# Seminars Academic Year 2017-2018

- 15 September 2017

Sundaram Thangavelu (Indian Institute of Science, Bangalore)

An extension problem and trace Hardy inequality for the sublaplacian on the Heisenberg group - 21 November 2017

Ivan Penkov (Jacobs University Bremen)

Primitive ideals of U(sl(∞)) - 22 January 2018

Amedeo Altavilla (Universita di Roma 2 "Tor Vergata")

New techniques in slice regularity - 26 January 2018

Amedeo Altavilla (Universita di Roma 2 "Tor Vergata")

More on slice regularity - 9 February 2018

Wouter Van De Vijver (UGent)

A discrete realization of the Racah algebra - 23 February 2018

Hendrik De Bie (UGent)

An n-variable realization of the rank n Racah algebra - 2 March 2018

Srđan Lazendić (UGent)

Denoising of color and multispectral images based on octonion sparse representation - 16 March 2018

Astrid Massé (UGent)

Discrete Segal-Bargmann transform - 20 April 2018

Sigiswald Barbier (UGent)

A minimal representation of the orthosymplectic Lie superalgebra - 9 May 2018

Alí Guzmán Adán (UGent)

Euclidean and Hermitian Clifford analysis on superspace - 15 May 2018

Gert Vercleyen (UGent)

Seminar - 25 May 2018

Zoë Gromotka (UGent)

Third order chemical kinetics: an initial analysis - 1 June 2018

Niels Bonneux (KULeuven)

Recurrence Relations for Wronskian Hermite Polynomials - 8 June 2018

Ren Hu (UGent)

Szegö-Radon transform for biaxial monogenic functions - 15 June 2018

Plamen Iliev (Georgia Institute of Technology)

Discrete orthogonal polynomials and superintegrability - 22 June 2018

Sijia Bao (Ghent University)

Special solutions for the parabolic Dirac operator - 29 June 2018

Roy Oste (Ghent University)

Sparse image representation using discrete moments - 30 August 2018

Phillip Isaac (University of Queensland)

On the construction of Casimir operators

Sigiswald Barbier (Ghent University)

The blocks of the periplectic Brauer algebra in positive characteristic - 14 September 2018

Cinzia Bisi (Università Degli Studi Di Ferrera)

The harmonicity of slice regular functions

## An extension problem and trace Hardy inequality for the sublaplacian on the Heisenberg group (Sundaram Thangavelu, Indian Institute of Science, Bangalore)

*Abstract:*

In this talk we plan to describe my recent work with Luz Roncal on the extension problem for the sublaplacian on Heisenberg gropus and how to use the solutions of the same to prove trace Hardy inequality for sublaplacian. From the trace Hardy inequality we show how to prove Hardy's inequality for fractional powers of the sublaplacian.

## Primitive ideals of U(sl(∞)) (Ivan Penkov, Jacobs University Bremen)

*Abstract:*

We describe explicitly the primitive ideals in U(sl(∞)). We show that all such ideals are the annihilators of integrable simple sl(∞)-modules. The only maximal ideal in U(sl(∞) is the augmentation ideal. We also prove a Duflo-type theorem describing primitive ideals as annihilators of highest-weight modules. Not all Borel subalgebras "realize" primitive ideals in this way: the ones that do, are "ideal Borel subalgebras". Finally, we provide an algorithm which computes the primitive ideal of an arbitrary simple highest weight module. This talk is based on several joint papers with Alexey Petukhov.

## New techniques in slice regularity (Amedeo Altavilla, Universita di Roma 2 "Tor Vergata")

*Abstract:*

In this talk I will overview some new techniques developed in the field of quaternionic slice regularity. In particular, I will present a new interpretation of the so-called *-product (that is a product which preserves regularity) and an interesting family of quaternionic functions, called "slice-polynomial functions", which naturally arises in a geometric application. Time permitting I will also say something about the harmonicity of slice regular functions and whether some of the result may or may not be generalized to other algebras.

References:

Altavilla, A. and Bisi, C., Log-biharmonicity and a Jensen formula in the space of quaternions, ArXiv e-prints 1708.04894

Altavilla, A. and Sarfatti, G., Slice-Polynomial Functions and Twistor Geometry of Ruled Surfaces in $\mathbb{CP}^3$, ArXiv e-prints 1712.09946

Altavilla, A. and de Fabritiis, C., S-regular functions which preserve a complex slice, ArXiv e-prints 1801.01318

## A discrete realization of the Racah algebra (Wouter van de Vijver, UGent)

*Abstract:*

The Racah algebra is an algebra encoding interesting properties of the univariate Racah polynomials. Recently it has been shown that this algebra can be generalized to the higher rank Racah algebra by considering the tensor product of n copies of su(1,1). Using Dunkl operators an explicit action of the generalized Racah algebra on the Dunkl-harmonics was established. Remarkably, the connection coefficients between bases diagonalized by labelling Abelian subalgebras are multivariate Racah polynomials as defined by M. V. Tratnik. One wonders if an action of the generalized Racah algebra can be realized on the multivariate Racah polynomials encoding their properties. We propose such a realization by making use of the bispectral shift operators defined by J.S. Geronimo and P. Iliev.

## An n-variable realization of the rank n Racah algebra (Hendrik De Bie, UGent)

*Abstract:*

The Racah algebra of rank n can be constructed within the (n+2)-fold tensor product of the universal enveloping algebra of su(1,1). So far two concrete realizations of this construction have received quite a bit of attention. I'll summarize them, and will also construct a third, new realization. On top of its simplicity (especially concerning its CK extension), it has the advantage of yielding an n-variable realization of the rank n case.

This is joint work with P. Iliev and L. Vinet.

## Denoising of color and multispectral images based on octonion sparse representation (Srđan Lazendić, UGent)

*Abstract:*

A recent trend in color image processing combines the quaternion algebra with dictionary learning methods. This talk aims to present a generalization of the quaternion dictionary learning method by using the octonion algebra. The octonion algebra combined with dictionary learning methods is well suited for representation of multispectral images with up to 7 color channels. Opposed to the classical dictionary learning techniques that treat multispectral images by concatenating spectral bands into a large monochrome image, we treat all the spectral bands simultaneously. Our approach leads to better preservation of color fidelity in true and false color images of the reconstructed multispectral image. To show the potential of the octonion based model, experiments are conducted for image reconstruction and denoising of color images as well as of extensively used Landsat 7 images.

## Discrete Segal-Bargmann transform (Astrid Massé, UGent)

*Abstract:*

The classical Segal-Bargmann transform is a unitary map between the space of square integrable functions and the Fock space, which is of great interest in e.g. quantum mechanics. In this presentation, we try to define an analogue version of this transform in the discrete Hermitian Clifford analysis, where functions are defined on a grid rather than the continuous space. This is done based on the classical definition, in combination with a discrete version of the Gaussian function and discrete counterparts of the classical Hermite polynomials. Furthermore, a discrete Hilbert space with appropriate inner product is constructed, for which the discrete Hermite polynomials form a basis. In this setting, we also investigate the behavior of discrete delta functions and delta distributions.

## Third order chemical kinetics: an initial analysis (Zoë Gromotka, UGent)

*Abstract*:

As a result of research into first and second order kinetics, our group (Denis Constales, Daniel Branco Pinto) has classified several invariants. These invariants are constant in time and are made up of functions of chemical concentrations where the concentrations are time dependent. To determine whether the idea of invariants can be extended to the domain of third order kinetics an analysis has been done into these kinetics. This talk will cover the background of third order kinetics and discuss the initial findings.

## Recurrence Relations for Wronskian Hermite Polynomials (Niels Bonneux, KULeuven)

*Abstract:*

We consider polynomials that are defined as Wronskians of certain sets of Hermite polynomials. Our main result is a recurrence relation for these polynomials in terms of those of one or two degrees smaller, which generalizes the well-known three term recurrence relation for Hermite polynomials. The polynomials are defined using partitions of natural numbers, and the coefficients in the recurrence relation can be expressed in terms of the number of standard Young tableaux of these partitions. Using the currence relation, we provide another recurrence relation and show that the average of the considered polynomials with respect to the Plancherel measure is very simple. Furthermore, we show that some existing results in the literature are easy corollaries of the recurrence relation.

## Szegö-Radon transform for biaxial monogenic functions (Ren Hu, UGent)

*Abstract:*

In this talk, I will introduce the Szegö-Radon transform for biaxially monogenic functions. Using the biaxial decomposition of homogeneous polynomials, we get two types of biaxially monogenic functions which satisfy a Vekua system, depending on the degree of homogeneity. Using these expressions, we can calculate explicit formulas for the Szegö-Radon transforms for biaxially monogenic functions according to their total degree of homogeneity. Furthermore, we simplify these results making use of the Funk-Hecke theorem and orthogonality results of appearing special functions.

## Discrete orthogonal polynomials and superintegrability (Plamen Iliev, Georgia Institute of Technology)

*Abstract:*

The spectral properties of the classical orthogonal polynomials of one and several variables have deep connections with the representation theory of Lie algebras and quantum superintegrable systems. Over the years, these connections have played a crucial role in the discovery of fundamental results with numerous applications in mathematics and physics. I will discuss recent results in a joint work with Yuan Xu relating spectral properties of discrete orthogonal polynomials to superintegrability.

## Special solutions for the parabolic Dirac operator (Sijia Bao, Ghent University)

*Abstract:*

During this talk, I will introduce the Fischer decomposition for the parabolic Dirac operator which factorizes the heat equation. We first establish and study the Fischer decomposition for the homogeneous part of the parabolic Dirac operator. Subsequently we construct null-solutions of the parabolic Dirac operator by means of series expansions in terms of cylindrical co-ordinates. At last, using the obtained recursive relations and through the use of the generalized hypergeometric functions we get a neat and simple form of the null-solutions of the parabolic Dirac operator.

## Sparse image representation using discrete moments (Roy Oste, Ghent University)

*Abstract:*

Sparse representations of images or signals by means of an overcomplete basis, usually called a dictionary, have seen successful application in image processing, denoising, compression, etc. An important aspect is the choice of the properties of the chosen dictionary, such as for instance the average coherence between its elements. In this talk we explore the idea of using moments based on discrete orthogonal polynomials to create suitable predesigned dictionaries. Unlike dictionaries trained on a set of representative examples, the predesigned dictionaries are generic and besides fast computation have supportive theoretical foundations.

Joint work with Srdan Lazendic, given by Roy Oste.

## On the construction of Casimir operators (Phillip Isaac, University of Queensland)

*Abstract:*

We present a straightforward algorithm for producing Casimir operators of a Lie algebra. This algorithm has been shown to be effective in determining Casimir operators of certain classes of non semisimple Lie algebras. We discuss some structural issues that arise.

## The blocks of the periplectic Brauer algebra in positive characteristic (Sigiswald Barbier, Ghent University)

*Abstract:*

Representation theory of the symmetric group can be related to representation theory of the general linear group via Schur-Weyl duality. Similarly, Schur-Weyl duality also relates the orthogonal Lie group, the symplectic Lie group and the encompassing orthosymplectic Lie supergroup to the Brauer algebra, and it relates the periplectic Lie supergroup to the periplectic Brauer algebra.

In this talk I will introduce this periplectic Brauer algebra and present recent work on the classification of the blocks of the Brauer algebra in positive characteristic.

This is joint work with A. Cox and M. Devisscher

## The harmonicity of slice regular functions (Cinzia Bisi, Università Degli Studi di Ferrera)

*Abstract:*

I will start improving the definition of slice regular function over the quaternions given by Gentili-Struppa in 2006-2007, and of monogenic function over a real Clifford algebra introduced by Colombo-Sabadini-Struppa in 2009. Then, bringing new ideas to the theory, I will answer positively to the question: is a slice regular function over the quaternions (analogously to a holomorphic function over the complex) ”harmonic” in some sense, i.e. is it in the kernel of some order-two differential operator over the quaternions? Finally, I will deduce novel integral formulas as applications. This is part of a project with J. Winkelmann