# Clifford Research Group

## Seminars Academic year 2013-2014

- 14 October 2013

Ghislain Franssens (BIRA)

The mathematics behind transversally polarized plane electromagnetic waves

- 21 October 2013

Lander Cnudde (UGent)

Algebraic approach to slice monogenic functions

- 9 December 2013

Michael Wutzig (UGent)

Reproducing Kernels in Hermitian Clifford Analysis

- 3 March 2014

Roy Oste (UGent, Vakgroep Toegepaste Wiskunde, Informatica en Statistiek)

Transforms based on characteristic properties of the Fourier transform

- 10 March 2013

Tom Mestdag (UGent, Vakgroep Wiskunde)

The Finsler metrizability problem

- 17 March 2014

Matthias Roels (Universiteit Antwerpen)

Higher spin generalisations of the Laplace operator

- 24 March 2014

Gert De Cooman (UGent, SYSTeMS Research Group)

Inference under exchangeability using sets of desirable gambles

- 31 March 2014

Thomas Batard (University Pompeu Fabra, Spain)

Generalizations of the Euclidean heat diffusion - Application to vector-valued fields regularization

- 28 April 2014

Klaus Rubbrecht (Master thesis student, UGent)

Convolutions in Clifford analysis (presentation of master thesis results)

Sigiswald Barbier (Master thesis student, UGent)

Segal-Bargmann transforms (presentation of master thesis results)

- 26 June 2014

Klaus Rubbrecht, Sigiswald Barbier (UGent)

Master thesis defense

#### The mathematics behind transversally polarized plane electromagnetic waves

Ghislain Franssens (BIRA)

Abstract:First, the concept of the polarization of a plane electromagnetic wave is explained. Then the Jones model for fully polarized light and the Stokes/Mueller model for partially polarized light is introduced. We will see that both models are related to different representations of the Lorentz group. The Jones model will be shown to possess a rich mathematical underpinning of homomorphic groups, which leads to alternative (not yet explored) descriptions of transversal polarization.

We will see that the Stokes/Mueller model involves exactly the same mathematics as the special theory of relativity, while the Jones model shares its underlying mathematics with certain early developments in quantum physics. This Stokes/Mueller model also contains an underlying group structure, which is not yet fully known. Interestingly, its parallel in the special theory of relativity is also still an open question, namely: what is the full causal group of our universe?

#### Algebraic approach to slice monogenic functions

Lander Cnudde (UGent)

Abstract:In recent years, the study of slice monogenic functions has attracted more and more attention in the literature. In this paper, an extension of the well-known Dirac operator is defined which allows to establish the Lie superalgebra structure behind the theory of slice monogenic functions. Subsequently, an inner product is defined corresponding to this slice Dirac operator and its polynomial null-solutions are determined. Finally analogues of the Hermite polynomials and Hermite functions are constructed in this context and their properties are studied.

#### Reproducing Kernels in Hermitian Clifford Analysis

Michael Wutzig (UGent)

Abstract:The study of homogeneous polynomials plays an important role in various settings of mathematical analysis. Harmonic homogenous polynomials (which are called spherical harmonics) are a powerful tool in Harmonic Analysis and provide a connection to Fourier Analysis. In Clifford Analysis one studies homogeneous polynomials that are null-solutions of the Dirac operator and are therefore named spherical monogenics. In all settings one essential property of homogeneous polynomials is the existence of a unique reproducing kernel. We will give an overview of these kernels, show a way to determine the (yet unknown) reproducing kernel in the hermitian setting of Clifford Analysis and show its connection to the case of (complex) Harmonic Analysis.

#### Transforms based on characteristic properties of the Fourier transform

Roy Oste (UGent, Vakgroep Toegepaste Wiskunde,
Informatica en Statistiek)

Abstract:The Fourier transform (FT) is of crucial importance in a whole range of areas such as harmonic analysis and signal processing as it has many interesting properties. Now, our intent is to investigate whether there are other transforms that satisfy an imposed set of favourable properties of the FT and moreover which properties are sufficient to uniquely characterise the classical FT.

In order to do this, we work in the framework of representation theory of the Lie algebra sl(2) and further expand to the Lie superalgebra osp(1|2) to find transforms in the setting of Clifford analysis.

#### The Finsler metrizability problem

Tom Mestdag (UGent, Vakgroep Wiskunde)

Abstract:Finsler geometry is a generalization of Riemannian geometry, where the role of the metric is played by a 1-homogenous function of a tangent bundle. I will discuss the following problem: When can a given collection of unparametrized paths on a manifold be regarded as geodesics of some (yet unknown) Finsler function.

#### Higher spin generalisations of the Laplace operator

Matthias Roels (Universiteit Antwerpen)

Abstract:Clifford analysis has become a branch of multi-dimensional analysis in which far-reaching generalisations of the classical Cauchy-Riemann operator in complex analysis are studied from a function theoretical point of view. Without claiming completeness, one could say that the theory focuses on first-order conformally invariant operators acting on functions taking values in irreducible representations for the spin group. This then leads to function theories refining (poly-)harmonic analysis on $R^m$.

In this seminar we will explain how to extend these results to a certain class of second-order conformally invariant operators, leading to analogues of the Rarita-Schwinger function theory for functions taking values in the space of harmonics (at the same time generalising classical harmonic analysis for the Laplace operator). After introducing some basic notions of representation theory, we will give a detailed description on how to define and construct this class of operators. As some equations of motion that often appear in physics are expressed in terms of these operators, we will explain how to translate our notation to the tensor notation used by physicists. Finally, some function theoretical results that were already obtained will be discussed.

#### Inference under exchangeability
using sets of desirable gambles

Gert De Cooman (UGent, UGent, SYSTeMS Research Group)

Abstract:I give an overview of one of the most powerful and elegant languages for representing and reasoning under uncertainty: coherent sets (cones) of desirable gambles. I relate it to less general models, such as classical propositional logic, lower prevision functionals and (sets of) probability measures. More importantly, perhaps, I provide evidence for its simplicity, mathematical elegance and power by zooming in on a few examples: dealing with symmetry in general and exchangeability in particular, leading to representations using cones of multivariate polynomials. This idea can be used to build a comprehensive mathematical framework for dealing for with conservative predictive inference.

#### Generalizations of the Euclidean heat diffusion - Application to vector-valued
fields regularization

Thomas Batard (University Pompeu Fabra, Spain)

Abstract:Signal/Image regularization is a fundamental preliminary stage for several applications. One of the seminal approaches was to perform an Euclidean heat diffusion on the signal/image to be regularized. Since then, this approach has been improved in several ways. In some previous works, my co-workers and I developed generalizations of the Euclidean heat diffusion for the purpose of vector-valued fields (e.g. color images) regularization. In this talk, I will present some particular cases that involve Clifford algebras.