Geometry and Fluids

Date: 7 - 11 April 2014

Location: Mathematical Institute, University of Oxford

The application of ideas from the theory of complex manifolds to fluids mechanics has revealed important connections betwen 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) partial differential equations, their transformation properties, and solutions. Recently, new connections have been established between these seemingly disparate 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 PDEs 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. Recently, several new tools have been developed, such as generalized geometry, flux compactifications, and string theory dualities for understanding the structure of these equations and their solutions.

The aim of this workshop is to bring together experts from complex manifolds and string theory with those from fluid mechanics to study the interplay between geometry, optimal transport, and applications.

The workshop will focus on the following topics

• Elliptic Monge-Ampère equations, hyper-Kähler geometry, and the classification of incompressible flows in two dimensions (4 dimensions in terms of the phase space of the fluid)
• Optimal transport on manifolds and the mathematics of quasi-equilibrium atmosphere/ocean flows
• Generalized complex structures and the local balance between vorticity and rate of strain in 3d turbulent flows (6 dimensions in terms of the phase space of the fluid)

Photograph by Sean O’Flaherty