The Euler equations of hydrodynamics, which model ideal fluid flows on some region $M$, admit a beautiful description as geodesic equations on the group of volume-preserving diffeomorphisms of $M$, as first shown by V. Arnold. A major difficulty in studying the geometry of this group is the fact that it is infinite-dimensional, so a natural idea is to try to approximate it in some sense by a finite-dimensional model. One such model was constructed by Zeitlin, based on a quantization scheme developed earlier by Hoppe. In this talk, I will give an overview of Zeitlin's model, and then describe some aspects related to its geometry, in particular conjugate points. This is joint work with Stephen Preston and Alice Le Brigant.

Conjugate points on Lie groups and Zeitlin's model of hydrodynamics Sponsored by the Meyer Fund