Étienne Ghys: Dynamics à la Dennis Sullivan

This lecture was held by Étienne Ghys at The University of Oslo, May 25, 2022 and was part of the Abel Prize lectures held in connection with the Abel Prize Week celebrations.
Étienne Ghys is a French mathematician known for his work in topology, geometry, and dynamical systems. He is the director of research at the École normale supérieure in Lyon, and is a member of the French Academy of Sciences. Ghys is also known for his passionate work in the promotion of mathematics in France and elsewhere, including making the popular Dimensions video series.⁠

[@lopezb]

Great talk Etienne! Merci beaucoup!
The space of vector fields is the Lie algebra of the group of diffeomorphisms of the manifold. When you integrate the vector field you get a flow, which is a geodesic in that group, tangent to the chosen vector field.
So in that sense any ODE comes from this geodesic flow. It has infinite dimensions so one has to define things properly to have it work. Now Dennis seems to be taking this abstraction one step further. A point is a vector field representing fluid flow, incompressible so divergence 0, as a path of vector fields this defines a time-varying ODE. But it's also a curve in the space of vector fields, and so should be a solution curve to a stationary ODE in the space of vector fields, give by the Euler equation. That's a great point of view. There's dynamics on like 3 levels....

[@ethien9093]

Its could be