Since meeting the Wasserstein metric during my PhD journey, I knew about the 1998 paper “The Variational Formulation of the Fokker-Planck Equation” by Richard Jordan, David Kinderlehrer and Felix Otto, thinking this pioneering article lives in its own little corner of mathematics surrounded by Wasserstein-2 gradient flow enthusiasts. Few mathematicians I talked to outside of this bubble seemed to have ever heard of a JKO scheme. Little did I know that this gem is the single most downloaded article of the SIAM Journal on Mathematical Analysis since its 50 volumes of existence. Wow.
Why is it so popular? It describes the gradient flow structure of the Fokker-Planck equation using an implicit time discretization in the form of a sequence of variational problems, called “minimizing movements”. In doing so, it builds a bridge between geometric analysis, optimal transportation, quasistatic dissipative evolutions, applied partial differential equations and rational mechanics.
The 10 most downloaded articles from the SIAM Journal on Mathematical Analysis. (source here)
For a short summary of the paper and why so many people are interested in it, have a look at this recent article in SIAM NEWS by Felix Otto, mentioning at least 10 mathematicians I’ve had the chance to personally interact with.
Army ants form colonies of millions and are able to solve logistical challenges as a group that are impressive in and by themselves, but even more so considering they have no leader and minimal cognitive resources. One individual ant is practically blind and has a minuscule brain. “There is no leader, no architect ant saying ‘we need to build here,’” says Simon Garnier, director of the Swarm Lab at the New Jersey Institute of Technology. And yet, they manage to perform complex tasks as a colony such as building bridges with their own bodies to overcome obstacles. How do they do it?
Here and here you can find a more detailed explanation to this question, including videos of bridge building challenges for army ants. In short, they follow three simple rules:
- If you come to a gap in your path, slow down.
- If you feel other ants walking on your back, freeze.
- If traffic over your back is above a certain level, stay put, but if it dips below some threshold, unfreeze and continue walking.
I have a feeling there is a PDE involved here…
A bridge of ants from the side. Credit: Christopher Reid, Matthew Lutz & New Jersey Institute of Technology
Amazing talk about how you can test physical concepts yourself. Replace “physics” with “maths”, and the main message still holds in my opinion.
Caltech… a beautiful bubble of carefully architectured buildings that transport its people into a different world. Arrival impressions.
When dealing with gradient flows, did you ever wonder why it is better to choose a certain energy rather than another? And how the choice of energy influences the choice of metric and vice versa? Did you every ask yourself what a certain entropy or metric means in terms of modelling choice?
Well, I did, many times. And finally I came across these lecture notes by Mark Peletier who are very enjoyable to read , very understandable even for people with little background in gradient flows, and they also make the connection to the world of probability and large deviations. Enjoy!
PS: If you are keen to know the connection between gradient flows and the Poincaré conjecture, have a look at this blog post by Terence Tao. You may also get a glimpse at some of Perelman’s ideas that helped solve it.
Also check out the interview with Sylvia Servaty in QUANTA Magazine!
A mathematician’s paradise…