This is a theoretical project at the intersection of Differential Geometry and the theory of Differential Equations. Potential Applications: Multidimensional dispersionless equations occur in a wide range of applications in fluid dynamics, averaging theory, general relativity, differential geometry, and the theory of integrable systems. Brief description: Dispersionless equations describe a broad variety of nonlinear models where dispersive effects can be neglected. Our group at Loughborough has proposed a novel approach to the study of such equations known as the method of hydrodynamic reductions [1]. In this project we will apply the method of hydrodynamic reductions to the study of higher-order dispersionless equations in 21 dimensions. The ultimate goal is a complete description of integrable equations within this class. Such equations are expected to have remarkable geometric/symmetry properties, and are of prime importance for the classification of multidimensional dispersive integrable models (which can be efficiently approached based on the method of deformations of hydrodynamic reductions [3]). We will utilise the necessary differential-geometric condition for integrability coming from the theory of double waves [2].