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Explore a 39-minute mathematics lecture that delves into Mañé's critical value for Hamiltonian PDEs, focusing on the two-component Hunter-Saxton system. Learn about the magnetic two-component Hunter-Saxton system (M2HS) as a magnetic geodesic equation on an infinite-dimensional Lie group, and discover how it magnetically maps to a system on an infinite-dimensional sphere. Understand the fascinating geometric property where magnetic geodesics become tangent to 3-spheres formed by intersecting the ambient sphere with complex planes. Master the explicit criteria for blow-ups, grasp the existence of global weak solutions, and examine how Mañé's critical value leads to an infinite-dimensional magnetic Hopf-Rinow theorem. Presented at the Erwin Schrödinger International Institute's Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications."
