The first step is to find the NPC that teaches you the gesture. Thops is a former student of the Academy who can be found at the Church of Irith, located to the far south of the Converted Fringe Tower itself. Thops will give you his sidequest upon asking him about the Academy Glintstone Key. The quest requires you to have progressed past the Red Wolf of Radagon and gained access to the Debate Parlour Site of Grace. Using this as a starting point, you’ll then have to traverse across the rooftops to grab the key from atop the chandeliers of the Church of the Cuckoo.
Once you hand over the key to Thops and learn the “Erudition” gesture, you’ll have to find a Glintstone Crown. Do note, however, that only the five crowns found within the Academy itself qualify for solving the puzzle: the Lazuli, Twinsage, Karolos, Olivinus, and Witch Glintstone Crowns. If you’ve been thorough in exploring so far, you’ll most likely have one of these already; if not, the Karolos crown can be looted from a crystallized crab just a short jog from the Debate Parlour (per Gamer Guru).
After this, all that’s left to do is equip the crown and perform the gesture in front of the locked door at the tower. There is another enemy past the door, so be careful, and enjoy your new spells.