Special functions in higher spin settings
13 June 2018
Campus Middelheim, G0.10 - Middelheimlaan 1 - 2020 Antwerpen (route: UAntwerpen, Campus Middelheim
Organization / co-organization:
Department of Mathematics & Computer Science
PhD defence Tim Janssens - Faculty of Science, Department of Mathematics and Computer Science
This thesis started with the search for a higher spin equivalent of the following well-known fact from classical harmonic analysis: Up to a multiplicative constant, there exists a unique polynomial solution for the Laplace operator R^m which is homogeneous of degree k and depends on the inner product of the position vector with a fixed (unit) vector.
This particular solution for the Laplace equation can be written in terms of the classical Gegenbauer polynomials and this particular solution plays a crucial role in both harmonic analysis and its refinement (known as Clifford analysis), e.g. these solutions are the basic building blocks in terms of which the Fueter images of holomorphic functions f(z) can be expressed. Moreover, the Gegenbauer harmonics of degree k can be constructed using a raising operator, which embeds the theory of Gegenbauer harmonics into the framework of so-called Appell sequences. We obtained similar results in the case of two vector variables and the special solutions found there are closely related to functions on the (oriented) Grassmann manifold Gr(2,m). This in turn lead to the appearance of the so-called wedge-system, a system with ties to the Cauchy-Kovalevskaya extension of polynomials depending on two vector variables.
Another source of motivation for this thesis is the aforementioned Fueter theorem, a theorem that describes how we can construct monogenic functions (solutions of the Dirac operator) on R^m starting from, the easier to describe, holomorphic functions in the complex plane. We gave a new proof for the classical theorem that lends itself well to generalisations and introduced the tools required to the higher spin setting.