Partial Differential Equations (PDEs) lie at the heart of nearly every area of science. Einstein’s theory of general relativity, quantum mechanics, complex weather patterns, the spread of disease, the turbulent flow of blood in the heart, the growth of tumors, the stability of bridges, the erratic patterns of stock options, the pulsing of electromagnetic waves, the flow of oceans and rivers, the flocking patterns of birds, the growth of bones as we develop, the spots of cheetahs, and the stripes of zebras, are all modeled by PDEs. Moreover, PDEs arise within mathematics itself, in areas such as differential geometry (the minimal surface equation), complex analysis (the Cauchy-Riemann equations), and harmonic functions (Laplace’s equation). Two of the seven famous $1,000,000 Clay Millennium Prize problems are directly about PDEs, and a third problem was solved by using PDEs as the major proof tool.

I will give many examples of PDEs, and then give you a tool to be able to understand much of the basic behavior of PDEs at a glance. We will see many visual demonstrations, and by the end, you will be able to understand some of the underlying dynamics of several important PDEs, including the 3D Navier-Stokes equations, and also the chaotic “flame” equation known as the Kuramoto-Sivashinsky equation. We will discuss the problem of singularities for these equations, that is, the question of whether solutions to the equation can become unphysical over time. Most of the talk should be accessible to students who have taken calculus.