Clifford analysis is a branch of classical analysis, which is both a generalisation to higher dimension of complex analysis in the plane, and thus offers an integrated alternative for multivariate complex analysis, and a refinement of classical harmonic analysis. Functions are considered taking values in a Clifford algebra.
De considered operator, the Dirac operator, factorises the Laplacian, and is invariant under the action of the conformal group.
This operator, its null solutions (monogenic functions) and a vast number of associated theories and results, where studied intensively during the past decades, and also further refinements and generalizations were considered, such as the ones to the hermitian context and in super space.
Een recently defended PhD studied discretisation in the Clifford context of an important integral transformation: the Weierstrass transform, which is used in image and signal processing, in quantum mechanics, and in particular for solving the heat equation.
Results indicate that further study of other integral operators in the discrete context, as well as the development of numerical methods, is very promising. Apart from that also the group invariance of the considered operators has to be precisely investigated, and to this end traditional actions on functions, such as rotations, translations, dilations and inversions, have to be carefully redefined. First steps were already made in this respect in another PhD, however this has to be profoundly studied further.
For this part I intend to collaborate with Prof H De Bie.

At the same time I also want to broaden my research scope through collaboration with Dr S Lazendic on numerical methods for partial differential equations with fractional operators.