Periodic orbits in algebraic billiards

Mathematical billiards is a mathematical model for anything that bounces: light, molecules, or the cue ball in the game of pool. The Ivrii conjecture (1980) predicts that no billiard has a positive-measure set of periodic points. In this talk, we prove the Ivrii conjecture for generic algebraic billiard tables by working with complex billiards, a correspondence that complexifies the classical map.