[UPDATE 2015-12-31. The good times turned out to be short-lived. Springer has now turned off open access for these books.]
Springer has made the full text of many of its math and statistics books freely available. Here are some books that I should read. Now there is one less excuse not to.
Mathematical Economics
Aubin. Optima and Equilibria. An abstract approach to the mathematics of microeconomics with an emphasis on nonsmooth analysis.
Aubin, Cellina. Differential Inclusions. Dynamics with set-valued functions.
Granas, Dugundji. Fixed Point Theory. A massive handbook of all fixed point theory you may want.
Algebra
Halmos. Finite-Dimensional Vector Spaces. A very good way to get a first taste of abstract mathematics.
Zhang. Matrix Theory. Halmos is good for aesthetics, but deep in the trenches you need weird matrix decompositions. A concise collection of useful results.
Analysis
Lang. Real and Functional Analysis. Maddening typos, but otherwise a very different presentation of graduate-level analysis.
Lang. Fundamentals of Differential Geometry Don’t know about differential geometry, but having implicit and inverse function theorems for infinite dimensional spaces is useful.
Cohn. Measure Theory. More details than the usual analysis book. This is an older edition.
Loeb, Wolff. Nonstandard Analysis for the Working Mathematician. Nonstandard analysis provides one framework for studying economies with continua of agents. The book has a chapter specifically on economic applications.
Dynamics
Aligood, Sauer, Yorke. Chaos: An Introduction to Dynamical Systems.
Probability Theory
Chung, AitSahlia. Elementary Probability Theory. A non-measure-theoretic but rigorous introduction to probability.
Kallenberg. Foundations of Modern Probability. A reference work giving a unified, elegant presentation of classical measure-theoretic probability.
Chow, Teicher. Probability Theory. A standard text. Unique for its results on martingales.
Kushner, Yin. Stochastic Approximation and Recursive Algorithms and Applications. Models of learning in game and macroeconomics are often formulated in terms of stochastic recursive algorithms. This book is a classic treatment of the so-called “ODE approach” to studying these models.
Arnold. Random Dynamical Systems.
Øksendal. Stochastic Differential Equations. If you want to study continuous time models with uncertainty you have to know about stochastic differential equations. This book attempts to teach you the basics with the least mathematical complications. Still, be prepared for a lot of math.
Karatzas, Shreve. Brownian Motion and Stochastic Calculus. At a higher level than Øksendal, referred to in a lot of the finance literature.
Protter. Stochastic Integration and Differential Equations. A non-mainstream approach to the high theory of stochastic differential equations which promises to reach useful results quickly.
Revuz, Yor. Continuous Martingales and Brownian Motion.
Statistics
Brockwell, Davis. Introduction to Time Series and Forecasting. Not much in the way of prerequisites, very clear treatment.
Wasserman. All of Statistics. A fine introduction to mathematical statistics. Few proofs but a broad coverage and modern outlook.
Schervish. Theory of Statistics. A rigorous treatment of mathematical statistics from a measure-theoretic point of view. The proofs you will not find in other places you will find here. Covers both frequentist and Bayesian statistics.
Brockwell, Davis. Time-Series: Theory and Methods. The rigorous theory.
Rosenbaum. Observational Studies. How to do statistics on non-experimental data, which is what economists most often have.
Berger. Statistical Decision Theory and Bayesian Analysis. I don’t think statistical methods can be justified without an underlying decision theory, and once you start from decision theory the natural outlook is the Bayesian one. A classic treatment.
Lehmann, Casella. Theory of Point Estimation. A classic reference work of frequentist statistics.
Robert, Casella. Monte Carlo Statistical Methods. Bayesian statistics has become popular because Monte Carlo methods have vastly expanded the range of models that can be fitted. A somewhat dated but still classic treatment.
Ghosh, Ramamoorthy. Bayesian Nonparametrics.
LeCam, Yang. Asymptotics in Statistics. LeCam is a master of this subject.
Manski. Partial Identification of Probability Distributions.
Vapnik. The Nature of Statistical Learning Theory. Vapnik’s work provides one of the foundations of modern machine learning theory. An intuitive exposition.
Venables, Ripley. Modern Applied Statistics with S. The R statistical language is a free implementation of the S language described in this book. The book is now somewhat dated, but still a good reference for R users.
History of Statistics
David, Edwards. Annoted Readings in the History of Statistics.
Kotz, Johnson. Breakthroughs in Statistics, Vol. I.
Kotz, Johnson. Breakthroughs in Statistics, Vol II.