# Overview

The ideal incompressible fluid is a natural example of a mechanical system with infinitely many degrees of freedom. Recently the Lagrangian point of view in terms of particle paths has become an active area of study for several purposes. Geometrically, the Lagrangian configuration space may be identified with the group of volume preserving diffeomorphisms of the flow domain, equipped with a right-invariant Riemannian metric. Our understanding of the geometry of this space is poor, in spite of considerable work in the last decades.

For example, although global existence in two dimensions of smooth solutions is well-known, the long-time behavior as a dynamical system is still mysterious. Numerical simulations suggest the existence of an attractor for the 2-d Euler equations, which is counterintuitive for a Hamiltonian system. A related phenomenon is the inverse energy cascade in 2-d fluid, whose connection with the geometry of the group of diffeomorphisms has not been studied at all.

The problem of (in)stability of fluid flows in connection with the curvature of the diffeomorphism group is also unclear; this was the original motivation for Arnold's geometric approach and is still being studied. Another question is the connection between the long-time behavior of flows and the structure of the diffeomorphism group at infinity (here we know only simple model examples).

The topics of the workshop include (but are not restricted to:

- - The existence and properties of possible attractors in the space of 2-d vector fields.
- - The properties of the dynamical system in the space of diffeomorphisms including the growth rate of "complexity" of individual solutions.
- - The large-scale structure of the diffeomorphism group and its connection with the

long-time behavior of trajectories ("asymptotic geometry and asymptotic dynamics"). - - Chirality and curvature in the geometry and dynamics on the group of diffeomorphisms, and instability of flows.
- - Blowup results for the 3-d Euler equation in terms of the geometry along particle paths.
- - Model systems for the fluid, including the ideal inextensible thread.
- - Regularity results in the two-point minimization problem on the space of volume-preserving maps.

These are examples of some of the most important emerging problems in fluid dynamics, and it is the aim of the conference to address those problems.