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# Villani’s "Birth of a Theorem"

Birth of a Theorem: A Mathematical Adventure. By Cédric Villani, Farrar, Straus and Giroux, New York, 2015, 272 pages, \$26.00.

Cédric Villani’s newly-translated popular work, Birth ofa Theorem: $$\frac{a}{m}$$athematical Adv(en)ture, is a book without precedent. It was apparently born of an encounter with Olivier Nora, who suggested to Villani that, in the wake of his 2010 Fields Medal, the general public would be fascinated to have the opportunity to read anything accessible he might be able to write on the nature of his award-winning work. The author took this to mean not only the mathematical content of what was produced, but also the process by which it was achieved, including the daily life and modus operandi of a leading mathematician. In response to this challenge, Villani offers the reader a window into the three years of his life during which he collaborated with Clement Mouhot on the problem of Landau damping. The result is equal parts diary, documentary, collage, stream of consciousness, mathematical history, biography, and exposition. The translator notes that “The book is meant chiefly as a work of literary imagination...The technical material, though not actually irrelevant, is in any case inessential to the story.”
The second chapter flashes back to a lunch at Oberwolfach two years earlier, during which a conversation with a pair of experts about the Landau damping phenomenon in plasma physics piques Villani’s curiosity. The discussion concerned a paradox Landau predicted based on a linearization of dynamics. In the paradox, the Vlasov equation (which gives a statistical description of a Coulomb gas  –  either attractive in the case of gravity, or repulsive as in the case of electrically-charged particles), despite being reversible in time, possesses certain equilibria which are stable, in the sense that the equation drives nearby initial  data back to them both as $$t \rightarrow +\infty$$ and $$t \rightarrow -\infty$$.  Many experiments observed this damping phenomenon, while there were others that seemed to violate it. Conservation of the Liouville measure precludes such behavior for a finite dimensional Hamiltonian system, but the Vlasov dynamics describe an infinite dimensional (statistical) limit. During a subsequent visit to Brown University, Mouhot learned that the mathematical crux of the matter, which had never been resolved, was whether or not linearization correctly predicted the nonlinear dynamics, and if so, in what sense.