What books or broad survey articles survey the mathematics of the last 50-100 years?
The ones I've read do a good job conveying mathematics from the ground up but typically assume a complete beginner or high school student audience and therefore reach only as far as the advanced undergraduate curriculum (middle of the 19th century).
I'm looking for something to pick up the thread from here, starting in the mid-1800s, and continuing through the 1930s and 40s, to the present. I'd like to see solid survey coverage, in the same accessible style, of 20th century mathematics: algebraic or differential topology, PDEs, algebraic geometry, number theory, calculus of variations, optimisation, analysis, abstract algebra, galois theory, functional analysis, etc.
It would be ideal if the intended audience were presumed to have an undergraduate degree in mathematics or even a graduate degree. Even better if the audience were presumed to include mathematicians specialising in one particular area but interested in the breadth of the field, or if the intent were to familiarize a technically specialized audience of statisticians, computer scientists, engineers, or physicists, with the breadth and directions of current active areas of investigation and research in mathematics.
Any particularly well-written such books & articles you have come across?
Examples of Surveys of the first kind (not modern):
- Kolmogorov, et.al., Mathematics: Contents, Methods & Meaning,
- Gellert, et.al., VNR Encyclopaedia of Mathematics
- FeliKlein's (Advanced Mathematics from an Elementary Point of View).
Edit (July 2016):
The Princeton Companion to Mathematics (ed. Timothy Gowers) came out in 2008 and turns out is probably the best possible such reference, see the accepted answer.
Super excited to see that in 2015, we now also have the Princeton Companion to Applied Mathematics (ed. Nicholas Higham), which covers the modern aspects of applied and applicable mathematics in the same format as Gowers' masterpiece!