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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 © Fusina Dominik.
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 story opens in the laboratory of the Ecole Normale Superiere de Lyon on a Sunday afternoon. Mouhot and Villani are in the latter’s office to discuss prospects for the Boltzmann equation, which describes the statistical evolution of a colliding gas. By the end of the conversation and several pages later, Villani has reconstructed a positivity result for a collisionless gas, which he had heard about from a postdoctoral fellow at Princeton University. Mouhot wonders aloud whether the argument might be relevant to some controversies concerning  the longtime behavior of a charged plasma interacting electrodynamically. This is followed by a two page aside on Ludwig Boltzmann, which displays his equation and his entropy law and relates the two by observing that the latter behaves monotonically along the dynamics of the former (exhibiting one particular mathematical incarnation of the second law of thermodynamics). The aside describes its subsequent impact and recalls some tragic details from Boltzmann’s life. There is also a depiction of Boltzmann, the first of a dozen or more sketches produced by Claude Gondard that tie together Villani’s story and illustrate many of its principal characters, living and dead. The sketches provide a sense of historical evolution in mathematics, portraying great minds revisiting persistent questions century after century, building on each other’s insights to peel back layers of some of the most profound mysteries in science. For example, why did structure emerge in the universe, and on what time scale?

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.

Their combined eventual resolution of this problem was among the principal contributions that garnered Villani his Fields Medal. (For those who want to know the answer: they show that the nonlinear equation indeed produces the local damping effect, but only for analytic or near-analytic initial data. For such data, the dynamics smooth the spatial density while producing oscillations in the velocity variable; the convergence back to equilibrium occurs in a weak topology which does not see these oscillations.)

However, the book attempts to trace the actual, meandering trajectory Villani traversed during the years he spent searching for this result, the many random (and some less random) encounters which shaped his thinking on this problem. There are certainly excursions into a variety of far-flung mathematical realms, including questions in number and graph theory, which have no relation to Landau damping. But there are equally many excursions to faraway conferences, children’s music lessons, a semester at the Institute for Advanced Study, a change in employer, and explorations of the author’s taste in poetry (e.g. William Blake), music (e.g. Catherine Ribiero), and books (e.g. Japanese comics). Also reproduced throughout the text are email correspondences from the period in question – many between Villani and his collaborator, Mouhot, but also others tangential to Villani’s story (including one from yours truly). Further additions include pages from a few different math papers, excerpts from Villani’s book on optimal transport, a figure illustrating one of the reviewer’s theorems, and slides from a lecture on an unrelated topic. Birth of a Theorem is a charming insider’s guide to a mathematician’s world, meant to convey the mystery, excitement, agony, ecstasy, and quotidian that seduce mathematicians into the profession. Designed to be accessible to outsiders, the original was a bestseller in France, titled Théoréme Vivant, meaning Living (the) Theorem or Theorem Alive! Insiders will find it delightful as well. Birth of a Theorem offers readers not only the chance to tag along with a future recipient of the Fields Medal as he does some of his best work, but also a delightful glimpse into the foreign land in which mathematicians seem to spend so much of their time.

Robert McCann (mccann@math.toronto.edu) is a professor of mathematics at the University of Toronto. He is the originator of displacement convexity and an authority on optimal transport. He has also co-authored publications on kinetic equilibration in granular media with Cédric Villani.