An algebraic approach to Boolean functions with a geometric domain: the everlasting friendship between algebra and combinatorics

Boolean functions on the hypercube have been studied intensively. Recently, Boolean functions defined on a different domain, e.g. a set of elements of finite geometries, have received particular interest from computer science and mathematics. However, as the theory is slowly unfolding, there are almost no results for Boolean functions of general degree on a geometric domain. It is my ambition to strengthen and generalize existing results to form a unifying theory of these functions. This goal will be achieved by partitioning the problem in the following main objectives. Initially, I intend to strengthen the existing results for the projective case by finding additional examples and non-existence conditions. I also intend to answer a long standing conjecture to ensure a full understanding of the objects. Secondly and thirdly, I will study Boolean functions of general degree on polar and affine spaces, while filling in existing gaps for degree 1. Finally, I will study Boolean functions on flags of finite geometries, a domain which is now a non-commutative association scheme. This investigation not only provides a new lane for this type of research, but also provides deeper insight in the underlying algebraic structures at play. These results directly play into the hands of a unifying theory that is required to properly understand these functions.