# Clifford Research Group

## Seminars Academic year 2010-2011

- Friday 16-09-2011

M. Abul-Ez (Sohag University, Egypt)

Generalized derivative and primitive of Cliffordian bases of polynomials constructed through Appell monomials

- Friday 19-08-2011

Michael Wutzig

A suggestion for generalized Cauchy formulae in Hermitean Clifford analysis

- Tuesday 09-08-2011

Luoqing Li (Hubei University)

Regularized least squares regression on the sphere

- Monday 25-07-2011

Tao Qian (University of Macau)

Phase derivative of monogenic signals in higher dimensional spaces

- Monday 20-06-2011

Master thesis defense of Timmy Clauwaert

*Hermite polynomials in the framework of Clifford analysis*

- Monday 06-06-2011

Gijs Tuynman (Univ. Lille 1)

*Super analysis and super representations, a different point of view*

- Monday 16-05-2011

Axel de Goursac (UCLouvain)

*Deformation quantization of Heisenberg supermanifolds*

- Monday 09-05-2011

Ricardo Ableu Blaya

*The jump problem in R*^{n}. Analysis versus geometry

- Monday 11-04-2011

Heikki Orelma (Tampere University, Finland)

*Tangential Dirac operators on 2-surfaces*

- Monday 28-02-2011

Nele De Schepper

The class of Clifford-Fourier transforms

- Monday 21-02-2011

Hendrik De Bie

*Clifford-Fourier transform and translation operator*

- Monday 07-02-2011

Kevin Coulembier

*Invariant integration over the orthosymplectic Lie supergroup*

- Wednesday 02-02-2011

Tim Raeymaekers

*Higher spin operators*

- Thursday 20-01-2011

David Eelbode (UA)

*From Gegenbauer polynomials to Fueter's Theorem*

- Wednesday 15-12-2010

Hilde De Ridder

*Taylor series for discrete functions*

- Wednesday 01-12-2010

Roman Lavicka (Charles University, Prague)

*The Gelfand-Tsetlin basis for Hodge-de Rham systems in Euclidean spaces*

- Wednesday 24-11-2010

Liesbet Van de Voorde

Construction of the embedding factors for null solutions of a higher spin Dirac operator

- Wednesday 10-11-2010

Uwe Kähler (University of Aveiro, Portugal)

Discrete Dirac operators and Clifford analysis

- Wednesday 03-11-2010

Hendrik De Bie

*The complex Dunkl operator and growth of logarithmic differences*

#### A suggestion for generalized Cauchy formulae in Hermitean Clifford analysis

Michael Wutzig (University of Ghent)

Abstract:In the talk I will present some results on the Cauchy-Pompeiu formula in Hermitean Clifford analysis (HCA) based on the PhD Thesis of Bram De Knock and show its connection to the Bochner-Martinelli formula in the case of several complex variables (SCV). Similar to the generalization of this Cauchy formula done by Walter Rudin for SCV and resulting in the Cauchy-Szego formula, I will propose an analogue for HCA.

#### Regularized least squares regression on
the sphere

Luoqing Li (Hubei University)

#### Phase derivative of monogenic signals in
higher dimensional spaces

Tao Qian (University of Macau)

*Abstract:*

In the Clifford algebra setting on the boundary of a domain it is natural to define a monogenic (analytic) signal to be the boundary value of a monogenic (analytic) function inside the domain. The question is how to define the corresponding phase and phase derivative. In this talk we give an answer to these questions in the unit ball and in the upper-half space. Among the possible candidates of phases and phase derivatives we decided that the right ones are those that give rise to, as in the one dimensional signal case, the equal relations between the mean of the Fourier frequency and the mean of the phase derivative, and the positivity of the phase derivative of the shifted Cauchy kernel. The talk is based on a joint paper between Yan Yang, Tao Qian and Frank Sommen.

#### Super analysis and super representations,
a different point of view

Gijs Tuynman (Univ. Lille 1)

*Abstract: *

In the first part of the talk,
which is accessible for a wide audience, I will introduce a version of
super analysis which is slightly different from the usual point of
view. I will in particular use a definition for C1 functions which does
not require derivatives. The second part is more specialized and
discusses the topic of super unitary representations of super Lie
groups. Here I will focus on the particular example of a Heisenberg
type supergroup. I will show how super symplectic varieties with
non-homogeneous symplectic form play an important role in this example.

#### Deformation quantization of Heisenberg
supermanifolds

Axel de Goursac (UCLouvain)

*Abstract:*

In this talk, we will present a
non-formal deformation quantization of the Heisenberg supergroup. The
construction of a universal deformation formula permits also to deform
the class of Heisenberg supermanifolds.

#### The jump problem in R^n. Analysis versus
geometry

Ricardo Ableu Blaya (Universidad de HolguÃn, Cuba)

*Abstract:*

The aim of the talk is to
present a survey of basic results on the socalled jump problem for monogenic
functions in R^n. We address the problem of finding the minimal
geometric requirements on the jumpsurface under which the problem
remains solvable.

#### Tangential Dirac operators on 2-surfaces

Heikki Orelma (Tampere University, Finland)

*Abstract:*

We will consider recent results
related to the tangential Dirac operators on embedded 2-surfaces. First
we recall definitions and some fundamental properties of these
operators. Then we study what happens in 2-surfaces. In this special
case our aim is to give complete characterization for the kernel of the
operator.

#### The class of Clifford-Fourier transforms

Nele De Schepper (UGent)

*Abstract:*

Similar to the classical case,
the kernel of the Clifford-Fourier transform (see talk Hendrik De Bie)
satisfies a system of differential equations, which we call "the
Clifford-Fourier system". In this talk, we will determine general
parabivector-valued solutions of this system, thus obtaining a whole
class of Clifford-Fourier transforms. Naturally, the original
Clifford-Fourier kernel is reobtained, but also the Fourier-Bessel
kernel belongs to this class. The latter kernel, which in the
two-dimensional case coincides with the Clifford-Fourier kernel, is
obtained by leaving an exponential factor out of the so-called
Bessel-exponential, introduced by Sommen who recently used it to
introduce Clifford generalizations of the classical Fourier-Borel
transform. Moreover, by expressing the newly obtained solutions of the
Clifford-Fourier system as derivatives of the Fourier-Bessel kernel, we
are able to determine the eigenvalues of an L2-basis consisting of
generalized Clifford-Hermite functions under the action of the new
Clifford-Fourier transforms.

#### Clifford-Fourier transform and translation
operator

Hendrik De Bie (UGent)

*Abstract:*

The Clifford-Fourier transform
was introduced a couple of years ago by Brackx et al. In this talk I
will explain how to obtained a closed formula for its kernel. I will
also show how one can construct a generalized translation operator
related to this integral transform and prove the important fact that
the translation of a radial function is again radial.

#### Invariant integration over the
orthosymplectic Lie supergroup

Kevin Coulembier (UGent)

*Abstract:*

In this talk we introduce Lie
supergroups and show the different methods to construct explicit
examples. The concept of invariant integration is also explained. Then
the orthosymplectic Lie supergroup OSp(m|2n) is introduced in a new and
mathematically rigorous way which connects the different approaches
explained in the introduction. This new approach allows to construct
the unique invariant integration on OSp(m|2n) for the first time in a
transparent and applicable formula. Finally some applications of this
integration are mentioned.

#### Higher spin operators

Tim Raeymaekers (UGent)

*Abstract:*

In this talk, I will explain the
notion of twisted operators. In connection with these, I will introduce some
representation theory, mainly
representations of the Spin(m) group, and thus come to an important part of my research: tensor product
decompositions. All this will be backed up by some examples in LiE, an
open source software program.

#### From Gegenbauer polynomials to Fueter's
Theorem

David Eelbode (University of Antwerp)

*Abstract:*

In this lecture, we will start
from the defnition of Gegenbauer polynomials and their relation to the branching
problem for harmonics (resp.
monogenics). Using two elementary relations for these polynomials, we will then obtain a very simple proof for
the Fueter Theorem, which
essentially says that classical complex analytic functions give rise to harmonic (or, even better, monogenic)
polynomials.

#### Taylor series for discrete functions

Hilde De Ridder (UGent)

*Abstract:*

I will start with a brief
introduction of the discrete Clifford setting and the basic notions
like discrete monogenic functions. The main aim of this talk is to
explain how we can develop a discrete (monogenic) function into its
discrete Taylor series. Hereby, I will point out some difficulties as
well as some differences with the continuous Clifford setting.

#### The Gelfand-Tsetslin basis for Hodge-de
Rham systems in Euclidean spaces

Roman Lavicka (Charles University, Prague, Czech Republic)

*Abstract: *

The main aim of this talk is to explain an explicit construction of
orthogonal bases of k-homogeneous s-vector valued solutions to the
Hodge-de Rham system in Euclidean spaces. Actually, we describe even
the so-called Gelfand-Tsetlin bases for such spaces. As an application,
we obtain an algorithm how to compute an orthogonal basis of the space
of homogeneous solutions of an arbitrary generalized Moisil-Thodoresco
system in any dimension. This is a joint work with R. Delanghe and V.
Soucek.

#### Construction of the embedding factors for
null solutions of a higher spin Dirac operator

Liesbet Van de Voorde (UGent)

*Abstract:*

We know how the kernel space of the higher spin Dirac operator Q decomposes into Spin(m)-irreducible vector spaces, labeled by their highest weight. There are many reasons to investigate how they are embedded in Ker Q. Here is one: some of those vector spaces appear to have multiplicity 2 or higher. This can be proved by showing the linear independence of the corresponding embedding factors. I will give a brief introduction to the world of the higher spin operator Q. The construction of the embedding factors will be explained by means of an example. Are the embedding factors part of an underlying algebra?

#### Discrete Dirac operators and Clifford
analysis

Uwe Kähler (University of Aveiro, Portugal)

*Abstract:*

In the last decade one can observe a growing interest in discrete structures equivalent to well-known continuous structures. From a Clifford analytic point of view a particular interesting object is a discrete analogue to the class of monogenic or regular functions. To this end one

needs to construct discrete Dirac operators which factorize the discrete Star- or Cross-Laplacian. But here a major problem arises: in general, two partial difference operators, forward and backward differences, are necessary in order to get a correspondence with each partial differential operator such that one is able to get a discrete equivalent of the Laplace operator. This means that, for instance, it is impossible to construct a discrete Dirac operator based on a quaternionic structure which factorizes the Star-Laplacian. In this talk we will discuss different

constructions for discrete Dirac operators. To this end we take a look at the construction of the necessary algebraic structures, such as pseudo-Clifford algebras and discrete differential forms.

#### The complex Dunkl operator and growth of
logarithmic differences

Hendrik De Bie (UGent)

*Abstract:*

First I will introduce the one-dimensional complex Dunkl operator and briefly discuss its space of null-solutions. Then I will discuss growth properties of these null-solutions and outline the procedure I want to follow. A basic tool here is the classical Poisson-Jensen formula in complex analysis, which we will adapt to a formula useful in the Dunkl

context.