Abstract: One way to understand incompressible fluids is to regard it as constrained free particle transport, a point of view realized in the Euler-Arnol'd formulation of fluid flow as a geodesic motion on the volume-preserving diffeomorphism group. In this talk we will discuss some observations that grew out of the further restriction that the fluid flow be affine. Sideris (2017) observed that such affine fluid flows can be described as geodesics on SL(n) with the Hilbert-Schmidt metric, and studied the properties of several explicit solutions when n = 3. Roberts, Shkoller, and Sideris (2020) then integrated the geodesic equations when n = 2 and obtained a complete classification. In this talk I will present some contrasting results obtained, in collaboration with my students Audrey Rosevear and Samuel Sottile, concerning the geodesic geometry of SL(n) for n > 2, and their applications towards stability and instability of the free boundary incompressible Euler flow.