We study wave maps from the circle to a general compact Riemannian manifold. We prove that the global controllability of this geometric equation is characterized precisely by the homotopy class of the data. As a remarkable intermediate result, we establish uniform-time global controllability between steady states, providing a partial answer to an open problem raised by Dehman, Lebeau and Zuazua (2003). Finally, we obtain quantitative exponential stability around closed geodesics with negative sectional curvature. This work highlights the rich interplay between partial differential equations, differential geometry, and control theory. Based on a recent joint work with Jean-Michel Coron and Joachim Krieger.

Speaker Profile: Xiang Shengquan, a researcher and doctoral supervisor at the School of Mathematical Sciences, Peking University. I graduated from Peking University with a bachelor's degree in Mathematics and Applied Mathematics in 2015, and obtained a master's degree in Mathematics from Paris VII University in the same year. In 2017, I received a diploma from Ecole Normale Superieure in Paris, and in 2019, I was awarded a doctorate in Applied Mathematics from Sorbonne University in France. From 2019 to 2022, I conducted postdoctoral research at the Ecole Polytechnique Federale de Lausanne in Switzerland. The research directions are cybernetics and partial differential equations, including control flow of geometric mappings, dispersion equations and quantitative control, quantitative fast stabilization, etc. The paper was published in Journal de Mathematiques Pures et Appliquees, SIAM J. Control Optim...