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This question is motivated by a little anecdote. I was at home teaching some secondary school math to a relative. At some relax time, he glanced at a book I had over the table - it was some text about analytical number theory that I had recently bought, second hand- and I explained him that that was an area of mathematics quite arcane to me, that I'd like to learn something about it in some future, but I had little hopes...

He looked puzzled for a moment, and then asked me: "But, wait a moment... You don't know all the mathematics?"

This happened months ago, and I'm still laughing. But also (and here comes the question) I'm still thinking about how to make some picture of this issue: the big extension of "the land of mathematics", in diverseness and ranges of depth - and the small regions that one has explored.

I was specifically looking for some kind of bidimensional (planar?) chart, perhaps with the most basic/known math kwowledge in the center, and with the main math areas as regions, placed according to its mutual relations or some kind of taxonomy.

(I guess this should go in community wiki)

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I seem to remember a graph which attempted to picture mathematical activity by area, with larger circles representing more activity. But I can't seem to locate it... – Arturo Magidin Jan 19 '11 at 22:04
@Arturo: Perhaps you mean this: – PEV Jan 19 '11 at 22:10
One thing that would be nice to do sometime is take all math papers in the arXiv, interpret the mathematical subject classifications as hex numbers, and produce a density plot of research interest over time. Unfortunately the arXiv doesn't like robots much so I don't see this happening anytime soon. – deoxygerbe Jan 19 '11 at 22:21
The chart of interrelationships between mathematical subjects cannot be plotted as a planar graph, I suspect. – Willie Wong Jan 19 '11 at 22:56
@PEV: Yes, that's the one I was thinking of. Thanks! – Arturo Magidin Jan 20 '11 at 2:35
up vote 7 down vote accepted

The Princeton Companion of Mathematics is a good resource. The mathematical atlas is good as well. The size of the bubbles are directly proportional to the amount of research activity in each area. This MO post might be useful also (looking at real world applications of arxiv areas).

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The Princeton Companion is a great resource, but unless you consider its Table of Contents a map... – Qiaochu Yuan Jan 19 '11 at 22:11

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