Homological mirror symmetry, curve counting, and a classical example: 27 lines on a nonsingular cubic surface (Reggie Anderson, CMC)

Though mirror symmetry requires much technical background, it gained traction in the mathematical community when physicists Candelas-de la Ossa-Green-Parkes discovered enumerative invariants counting the number of rational degree d curves inside of certain space called a “quintic threefold.” This answered longstanding problems in enumerative geometry from antiquity. In particular, the number of rational degree d=1 curves inside of the space counts the number of lines. We will review a simpler, classical example: any nonsingular cubic surface contains exactly 27 lines.