Just going to be mercifully brief with this one. The notes at the end of the book say this:
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.
They also say this:
Whatever else it may be, it is in no way, shape, or form a scientific treatise.
Yet anyone who opens the book will notice quite enormous equations, and proofs (or at least proof outlines) spanning multiple pages. He or she will also notice long transcripts of email exchanges containing — and this must be a first for any book not written by Leslie Lamport or Donald Knuth — lots of un-rendered TeX source code.
I don’t know who the intended audience for this book is. I don’t know who the intended audience for this book could be. Does anyone want to read emails with un-rendered TeX? Mathematical amateurs will get virtually nothing out of this book. Mathematical professionals, as mentioned in the note I quoted, will get very little out of this book. The book itself doesn’t even know who its audience is. It gives brief, mildly fun biographical sketches of brilliant mathematicians, which suggests that maybe the book is intended for non-professionals. But the volume is filled with so much dense mathematics that’s only professionals could read it. Complicated mathematical concepts are used, but then mostly explained many pages after they’re first mentioned.
The book is mostly an exercise for the author, who recently won a Fields Medal, to talk about how clever he is in the guise of just showing “a day in the life” of a professional mathematician. So on page 15, Villani grants us the favor of letting us listen to this conversation with his colleague:
“Violent relaxation, Cédric, is like Landau damping. Except that Landau damping is a perturbative regime and violent relaxation is a highly nonlinear regime.”
These sentences don’t make any sense to me. The author has done nothing to ensure that they’ll make sense to anyone apart from, I suppose, specialists within his particular corner of mathematics.
The book generally is supposed to show the thought process that led the author to prove something about Landau damping, whatever that might be. So it’s got diary entries, emails between him and his colleague, and scenes from the author’s life with his family (including recounting stories he tells his kids). So it feels like a bunch of hastily bolted together snippets that are supposed to form a book. If I had to guess, I would assume that his Fields Medal brought him a lot of press in France (which I’m told honors scholars much more than the United States does), and that his publishers decided to capitalize on it by throwing this out to the world as quickly as possible.
All the paper copies of this book should be pulped, and the hard drives on which the electronic copies exist should be subjected to strong magnets.