# Clifford Research Group

## Seminars Academic year 2012-2013

- 15 October 2012

Denis Constales (UGent)

Mathematical Modelling and Data Analysis of Temporal Analysis of Products (TAP) Experiments

- 5 November 2012

Alain Sarlette (Systems, UGent)

Quantum control through reservoir engineering: introduction and applications

- 12 November 2012

Tim Raeymaekers (UGent)

The kernel of the higher spin Dirac operator

- 17 December 2012

Kevin Coulembier (UGent)

The Joseph ideal for the orthogonal algebra

- 4 March 2013

Hilde De Ridder (UGent)

Discrete Clifford analysis

- 18 March 2013

Bart Goossens (TELIN, UGent)

From Wavelets to Shearlets: Development of Sparse Directional Representation Systems and Application to Digital Image Reconstruction

- 4 April 2013

Howard Cohl (NIST, Washington DC)

Fourier and Gegenbauer expansions of a fundamental solution of Laplace's equation on Riemannian spaces of constant curvature

- 8 April 2013

Marie Dostalova (Charles University, Prague)

Symplectic analogues of the Dirac and Twistor operators and their solution spaces on Euclidean space $R^2$ and elliptic curves (2-dimensional tori.)

- 22 April 2013

John Ryan

Clifford Analysis: A Brand New Retrospective

- 3 May 2013

Matthias Roels (Master thesis student, UA), Roy Oste (Master thesis student, UGent)

Presentation of master thesis results

- 13 May 2013

Bernardo Wolf (UNAM, Mexico)

Discrete systems on phase space and some of its applications

- 27 May 2013

Jasson Vindas (UGent)

Generalized functions, analytic representations, and applications to generalized prime number theory.

Stevan Pilipović (University of Novi Sad)

Boundary values of holomorphic functions in spaces of ultradistributions and Fourier hyperfunctionss

- 27 June 2013

Roy Oste (UGent)

Master thesis defense

- 1 July 2013

Joachim Hilgert (University of Paderbon)

A geometric approach to Lie supergroups.

Kevin Coulembier (UGent)

Bott-Borel-Weil theory for basic classical Lie superalgebras.

- 4 July 2013

Mario Kieburg (University of Bielefeld)

The Supersymmetry Approach in Random Matrix Theory.

- 30 July 2013

Jan Möllers (Aarhus University)

The Segal-Bargmann transform via Jordan algebras

Fabian Radoux (Université de Liège)

Higher symmetries of the conformal Laplacian

- 7 August 2013

Steve Sangwine (University of Essex)

Toward a Clifford toolbox in MATLAB

#### Mathematical Modelling and Data Analysis of Temporal Analysis
of Products (TAP) Experiments

Denis Constales (UGent)

Abstract:Temporal Analysis of Products (co-invented by J.T. Gleaves at the Monsanto Company in 1988) is a specially designed experimental procedure in heterogeneous catalysis research. It relies on a small reactor that incorporates many deliberate simplifications for its mathematical modelling, so that a general theoretical framework for it can be based on the Laplace transform. This will be presented during the present talk, along with its implications for the data analysis methods used in practice.

#### Quantum control through reservoir engineering: introduction
and applications

Alain Sarlette (Systems, UGent)

Abstract:The goal of this talk is to give an accessible introduction to stabilization by tailored interaction of quantum dynamical systems. We first introduce the model of quantum dynamical interaction in general mathematical terms. We relate both disturbing and controlling effects to these interactions. We then consider an application where the quantum state of 'trapped' light is controlled in this way through its interaction with atoms. Time permitting, we will also show how adding classical feedback can improve the stabilization. This work has been proposed for the 2012 Nobel prize winning experiment of the LKB team (Serge Haroche), jointly with Pierre Rouchon and the LKB team.

#### The kernel of the higher spin Dirac operator

Tim Raeymaekers (UGent)

Abstract:In this presentation, I will explain the idea of higher spin Dirac operators, which should be seen as generalizations of the well-known classical Dirac operator. Moreover, I will give some ideas on how to determine the structure of the kernel of these operators, using representation theory of Lie algebras.

#### The Joseph ideal for the orthogonal algebra

Kevin Coulembier (UGent)

Abstract: In this talk we review some aspects of the Joseph ideal for so(m), the corresponding minimal representation and its appearance in conformal geometry, based on results by A. Joseph, M. Eastwood, B. Binegar, M. Zierau, J. Möllers, J. Hilgert and T. Kobayashi. These structures are currently being extended to the theories of Clifford analysis and super conformal geometry.

We introduce some equivalent definitions of the Joseph ideal in the universal enveloping algebra of a simple Lie algebra. For $so(m)$ we explicitly construct the representation which has as its annihilator ideal this Joseph ideal. This is a representation on the spherical harmonics on ℝ

^{m-2}. Then we show how this ideal appears in the characterization of the algebra of symmetries of the conformal Laplace operator on ℝ

^{m-2}.

#### Discrete Clifford analysis

Hilde De Ridder (UGent)

Abstract: Roughly speaking, there are two options to generalize the theory of holomorphic functions in the complex plane to the higher-dimensional setting: either Euclidean Clifford analysis, or the theory of holomorphic functions of several complex variables, both being moreover connected to each other. Discrete Clifford analysis is the discrete counterpart of Euclidean Clifford analysis, studying the null functions of a discrete Dirac operator, which are called discrete monogenic functions. As in the continuous setting, the discrete Dirac operator factorizes the Laplacian, whence discrete monogenic functions may not only be seen as a generalization of discrete holomorphic functions, but also as a refinement of discrete harmonic ones.

Most settings for discrete Clifford analysis focus immediately on applications, namely the numerical treatment of problems from potential and boundary value theory, without establishing the foundations of the underlying discrete function theory. In this presentation, I will give an overview of a concrete discrete function-theoretic Clifford framework, which has been developed during my PhD, and which incorporates discrete counterparts of important integral theorems (such as Cauchy's theorem, Cauchy's integral,...) while also being suitable for studying polynomial solutions.

Developing this discrete framework has proven to be a capacious work, where every new topic constitutes a necessary building block to continue the further development. I will give a summary of the obtained results so far, which include a discrete Fischer decomposition, a CK-extension theorem and the expansion of discrete functions in a discrete Taylor series, and also go into detail about some open problems.

#### From Wavelets to Shearlets: Development of Sparse Directional
Representation Systems and Application to Digital Image Reconstruction

Bart Goossens (TELIN, UGent)

Abstract: In the past decades, wavelets have successfully been used for many purposes, such as compression, data deconvolution and filtering. Even though the wavelet transform is well matched for processing one-dimensional signals (for example, piecewise smooth functions of low-order polynomials), the transform turns out to be sub-optimal for representing 2D (or higher-dimensional) images, because of its ineffectiveness to represent non-horizontal and/or non-vertical edges. For this reason, there has been a lot of interest recently in multi-resolution representations that better adapt to different edge directions, i.e., transforms that also perform a multi-directional analysis. The shearlet transform is one of the most recent siblings in the family of multi-directional transforms. The shearlet transform provides a traditional multi-resolution analysis (such as the wavelet transform) combined with a multi-directional analysis in an arbitrary number of directions. In this talk, we first give a brief overview of the shearlet representation system and its properties and we compare to the properties of wavelets. Then, we consider the application of the shearlet transform to improve digital image reconstruction. In practice, digital images are often not fully sampled due to cost and power reasons: not all data samples of the images are acquired, leading to missing information. Therefore, a reconstruction technique is needed to recover the missing data samples. Solving this ill-posed inverse problem is generally a challenging task. Here, we demonstrate that shearlet-based regularization leads to very promising results for medical (MRI) and non-medical (digital still camera) reconstruction.

#### Fourier and Gegenbauer expansions of a fundamental solution
of Laplace's equation on Riemannian spaces of constant curvature

Howard Cohl (NIST, Washington DC)

Abstract: A fundamental solution of Laplace's equation is derived on Riemannian spaces of constant curvature, namely in hyperspherical geometry and in the hyperboloid model of hyperbolic geometry. These fundamental solutions are given in terms of finite-summation expressions, Gauss hypergeometric function, definite integrals and associated Legendre functions with argument given in terms of the geodesic distance on these manifolds. Fourier and Gegenbauer expansions of these fundamental solutions are derived and discussed.

#### Symplectic analogues of the Dirac and Twistor operators and their
solution spaces on Euclidean space $R^2$ and elliptic curves (2-dimensional tori.)

Marie Dostalova (Charles University, Prague)

Abstract: In the lecture I will introduce the symplectic Dirac and the symplectic Twistor operators as Spin-symplectic analogues of the Dirac and the Twistor operators in Riemannian Spin geometry. Then I will focus on the real dimension 2 and describe their solution spaces on ℝ

^{2}and elliptic curves. The main technical tool used in the analysis is based on the metaplectic Howe duality.

#### Clifford Analysis: A Brand New Retrospective

John Ryan

Abstract:I will give a review of Clifford analysis from my perspective focusing on conformal structure, Hardy spaces and operators of Dirac type. Time permitting I will discuss CR Dirac operators on odd dimensional spheres.

#### Discrete systems on phase space and some of its applications

Bernardo Wolf (UNAM, Mexico)

Abstract: The analysis of discrete signals -in particular finite N-point signals- is done in terms of the eigenstates of discrete Hamiltonian systems, which are built in the context of Lie algebras and groups. These systems are in correspondence, through a `discrete-quantization' process, with the quadratic potentials in classical mechanics: the `Kravchuk' harmonic oscillator, the repulsive oscillator, and the free particle. Discrete quantization is achieved through selecting the position operator to be a compact generator within the algebra, so that its eigenvalues are discrete. The discrete harmonic oscillator model is contained in the `rotation' Lie algebra so(3), and applies to finite discrete systems, where the positions are

{ -j,-j+1,...,j} in a representation of dimension N=2j+1. The discrete radial and the repulsive oscillator are contained in the complementary and principal representation series of the Lorentz algebra so(2,1), while the discrete free particle leads to the Fourier series in the Euclidean algebra iso(2) For the finite case of so(3) we give a digest of results in the treatment of aberrations as unitary U(N) transformations of the signals on phase space. Finally, we show two-dimensional signals (pixellated images) on square and round screens, and their unitary transformations.

#### Generalized functions, analytic representations, and applications
to generalized prime number theory.

Jasson Vindas (UGent)

Abstract:The first part of this talk is an overview of results about analytic representations of various generalized function spaces over the real line (distributions, ultradistributions, and hyperfunctions). We then discuss recent applications of these ideas to the analysis of the asymptotic distribution of Beurling's generalized prime numbers.

#### Boundary values of holomorphic functions in spaces of
ultradistributions and Fourier hyperfunctions.

Stevan Pilipović (University of Novi Sad)

Abstract:We discuss Cauchy and Poisson integral representations of ultradistributions from 𝒟′(∗,L

^{s}). The Cauchy integrals are holomorphic functions of the complex variable z ∈ T

^{C}= ℝ

^{n}+ i C, where C is a regular cone in ℝ

^{n}. Then, boundary values of holomorphic functions will be considered in the general framework of ultradistribution and Fourier hyperfunction spaces.

#### A geometric approach to Lie supergroups.

Joachim Hilgert (University of Paderbon)

Abstract:In this talk I will explain the category of supermanifolds as manifolds equipped with structure sheaves of superalgebras and the corresponding group objects: Lie supergroups. Moreover, we explain how to relate this to supergroup pairs consisting of a Lie group and a compatible Lie super algebra.

#### Bott-Borel-Weil theory for basic classical Lie superalgebras.

Kevin Coulembier (UGent)

Abstract:A classical result of Borel and Weil provides a unified method to construct all finite dimensional representations of semisimple Lie algebras as holomorphic sections on a line bundle of the flag manifold of a Lie group. Bott extended this result to identify the higher cohomology groups on such line bundles. For algebraic Lie groups this problem can be reformulated in terms of the Zuckerman functor and its derived functors. We take this approach for Lie supergroups. We review the results obtained on this subject by I. Penkov, V. Serganova and R.B. Zhang before proceeding to some new recently obtained progress on this subject.

#### The Supersymmetry Approach in Random Matrix Theory.

Mario Kieburg (University of Bielefeld)

Abstract:Random Matrix Theory is a big branch of statistics and statistical physics with a broad range of applications ranging from physics over mathematics to engineering and social science. One particular mathematical technique in this field is the supersymmetry method. It was introduced in the early 80's and enjoys a wide popularity in Random Matrix Theory. Its big advantage is the impressive reduction of number of integrals to be performed by employing dualities between different matrix spaces. In my talk I will give an introduction to this map. In particular I will present two important mappings to superspace known as the generalized Hubbard-Stratonovich transformation and the superbosonization formula. Moreover I will sketch what the main problem with supermatrices are (namely the changes of coordinates like eigenvalue and singular value decomposition) and what the solution for particular cases are.

#### The Segal-Bargmann transform via Jordan algebras

Jan Möllers (Aarhus University)

Abstract:The classical Segal-Bargmann transform is a unitary isomorphism between L

^{2}(ℝ

^{n}) and the space of holomorphic functions on ℂ

^{n}which are square integrable with respect to the Gaussian measure e

^{-|z|2}dz. It has applications in physics or PDEs where it can e.g. be used to characterize the unitary image of the heat kernel transform. In representation theory the Segal-Bargmann transform connects two different realizations of the same representation, the Segal-Shale-Weil representation. In this talk we will try to explain a generalization of the Segal-Bargmann transform in the setting of Jordan algebras.

#### Higher symmetries of the conformal Laplacian

Fabian Radoux (Université de Liège)

Abstract:In this talk, we will study the symmetries and the conformal symmetries of the conformal Laplacian on an arbitrary pseudo-Riemannian manifold (M,g).

On an m-dimensional pseudo-Riemannian manifold (M,g), this operator, which we denote here by Δ

_{Y}, is given by

^{ij}∇

_{i}∇

_{j}-

^{m-2}⁄

_{4(m-1)}R

A symmetry of Δ

_{Y}is a differential operator which commutes with Δ

_{Y}. A conformal symmetry of Δ

_{Y}is a differential operator D

_{1}such that there exists a differential operator D

_{2}giving rise to the relation Δ

_{Y}D

_{1}= D

_{2}Δ

_{Y}.

We will describe in this talk all the second-order (conformal) symmetries of Δ

_{Y}on an arbitrary pseudo-Riemannian manifold (M,g). The principal symbol of such a (conformal) symmetry has to be a symmetric (conformal) Killing 2-tensor that satisfies some additional condition.

We will determine whether this condition is verified on some pseudo-Riemannian manifolds endowed with some (conformal) Killing tensors, determining in this way whether there exists an obstruction to the existence of (conformal) symmetries in these particular situations.

At the end of the talk, we will show how the study of the conformal symmetries (resp. symmetries) of the conformal Laplacian is related to the study of the R-separation of variables in the Laplace (resp. Helmholtz) equation.