The application of differential geometry to the equations of fluid mechanics has a long and distinguished history. Contemporary geometric descriptions of nonlinear partial differential equations appear to play a key role in studying vortex tubes and sheets, the fundamental building blocks of turbulence.
The application of ideas from the theory of complex manifolds to fluid mechanics has revealed important connections between complex structures and the dynamics of vortices in many different fluid flows. Large-scale atmospheric flows, optimal transport, and complex geometry have each provided a framework for studying (Monge—Ampère) differential equations, their transformation properties, and solutions.
For instance, in the semi-geostrophic equations, which are a family of nonlinear partial differential equations used by meteorologists to describe the formation of weather fronts, the fronts manifest themselves as singularities in solutions to a Monge—Ampère equation. This Monge—Ampère equation, in turn, governs the relationship between wind, temperature, and pressure fields in atmosphere, and understanding this equation has applications in the numerical modelling of atmospheric flows.
Recently, further new connections have been established between these areas, as well as between coherent vortices in incompressible Navier—Stokes flows and almost complex structures. The application of geometry to fluid mechanics has opened up promising new perspectives on some enduring problems, and facilitates a unification of otherwise ostensibly disparate topics, including singular behaviour, conservation laws, and the partial differential equations describing vortex dynamics.
The interplay between hyper-Kähler geometry and the Monge—Ampère equation also has a long cherished history in string theory, a subject far-removed from fluid dynamics. Amongst other things, we develop and apply tools such as generalised and extended geometry and higher structures to gain further insights into the structure of these equations and their solutions.