Scott Sheffield examines conformal invariant objects which arise in the study of two-dimensional statistical physics models. He studies the Schramm–Loewner evolution SLE(K) and its relations to a variety of other random objects. For example, he proved that SLE describes the interface between two Liouville quantum gravity surfaces that have been conformally welded together. In joint work with Oded Schramm, he showed that contour lines of the Gaussian free field are related to SLE(4). With Jason Miller, he developed the theory of Gaussian free field flow lines, which include SLE(K) for all values of K, as well as many variants of SLE.
Sheffield and Bertrand Duplantier proved the Knizhnik-Polyakov-Zamolodchikov (KPZ) relation for fractal scaling dimensions in Liouville quantum gravity. Sheffield also defined the conformal loop ensembles, which serve as scaling limits of the collection of all interfaces in various statistical physics models. In joint work with Wendelin Werner, he described the conformal loop ensembles as the outer boundaries of clusters of Brownian loops.
In addition to these contributions, Sheffield has also proved results regarding internal diffusion limited aggregation, dimers, game theory, partial differential equations and Lipschitz extension theory.