'Complexes of Differential Operators' by Michael Eastwood - June 8, 2023 - Session 1

The classical differential operators of grad, curl, and div in Euclidean 3-space can be substantially generalised to obtain the de Rham complex on an arbitrary smooth manifold. In 1864 Saint-Venant, motivated by continuum mechanics, introduced another complex of differential operators in Euclidean 3-space including a second order operator, often called the double-curl. This complex also admits a substantial generalisation in the realm of projective differential geometry. In 1990 Rumin introduced a complex of differential operators on a contact manifold, effectively replacing the de Rham complex but including a second order operator. There are many such variations and this course will describe several of them and how they are linked. We shall start from scratch. The only real ingredient is the equality of mixed partial derivatives.