During the earlier phase of his career, Geoff Smith did research in asymptotic group theory, including the theory of Dirichlet series associated with finitely generated groups. He has latterly become interested in both infinite permutation groups and problems of classical geometry. Two of his PhD students, Simon Turner and Carolyn Ashurst, have recently been awarded their doctorates, and he currently has one student, Dorian Lidell.
Geoff is an activist in the mathematics olympiad movement, and is the author of many problems, including the well-known IMO 2011/2 Windmill question which was analyzed as one of Terry Tao's mini-polymath projects.
He instigated the European Girls' Mathematical Olympiad, and is President of the International Mathematical Olympiad.