the classical borsuk-ulam theorem states that a continuous map from a n-dimensional sphere to m-dimensional sphere which preserves the antipodal z/2-actions only exists when m is greater than or equal to n. one can ask a similar question, by replacing the antipodal z/2-action with an action of the lie group pin(2). on a seemingly unrelated side, the geography problem of 4-manifolds asks which simply connected topological 4-manifolds admits a smooth structure. by the celebrated works of kirby-siebenmann, freedman, donaldson, seiberg-witten and furuta, there is a surprising connection between these two questions. in this talk, i will: 1. explain this beautiful connection between the two problems. 2. present a solution to the pin(2)-equivariant borsuk–ulam problem. 3. state its application to the geography problem. in particular, a partial result on the famous 11/8-conjecture. 4. describe the ideas of our proof, which uses pin(2)-equivariant stable homotopy theory. this talk is based on a joint work with mike hopkins, xiaolin danny shi and zhouli xu. no familiarity of homotopy theory or 4-dimensional topology will be assumed.