We show that the wave map equation in ${\bf R}^{1+1}$ is in general ill posed in the critical space $\dot{H}^{1/2}$, and the Besov space $\dot{B}_{2}^{1/2,1}$. The ...

BUFFALO, N.Y. — Last year, onlookers observed a startling site on China’s Qiantang River: waves forming a grid-like pattern. Dubbed the “matrix tide,” this complex wave pattern was caused by the river ...