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Are there any advanced topics--preferably at the research frontier--which can nevertheless be explained accurately, if not efficiently, using very down-to-earth ideas which would be accessible to most college students, not necessarily even math majors?

As a prototypical example of what I mean, I offer the book QED by Feynman, which explains the ideas of quantum electrodynamics without even using the word "complex number." In mathematics, the book Naive Lie Theory by John Stillwell comes close to what I'm looking for. Visual Complex Analysis by Tristan Needham is a pseudo-example, the main issue being that its subject matter is very standard undergraduate material.

Of course, any such explanation cannot be a fully faithful representation of the original work, but ideally it would still be possible to convey the essence of the methods without introducing any real inaccuracies.

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Depending on your definition of the word "accurately"... there are some recommendations here: mathoverflow.net/questions/8609/favorite-popular-math-book –  Qiaochu Yuan Dec 4 '10 at 17:59
By accurate I mean more or less that false statements aren't made. Some popular treatments attain their accessible level by communicating half-truths. Again, Feynman's QED is a good example of what I mean: it doesn't cover the details, and it won't teach you how to do the calculations yourself, but it does communicate the ideas of the subject, it does explain some of its more striking applications, and it doesn't make false statements. –  Zach Conn Dec 4 '10 at 18:12
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up vote 5 down vote accepted

I can think of a couple of books that may be close to what you claim to be looking for. One is Fearless Symmetry by Avner Ash and Robert Gross.

They try to "explain" (without entering in the details of course) the general ideas behind Wiles' proof of Fermat's Last Theorem. So they begin in chapter 1 with an intuitive explanation of the purpose and aim of the idea of a representation and follow their way through elliptic curves, reciprocity laws and even Galois Representations.

This is certainly a book that "explains" very advanced stuff, but that is aimed at a non-specialist audience. It is supposed to be a popular math book, but I'm not sure how accurate that is since it is not that easy to digest if you don't have at least some undergraduate mathematics under your belt. At least that's how I see it.

The other book I have in mind is Visual Group Theory, but this is closer to the spirit of Visual Complex Analysis, I suppose, and as in that case it explains standard undergraduate mathematics.

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I have heard good things about Fearless Symmetry from one of my colleagues, who wrote a review for it. It seems to include about an order of magnitude more mathematical content than most books in that genre, while still being reasonably accessible to a general audience. –  Pete L. Clark Dec 4 '10 at 18:39
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