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I'm wondering if there are any books or broad survey articles that survey current mathematical areas, i.e. mathematics of the last 50 years.

Ideally, these would continue in the style of the various classics that survey mathematics, but do so from "the ground up" and so only reach the advanced undergraduate curriculum (topology, PDEs, algebraic geometry, number theory, calculus of variations, optimisation, analysis, abstract algebra, galois theory, functional analysis, etc.)

These existing survey books I've seen typically bring one up to the edge of 19th century and early 20th century mathematics. For example, the surveys by Kolmogorov, (Mathematics: Contents, Methods & Meaning), the VNR Encyclopaedia of Mathematics (Gellert,, or Klein's (Advanced Mathematics from an Elementary Point of View).

I'm now looking for something that ideally would pick up the thread beyond these, starting in the 1930s and 40s, to the present.

It is fine if the audience is expected to have an undergraduate degree in mathematics or even a graduate degree.

The intended audience would perhaps not be a complete beginner or high school student, but perhaps a mathematician specialising in one particular area but interested in the breadth of the field, or a statistician / computer scientist / engineer / physicist, with the intent to familiarise the reader with the breadth and directions of current active areas of investigation / research.

Any particularly well-written books / articles you have come across?

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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: Thanks -- just checked it out using Amazon's "Look Inside!" feature Princeton Companion to Mathematics--- perfect --- exactly what I was hoping for. If you want to put your one liner into an answer, happy to accept this and close the topic. – Assad Ebrahim Feb 2 '14 at 3:01
up vote 5 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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