# Seminars Academic Year 2021-2022

- 11 February 2022

Alexis Langlois-Remillard (Ghent University)

Bases for Dunkl monogenics by generalised symmetries - 18 February 2022

Sam Claerebout (Ghent University)

A minimal representation of the exceptional Lie superalgebra D(2, 1; ɑ) - 25 February 2022

Alexi Morin-Duchesne (Ghent University)

Universality and conformal invariance in percolation models - 4 March 2022

Michiel Smet (Ghent University)

Structurable algebras over arbitrary rings - 25 March 2022

Marcelo De Marino (Ghent University)

Symplectic Dirac Operators for Lie Algebras and graded affine Hecke algebras - 1 April 2022

Srđan Lazendić (Ghent University)

On Interpretability of CNNs for Multimodal Medical Image Segmentation - 22 April 2022

Frederick Maes (Ghent University)

A thermoelastic problem in the dual-phase-lag setting - 29 April 2022

Astrid Massé (Ghent University)

Generalisations of the discrete Weierstrass transform - 6 May 2022

Sigiswald Barbier (Ghent University)

A deformed periplectic Brauer category - 13 May 2022

Yang Ze (Ghent University)

Solutions of the Clifford-Helmholtz system - 20 May 2022

Kamal Diki (Chapman University, California, USA)

An approach to the Gaussian RBF kernels via Fock spaces - 17 June 2022

Nicolas Vercheval (Ghent University)

Betting Networks

## Bases for Dunkl monogenics by generalised symmetries (Alexis Langlois-Remillard, Ghent University)

*Abstract:*

From a root system and a function invariant under the orbit of the related reflection group, Dunkl operators generalise partial derivatives. We are interested in understanding the symmetry algebra of the Dunkl version of the Dirac operator. To this purpose, we introduce a family of commuting generalised symmetries of the Dunkl−Dirac operator inspired by the Maxwell construction in harmonic analysis. These symmetries are then expressed using a Kelvin-type transform in the Dunkl setting and they are linked to projection operators. We use these generalised symmetries to construct bases of the polynomial null-solutions of the Dunkl−Dirac operator, which form a representation of the Dunkl−Dirac symmetry algebra. In the case where the root system is a direct sum of rank one root systems, we retrieve, up to a sign and a permutation of the spinors, the basis of De Bie, Genest, Venet realised by the Cauchy−Kovaleskaya extension.

## A minimal representation of the exceptional Lie superalgebra D(2, 1; ɑ) (Sam Claerebout, Ghent University)

*Abstract:*

We construct a representations for D(2, 1; ɑ) and integrate it to the group level. This representation is a superversion of minimal representations constructed for Hermitian Lie groups of tube type. However, like many other super version of classical unitary representations, this representation is not superunitary. Therefore, we look at a proposed alternative definition of superunitarity and prove that our representation is unitary with respect to this new definition.

## Universality and conformal invariance in percolation models (Alexi Morin-Duchesne, Ghent University)

*Abstract:*

In this talk, I will describe our investigations of the universal behaviour of two critical percolation models: site percolation on the triangular lattice and bond percolation on the square lattice. Both are Yang-Baxter integrable models that can in principle be solved exactly. In the scaling limit, they are conformally invariant and described by non-unitary representations of the Virasoro algebra. I will describe our calculation of the models' partition functions on the cylinder and torus, and how this is related to these Virasoro representations.

This is joint work with A. Klümper and P.A. Pearce.

## Structurable algebras over arbitrary rings (Michiel Smet, Ghent University)

*Abstract:*

Structurable algebras are generalizations of linear Jordan algebras, with which one can also associate a Tits-Kantor-Koecher Lie algebra. However, there is no suitable definition yet for structurable algebras if 1/6 is not contained in the basering. We introduce the more general notion of a quasi-inverse pair, over an arbitrary ring, and associate a Z-graded commutative Hopf algebra with such a pair. We also construct quasi-inverse pairs associated to certain families of structurable algebras and indicate what parts are necessary for a definition of structurable algebras over arbitrary rings which resembles the one for quadratic Jordan algebras a bit more.

## Symplectic Dirac Operators for Lie Algebras and graded affine Hecke algebras (Marcelo De Marino, Ghent University)

*Abstract:*

We define a pair of symplectic Dirac operators in an algebraic setting motivated by the analogy with the algebraic theory of orthogonal Dirac operators. We compute and study the commutator of these two elements in the context quadratic Lie algebras and graded affine Hecke algebras. This commutator can be seen as an analogue of Parthasarathy's formula for the square of orthogonal Dirac operators.

This is a joint work with D. Ciubotaru and P. Meyer.

## On Interpretability of CNNs for Multimodal Medical Image Segmentation (Srđan Lazendić, Ghent University)

*Abstract:*

Despite their huge potential, deep learning-based models are still not trustful enough to warrant their adoption in clinical practice. The research on the explainability of deep learning is currently attracting huge attention. Multilayer Convolutional Sparse Coding (ML-CSC) data model, provides a model based explanation of convolutional neural networks (CNNs). In this talk, we will explain the extension of the ML-CSC framework towards multimodal data for medical image segmentation, and propose a merged joint feature extraction ML-CSC model, with special emphasis on theoretical and practical analysis of the proposed multimodal extension. We will derive a sparse coding algorithm which enables the systematic design of an interpretable multimodal CNN segmentation model. The experimental analysis shows that the segmentation results obtained are consistent with the derived theoretical results. Furthermore, we will present the conducted interpretability study verifying whether the theoretical interpretability claims for the CNNs are also valid in practice.

## A thermoelastic problem in the dual-phase-lag setting (Frederick Maes, Ghent University)

*Abstract:*

Thermoelastic systems aim to model the changes in the shape of an object and the fluctuations in temperature. These models consist of two equations, a vectorial hyperbolic equation for the displacement and a parabolic heat equation, that are coupled. We will be interested in the use of Tzou's model for the dual-phase-lag heat conduction in a thermoelastic setting. The arising coupled system will be of hyperbolic nature, yielding finite speed of heat propagation. In this talk, we will discuss the existence and uniqueness of a weak solution to this system. Our results are based on a variational approach and Rothe's method, and yield low regularity conditions on initial data and sources

The presented work is a joint work with Karel Van Bockstal.

## Generalisations of the discrete Weierstrass transform (Astrid Massé, Ghent University)

*Abstract:*

The classical Weierstrass transform is an isometric operator mapping elements of the weighted L_2-space to the Fock space. We defined an analogue version of this transform in discrete Hermitian Clifford analysis, where functions are defined on a grid rather than the continuous space, in dimension 1. This transform is based on the discrete version of the Gaussian function and discrete counterparts of the classical Hermite polynomials. The aim of this talk is to extend the definition to higher dimensions, where we must take into account the anticommutativity of the basic Clifford elements and make use the generalised discrete Hermite polynomials. Furthermore, we investigate in dimension 1, what happens if the mesh width approaches 0.

## A deformed periplectic Brauer category (Sigiswald Barbier, Ghent University)

*Abstract:*

The symmetric group algebra, the Brauer algebra and the periplectic Brauer algebra all can be represented using diagrams. This interpretation allows us to construct for each class of these diagram algebras a corresponding category. Its objects are the natural numbers and homomorphisms between m and n in this category are given by (m,n)-diagrams. The diagram algebras themselves can then be recovered by the endomorphism algebras in this category.

For the symmetric group algebra and the Brauer algebra there exists deformations, namely the Iwahori-Hecke algebra and the Birman-Wenzl-Murakami algebra. These deformed algebras can also be represented using diagrams. However, until recently a deformation of the periplectic Brauer algebra was not known.

In this talk I will introduce a new monoidal supercategory using diagrams. The endomorphism algebras of this category recover the quantum periplectic Brauer algebra recently introduced in the setting of Schur-Weyl duality for the quantized enveloping superalgebra of type P by Ahmed, Grantcharov and Guay.

## Solutions of the Clifford-Helmholtz system (Yang Ze, Ghent University)

*Abstract:*

The Clifford-Helmholtz system as a system of PDEs in a Clifford algebra forms a refinement of the classical Helmholtz equation. Recently we use Laplace transform method to consider this system and try to get the exponential generating functions for the kernels in even-dimensional case.

The Laplace transforms of the solutions are obtained easily, we then equally derive an explicit formula of kernels, which can be further simplified to an infinite sum of Gamma functions and hypergeometric functions. After getting the recursion relations of kernels, it’s possible to find the exponential generating functions of kernels. In this talk we will discuss some attempts on this goals.

The presented work is a joint work with Hendrik De Bie and Roy Oste.

## An approach to the Gaussian RBF kernels via Fock spaces (Chapman University, USA)

*Abstract:*

The Gaussian RBF kernel is one of the most used kernels in machine learning kernel methods such as support vector machines (SVMs) algorithms. In this talk we apply Fock spaces to study Gaussian RBF kernels of a complex variable. It turns out that complex analysis techniques allow us to consider feature spaces and feature maps of these kernels using the Segal-Bargmann transform. We will also discuss how these RBF kernels can be related to some important operators in quantum mechanics and time frequency analysis, specifically, we study different connections of Gaussian RBF kernels with creation, annihilation, Fourier, translation and Weyl operators. In particular, a semi-group property will be proved in the case of Weyl operators. Finally, we use quaternionic Fock spaces in order to introduce RBF kernels of a quaternionic variable.

## Betting Networks (Nicolas Vercheval, Ghent University)

*Abstract:*

Neural network classifiers are usually probabilistic, in that they typically select their prediction(s) based on an output discrete probability distribution of the label given the sample called confidence scores. On the one hand, confidence scores are helpful to the final user as they present a quantitative risk estimation when accepting the prediction on never-seen-before data. On the other hand, both the frequentist and the Bayesian interpretation of probability are incompatible with current machine learning challenges. In this seminar, I will discuss the De Finetti definition of probability and its implementation as a bet between two models. The related betting loss shows good performances and offers some connections to other loss functions.