After a brief review of the history, some applications of these models will be reviewed. This will include descriptions of rogue waves, tsunami propagation, internal waves and blood flow. Some of the theory emanaging from these applications will then be sketched.

In dimension 4, the works of Freedman and Donaldson led us to the striking discovery that the smooth category is drastically different from the topological category, compared to other dimensions. Since then, it has been extraordinarily successful in investigating the difference in various contexts. In contrast, our understanding of when smooth and topological categories would exhibit similarity in dimension 4 remained, at best, minimal. In this talk, we will introduce some recent progress on new “topological = smooth” results in dimension 4, focusing on embedded disks.

A rational map, like f(z) = (1+z^2)/(1-z^2), gives a map from the (extended) complex plane to itself. Studying the dynamics under iteration yields beautiful Julia set fractals with intricate nested structure. How can that structure be best understood? One approach is combinatorial or topological, giving concrete models for the Julia set and tools for cataloguing the possibilities.