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.