Unexpectedly, it turned out that this year, I gave twice an introductive mini-course on logarithmic Sobolev inequalities, for distinct audiences, at Institut Henri Poincaré, in April, and at École normale supérieure, in December. Here are the short notes : PDF.
There is something fun about it. I like very much to connect logarithmic Sobolev inequalities to the work of Boltzmann. This is not what I have learned, since my teachers Dominique Bakry, Michel Ledoux, and Laurent Saloff-Coste were not particularly versed in or attracted by Physics. Meanwhile, Cédric Villani, who knows very well Boltzmann, is a limited probabilist. Personally, I like very much the Physics of Boltzmann, however, I am not excited and seduced by the technical mathematics that have been constructed around his evolution equation. I have discovered that many of the mathematicians specialists of this equation are not even aware of what is really a Boltzmann-Gibbs measure. All these facts are striking effects of hyperspecialization in science. A somewhat similar phenomenon can be observed for the Schrödinger equation, which has a deep probabilistic and physical content, but which is studied by several high profile mathematicians from the mathematical analysis perspective, with often a very limited culture in probability and even in Physics. All this should be seen as a motivation to give more introductory talks, across fields, to reduce the effects of hyperspecialization. At least this is what I try to do !
In a way culture is the result of curiosity and memorization. A limited culture comes from a limited curiosity and/or memorization. Curiosity is instinctual, takes time, requires effort, and implies a sort of intellectual appetite for danger, since it could force to revisit some foundational certainties. Also it is often replaced by a sort of contempt, hiding an intellectual laziness. These mechanisms can operate at various scales/scopes, with various intensities, and vary over time.
Even if it can help, a pluridisciplinary culture is neither necessary nor sufficient for being deep and creative, it is something else. Curiosity can be even defocussing, time consuming, and counterproductive in the short-term. This plays into the game of hyperspecialization, which is essentially the result of a sort of short-term utilitarianism, something unconscious which is implicitly encouraged by the selective educational and academic systems. However, an advantage of culture is that it always helps to be a better teacher and guide for young people.
Everything I’ve ever known is wrong
Oh, what’s the matter with me?
Did I even want it?
Did I just assume that’s how it had to be?Throw a ball, it’s bouncing off the wall
That’s how I heard music first
Did I really need it?
How can something so free feel so rehearsed?Calculate the entropy
Running out of energy
A lack of love or empathy
Leave me lonely
Calculate the entropy
Bitter path in front of me
Oh neither friend or enemy
Leave me lonelyEvery day I let the music play
It seems to play by itself
Did I let it bleed me?
Did I hit it back until it bled itself?But Daddy I Love Him
Taylor Swift
I Can Do It With a Broken Heart
Taylor SwiftHow the birds can sing a tuneless song?
How can they stay in the sky?
Maybe they’re just screaming
Maybe it’s not music and it’s all a lieCalculate the entropy
Running out of energy
A lack of love or empathy
Leave me lonely
Calculate the entropy
Bitter path in front of me
Oh neither friend or enemy
Leave me lonelyE-eh, e-eh, e-eh, eh
E-e-eh, eh
E-eh, e-eh, e-eh, eh
E-e-eh, ehCalculate the entropy
Running out of energy
A lack of love or empathy
Leave me lonely
Calculate the entropy
Bitter path in front of me
Oh neither friend or enemy
Leave me lonelyKeep me lonely
Keep me lonely
Keep me lonely
Keep me lonely
Keep me lonely
Keep me lonely
Keep me lonely
Keep me lonely
PS (January 2025) : Cédric Villani has just posted on arXiv a course on Fisher information in kinetic theory, in which he makes the effort to start from the maximum likelihood of Ronald Aymler Fisher in Statistics, and in which he uses crucially a logarithmic Sobolev inequality, on the real projective space, for the Boltzmann equation. Thank you and bravo, Cédric !