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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,, Mathematics: Contents, Methods & Meaning,
  • Gellert,, 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!

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Not sure about your question, but are you familiar with MSC? – Amzoti Jan 30 '14 at 12:52
Yes -- that's a listing of the active areas, broken down to an extremely granular level in many cases. But that would be an excellent outline for such a summary. I'm looking for prose coverage of recent developments in these areas, preferably drawing links between them, etc. --- something in the model of the classics I mentioned. For these reasons lists such as MSC and also the detailed technical nature of most professional journals aren't much help. – Assad Ebrahim Jan 30 '14 at 14:37
You should take a look at "The Princeton Companion to Mathematics". – Michael Greinecker Feb 2 '14 at 2:37
@MichaelGreinecker 's suggestion is probably the best possible. There has been an enormous amount of mathematics done in the past 50 years. – Mike Miller Feb 2 '14 at 2:55
@MichaelGreinecker: Super excited to see now the "Princeton Companion to Applied Mathematics" (ed. Nicholas Higham) printed in 2015. – Assad Ebrahim Jul 12 at 21:41
up vote 7 down vote accepted

A fairly comprehensive survey is given in the Princeton Companion to Mathematics, edited by Timothy Gowers. The book contains a lot of material by many great mathematicians. In terms of the level, early undergraduates can benefit from the book, but probably everyone could learn something new. The only downside is that it is really, really heavy.

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This is of course personal, but there are other downsides of this book. To name just one, there is no mention of Physics in the chapter entitled "Influence of Mathematics," which is, hmm, quite strange. – Artem Feb 2 '14 at 3:23
@Artem I'm pretty sure they did this because physics is covered in many other places. General relativity has its own chapter. – Michael Greinecker Feb 2 '14 at 3:26
o, here is another downside of the book: General relativity is a branch of mathematics according to this book :) – Artem Feb 2 '14 at 3:31

These are the ones I had come across before posing the question. Though none of them were quite what I was after (see accepted answer of Princeton Companion to Mathematics), they're recorded here in case they are useful to others:

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