About me

In a nutshell

Since 1/10/2019 I am affiliated to the departement of Mathematics (research group Fundamental Mathematics). On September 5, 2023 I obtained the degree of doctor in mathematics after defending my thesis titled Vortex dynamics and symplectic Dirac operators under the supervision of prof. dr. Hohloch and prof. dr. Eelbode. My research is mostly centered around symplectic techniques in analysis supported by methods from algebra. The latter includes representation theory and Lie theory in order to reveal the underlying symmetries of operators (such as the Dirac operator and generalisations thereof) and dynamical systems (in particular Hamiltonian systems, e.g. vortices) and to study their internal interaction.

Since October 2023 I'm working at Argenta in Antwerp, Belgium as a quantitative credit risk analyst.

Abstract. The CCR (canonical commuting relation) and CAR algebras (canonical anticommuting relation) are fundamental algebras in theoretical physics used for the study of bosons and fermions. From a mathematical viewpoint, these algebras are named the Weyl algebra (or symplectic Clifford algebra) and Clifford algebra. These algebras can be constructed in a very analogous way. The Clifford algebra is constructed on a vector space equipped with a symmetric bilinear form, whereas the Weyl algebra requires an even dimensional vector space equipped with a skew-symmetric bilinear form (or symplectic form). There is, however, a fundamental difference: the Clifford algebra is finite-dimensional, whereas the Weyl algebra is infinite-dimensional. Using the generators of the Clifford (resp. Weyl) algebra, one can associate a natural first order spin (resp. metaplectic) invariant differential operator by contracting the Clifford algebra elements using the bilinear form (resp. the symplectic form) with derivatives. The theory which studies the solutions of the Dirac operator is known as Clifford analysis and can be seen as a hypercomplex function theory. In the first part of the thesis, we will study the symplectic Dirac operator, from an orthogonal point of view. By this we mean that we will focus on the subalgebra so(m), as this will allow us to derive branching rules for the space of k-homogeneous polynomial solutions for the symplectic Dirac operator. To arrive at this result we use techniques from representation theory, including the notion of transvector algebras and tensor products of (Verma modules). In addition, we provide the foundations of what we will call a 'hermitian variant' of symplectic Clifford analysis, where we incorporate the additional datum of a compatible complex structure and study the associated solution space using algebraic techniques and arrive at a Fischer decomposition.

In the second part of the thesis, we provide tools to study the dynamics of point vortex dynamics on the complex projective spaces and the six-dimensional flag manifold. These are the only Kähler twistor spaces arising from 4-manifolds. We give an explicit expression for Green's function on the projective space which enables us to determine the Hamiltonian (the energy of the system) and the equations of motions for the point vortex problem. Moreover, we determine the momentum map on the flag manifold, which is a key ingredient in understand the dynamics better.

The original Fueter theorem knows his origin in quaternionic analysis. It was a non-trivial question to find the analogue of a holomorphic function from complex analysis. This was triggered by the fact that the quaternionic powers are not regular, this with respect to the Fueter operator. The question about generating Fueter-regular functions was thus non-trivial and the Fueter theorem gave a mechanism to construct them. In my master thesis we obtained a symplectic version of this result, which was a new contribution in the research field of symplectic Clifford analysis.

In bijlage kan u mijn bachelorproef terugvinden (in het Nederlands). 

1. De fundamentele bachelorproef behandelt de zgn. Haarmaat: een translatie-invariante maat op lokaal compacte Hausdorff groepen.
2. De toegepaste bachelorproef is eerder fysisch van aard en beschrijft op een wiskundige manier van het waterstofatoom a.d.h.v. kwantummechanica en computerende relaties.