It's a best-seller in France, apparently: the wild-haired but immaculately-dressed Fields Medallist's story of how he and his colleague solved the Landau damping problem.
But therein lies my difficulty with it: I don't know about, or especially care about, the Landau damping problem. It's a McGuffin: it might as well be an articulate baseballer's quest for theTy Cobb prize, or an alien from planet Zorg trying to decipher strange signals from Earth. It's clearly very important to M. Villani and his colleague, and obviously rather difficult and involved -- but it's so far removed from my knowledge and interests as to be practically meaningless.
That's not to say Villani doesn't try. He recounts his conversations ("The hoojimaflip? Surely you can't mean that if we assume the thingummy is whatchamacallit, that the issue with flapdoodliness goes away?), includes his emails with his colleague Clément, the odd theorem or excerpt from a paper (context be damned) and -- showing that he's a human and not a mathematics machine -- a poem here, a story there, a piece of a prospectus there.
So, let's set aside the bits at the end of the chapters and focus on the story part. We look on as Villani hops from meeting to meeting, from visit to visit, trying to carve out time for the problem he's interested in, banging his head against the frustrations, following blind alleys, cursing Clément for finding yet another difficulty. That is something the average reader might get something out of: research mathematicians will likely empathise.
Towards the end of the book, Villani points out a media creation he called Cédricvillani, something that's not really him and that he's not really in control of. To me, that's one of the most revealing things in the book. He's at pains throughout to highlight his diverse and eclectic interests, to shower praise on his cohort of professors and colleagues, and generally try to reclaim himself for himself. As with all autobiographical work, it's rather self-serving, but I would expect the Villani living on the pages of his book is at least more like the Villani he feels he is than the 'rock star mathematician' Cédricvillani that lives in the French media.
Villani builds up a picture, in snapshots, of what it's like to be a top-rate research mathematician. Snapshotting is a style of autobiography that can be pulled off -- Dylan's Chronicles is a fine example -- but for me, Villani's pictures just don't quite hang together as a gallery.
I think Birth of a Theorem probably best compared to non-alcoholic beer: people who like beer wouldn't like it much, and people who don't like beer wouldn't like it much, either. If you're a Landau damper, or whatever the people who deal with that sort of thing are called, there may be a few morsels in here for you; I suspect that's as niche a market as that of non-alcoholic beer aficionados (any members of CAMRNAA out there?). If you're a mathematician, there's not enough to help you get get your teeth into the problem; if you're looking for a science biography, there are better-told ones out there.
* Edited 2015-11-24 to add links.