My talk consists of two parts.
Part 1 is a 10-minute ignite talk, in which I exhibit several beautiful spiral patterns, explain two mechanisms that trigger spiral waves, and introduce three mathematical viewpoints to treat spiral waves.
Part 2 focus on Ginzburg-Landau spiral waves and I solve the existence problem of spiral waves in the circular and spherical geometries. Instead of applying the shooting method in the literature, I establish a functional approach and generalize the known results of existence for rigidly-rotating spiral waves. Moreover, I prove the existence of two new patterns: frozen spirals in the circular and spherical geometries, and 2-tip spirals in the spherical geometry.