Ivan Smith works in symplectic topology, a branch of pure mathematics originating from the search for periodic orbits of mechanical systems. Alongside its roots in classical dynamics, the subject has deep interactions with theoretical physics.

Much of Ivan's work concerns Floer theory and the Fukaya category, which packages counts of solutions to a partial differential equation into a global invariant governed by non-commutative algebra. The Fukaya category is one pillar in homological mirror symmetry, a partly conjectural relation between analytic and algebraic aspects of geometry first predicted in string theory. With various co-authors, Ivan has used Floer theory to define new invariants of knots and links, to find universal constraints on billiard trajectories on flat surfaces, to show that symmetry groups of symplectic spaces can have infinite complexity, and to establish new cases of mirror symmetry.

Ivan was awarded a London Mathematical Society Whitehead Prize (2007), the Adams Prize (2013), was an invited speaker at the International Congress of Mathematicians (2018) and was a Clay Senior Scholar (2022).