Explore conjugate points on volume-preserving diffeomorphism groups in this 44-minute lecture from the Workshop on Geometry and Analysis of Fluid Flows. Delve into new examples discovered using the MC criterion, a curvature-like functional introduced by Misiolek. Examine the Euler equations, geometric framework, and properties of the exponential map. Investigate generalized zonal flows, 2D and 3D exponential maps, and their normal forms. Gain insights into the applications of the M-Criterion and recent developments in the field of fluid flow geometry and analysis.