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Wave maps with large dataPDE and Applied Math Seminar
The wave maps equation is perhaps the simplest incarnation of a geometric wave equation -- the nonlinearity arises naturally from the Riemannian structure of the target manifold. It is the hyperbolic analogue of the elliptic harmonic maps equation and the parabolic harmonic map heat flow. We will review some of the significant developments over the past decade concerning the asymptotic dynamics of solutions to the energy critical wave maps equation, emphasizing the crucial role that harmonic maps play in singularity formation and drawing analogies to the classical bubbling phenomena for the 2d harmonic map heat flow. We will also discuss different nonlinear phenomena that arise when curvature is introduced in the domain, in particular for the case of wave maps on hyperbolic space in joint work with Sung-Jin Oh and Sohrab Shahshahani.