I am lucky enough to work with a math-enthusiastic bunch of folks. We often talk about great papers and books for the student of Math. Here’s my list of my favorite books from my graduate school days.

So often subjects blend that I am not going to do the usual Algebra/Analysis/Topology/etc. breakdown. Is Linear Algebra algebra or analysis? How can you tell in functional analysis or harmonic analysis? I can’t so I won’t even try! Instead I am grouping them “thematically” e.g. the way I see them fitting well together! I am only listing the exceptional books here as there are lots of books we know as classics and the list will be identical to all of those - Baby Rudin! Lang or Hungerford’s Algebra! Alfor’s Complex Analysis!

Moreover, I am linking to publisher sites rather than big vendors. You are fully capable of finding sellers.

• Linear Algebra Done Right by Sheldon Axler. This book basically punts on determinants and instead talks about the properties of operators!

• Finite Dimensional Vector Spaces by Paul Halmos. Some of the clearest and most elegant prose giving intuition to linear algebra I have ever read.

• A First Course in Wavelets with Fourier Analysis by Albert Boggess and Francis J. Narcowich. A great and comprehensible introduction to Fourier Analysis leading into a gentle building of wavelet theory.

• Fourier Analysis - An Introduction by Elias M. Stein & Rami Shakarachi. For focussed and gentle discussion of Fourier Analysis I have never found a clearer explanation than this book.

• Measure and Integration Theory by Heinz Bauer. Measure theory is pretty dry but so critical. Bauer’s exposition is concise, to the point, and builds the theory from foundations in such a way as to never leave difficult gaps.

• Beginning Functional Analysis by Karen Saxe. So many Functional Analysis books are built in exactly the same way around linear algebra and functionals and duals but Saxe gets you there with extreme grace.

• Measure, Topology, and Fractal Geometry by Gerald Edgar. This is a rare-ish book that provides a very gentle introduction to Geometric Measure Theory.

• Ideals, Varieties, and Algorithms - An Introduction to Computational Algebraic Geometry and Commutative Algebra by David Cox, John Little, and Donal O’Shea. I am not an algebraist by any stretch of the imagination but this book is such a nice introduction to commutative algebra that everyone should read and play with it at least once.

• Weighing the Odds - A course in Probability and Statistics by David Williams. A course in probability that does not simply jump into Expectation and Distributions but instead wants to lend intuition to the process.

• A Course in Machine Learning by Hal Daume III. This book, while incomplete, and I have not finished working through it, provides a great, intuitive, introduction to Machine Learning without a need for a strong probability background. Intuitive descriptions and gentle prose make for a fun read.