Clifford Research Group

Seminars Academic year 2010-2011

A suggestion for generalized Cauchy formulae in Hermitean Clifford analysis
Michael Wutzig (University of Ghent)

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)

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)

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)

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 Holguín, Cuba)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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

Department of Mathematical AnalysisDepartment of Mathematical Analysis