed out any issues you have with the ideas.
  • Steve Awodey, Category Theory is the next logical step after Conceptual Mathematics. It’s a modern introduction to “proper” category theory with fewer prerequisites than Mac Lane, and will get you to a level with category theory where you can understand examples from all across mathematics categorically, which is a very enlightening experience.
  • Francis Borceux, Handbook of Categorical Algebra is where to go after this. This is an amazingly detailed and extensive three-part series on everything you could possibly want to know about category theory. I’m currently reading the first volume and learning about “localising categories”. The goal of the books is that once you’re finished reading them all, you should be just about ready to start doing research in category theory (or related areas).
  • Silverman and Tate, Rational Points on Elliptic Curves is one of the best maths books I’ve read. It introduces an important and modern subject (the beginnings of arithmetic geometry!) at a level where dedicated undergrads can read it and really feel like they have come away with some deep knowledge. I had my mind blown a few times with this book! And after reading this (plus some knowledge of commutative algebra) you can go on to read Silverman’s Arithmetic of Elliptic Curves.
  • Liu’s Algebraic Geometry and Arithmetic Curves is the principal text I am following on scheme theory. It has far fewer commutative algebra prerequisites than Hartshorne’s classic text, and is more suited for arithmetic geometry as it considers the more general case of nonzero-characteristic/non-algebraically closed fields. So far it is an excellent read and the problems are pitched at just the right level.
  • Algebraic Geometry 1 is a very readable and friendly text on schemes. I’d actually see it as probably the best one out there for learning the material for the first time.
  • Rotman’s An introduction to Homological Algebra is another friendly book where derived functors and sheaf cohomology seem way less scary.
  • Szamuely’s Galois Groups and Fundamental Groups is a fascinating and readable account of a modern and deep area of mathematics – fundamental groups of schemes.