The main interest of van der Kallen is in representations of reductive algebraic groups like the group of n by n matrices of determinant one. An important tool in his work has been the method of Frobenius splittings from algebraic geometry. Van der Kallen conjectures connections between invariant theory, a topic from the nineteenth century, and the homological algebra of the represesentations of reductive algebraic groups. To find an explanation of Demazure operators, click on the link in the picture.