I almost did not bother to read most of Birth of a Theorem: A Mathematical Adventure, by Cédric Villani. After reading several chapters, it was clear that I wasn't going to understand the mathematics in this book, and via Twitter I came across some negative reviews that emphasized the book's inclusion of incomprehensible material. Moreover, in keeping with my recent hobby of studying math, I've been very much in the mode of wanting to actually do math, rather than just observe it in some vague way. But I plowed ahead with Villani's book, and I am certainly glad I did.
Birth of a Theorem gives a compelling and personal picture of what it is like to do math at an extremely high level; for example, to think you've solved a longstanding problem and then find that you haven't, or to wake up with a momentous realization that "You've got to bring over the second term from the other side, take the Fourier transform, and invert in L2"--and then write a note on a scrap of paper before rushing to get the kids dressed and onto the school bus.
Some pages of the book are laden with equations, and a note on translation at the end states: "No attempt has been made to expand upon, much less to explain, fine points of mathematical detail, many of which will be unfamiliar even to professional mathematicians. The technical material, though not actually irrelevant, is in any case inessential to the story Cédric Villani tells in this book."
I would have preferred it if, at some point, there had been a diagram with annotations summarizing, term by term, what a key equation means. As it was, though, I had some fun picking out the symbols I did understand--an ∈ here, an ∀ there, a sup somewhere (all of which I learned fairly recently), and I agree that this book tells a valuable story even while displaying so much unexplained math. The only pages that I didn't find interesting were ones devoted to a long listing of musicians and bands the author likes. (Some readers may like this part, however, especially if they were attracted to the book by Patti Smith's blurb on the back cover.)
So, Birth of a Theorem is recommended. It is a very different offering from Edward Frenkel's Love and Math: The Heart of Hidden Reality, which strives to make some very difficult math comprehensible to a lay audience. Still, I suspect that some people will pick up Villani's book and end up being drawn further into mathematics, as well.