My research may be described as an exploration of the mathematical landscape of M-theory. Much of it deals with supersymmetry, which can be understood as adding extra quantum dimensions to spacetime. The bulk of my work touches upon the gauge/gravity correspondence, most recently on the one associated to M2-branes, whose understanding might shed light on one of the outstanding problems in string theory: to nd a formulation of eleven-dimensional M-theory. Currently I am involved in a research programme based on a recent proposal for a nonabelian theory of membranes. The current view is that membranes are described by three-dimensional superconformal Chern-Simons theories, which can be formulated in terms of somewhat exotic mathematical objects known as 3-algebras. Together with my collaborators I have obtained a number of structure results about these objects and continue to apply them to the study of the superconformal theories.