In the series we are in the Song dynasty period. The hat the scholars are wearing are called Futou!
Futou gradually first appeared in the Tang dynasty (previous to Song) and then Futou became a common accessory during the Song Dynasty, although most of them had less cumbersome extensions—more feet than wings. People of all classes wore these hats, and there were five styles that individuals donned, according to their status or for various occasions.
The shape of the wings indicated the wearer’s rank and the longest were reserved for the emperor and other high-ranking officials.