What is the smallest total width of a collection of strips that cover a disk in the plane? How many lines through the origin pairwise separated by the same angle can be placed in 3-dimensional space? What about higher-dimensions?

These extremal problems in Discrete Geometry look deceitfully simple, yet some of them remain unsolved for an extended period or have been partly solved only recently following great efforts. In this talk, I will discuss two longstanding problems: Fejes Tóth’s zone conjecture and a problem on equiangular lines with a fixed angle.

No specific background will be needed to enjoy the talk.