Lecture: "The survival of basic spreading PDE models"

Professor Lenya Ryzhik is ​2022/2023 Nirit and Michael Shaoul Fellow of the Mortimer and Raymond Sackler Institute of Advanced Studies.

Abstract

The Fisher-KPP equation has been proposed, independently, by Ronald Fisher and Andrey Kolmogorov, Ivan Petrovskii and Nikolai Piskunov in 1937, as a basic partial differential equations model of an invasion in mathematical biology. It became a fundamental model for various spreading phenomena ever since, and was studied very extensively both in the PDE and mathematical biology literature. In 1975, McKean discovered an exact connection between the Fisher-KPP equation and branching Brownian motion, the basic probabilistic spreading model. This was the first of the surprising connections between this partial differential equation and the probability theory. Later, it was realized that this connects the Fisher-KPP equation to the much larger class of logarithmically correlated random fields. They appear in the seemingly disconnected contexts of random matrix theory, extrema of the Gaussian free field, and the maxima of the Riemann zeta function on the critical line, and all exhibit a certain universal behavior. Availability of the analytic methods makes the Fisher-KPP equation the easiest model to study in this class and allows to compute various universal quantities conjectured to be  common to all such models. Independently, other PDE modeling spreading appeared in such diverse problems as voting models on random trees and mean field games models of knowledge diffusion and growth in macroeconomics. We will try to review and explain these connections.