# Seminars Academic Year 2020-2021

- 5 March 2021

Sigiswald Barbier (Ghent University)

Unitary representations of Lie supergroups - 12 March 2021

Kamal Diki (Politecnico di Milano)

A new family of QRKHS in the Clifford-Appell case - 26 March 2021

Teppo Mertens (Ghent University)

Radon type transforms in the Clifford setting - 2 April 2021

Alexis Langlois-Rémillard (Ghent University)

Generalizing the Deligne category via cobordisms, Khovanov's and Sazdanovic's approach - 23 April 2021

Sam Claerebout (Ghent University)

A Schrödinger model, Fock model and intertwining Segal-Bargmann transform for the exceptional Lie superalgebra D(2,1;ɑ) - 30 April 2021

Frederick Maes (Ghent University)

Inverse Source Problems in Fractional Dual-Phase-Lag heat conduction - 21 May 2021

Kieran Calvert (University of Manchester)

Clifford invariants of symmetric Lie algebra pairs - 28 May 2021

Asmus Bisbo (Ghent University)

PBW-type bases for simple osp(1|2n)-modules - 4 June 2021

Srđan Lazendić (Ghent University)

Distributed order fractional wave equations with irregular coefficients - 11 June 2021

Alí Guzmán Adán (Ghent University) - 18 June 2021

Hadewijch De Clercq (Ghent University) - 25 June 2021

Kristof Rose - 2 July 2021

Nicolas Vercheval (Ghent University)

## Unitary representations of Lie supergroups (Sigiswald Barbier, Ghent University)

*Abstract:*

Understanding all unitary irreducible representations of a Lie group is an important goal in representation theory. Likewise we also want to understand all unitary irreducible representations of a Lie supergroup. Unfortunately, the standard definition of unitarity in the super sense has the disadvantage that a lot of a Lie supergroups do not have any unitary representations at all. Therefore people have been proposing alternative definitions of unitarity. In this talk we will take a look at some of these alternatives.

## A new family of QRKHS in the Clifford-Appell case (Kamal Diki, Politecnico di Milano)

*Abstract:*

We will use the classical Fueter mapping theorem in order to construct a special system of Clifford-Appell polynomials in the quaternionic setting. We will discuss various properties of such polynomials and study their algebraic behaviour with respect to the well-known Cauchy-Kowalevski product. Then, based on this system we can introduce a new family of quaternionic reproducing kernel Hilbert spaces in this framework. As particular examples we will treat further results on the Hardy and Fock spaces and their related operators such as creation, annihilation, shift and backward shift operators. Finally, if time allows we will discuss some connections and applications of this new approach to the recent theory of quaternionic slice polyanalytic functions and Schur analysis. This talk is based on recent works that are in collaboration with Daniel Alpay, Fabrizio Colombo, Soeren Krausshar and Irene Sabadini.

## Radon type transforms in the Clifford setting (Teppo Mertens, Ghent University)

*Abstract:*

The Radon transform is an integral transform with many applications in both pure and applied mathematics. The Szegö-Radon transform, an integral transform of monogenic functions over the unit sphere, is its counterpart in the Clifford setting. It can moreover be generalized to the framework of complex analysis by replacing the monogenic functions by their holomorphic extensions. This gives rise to several Radon type transforms over the Lie sphere, of which we will give an overview. Afterwards, we will discuss how to invert these transforms by means of so-called dual Radon transforms.

## Generalizing the Deligne category via cobordisms, Khovanov's and Sazdanovic's approach (Alexis Langlois-Rémillard, Ghent University)

*Abstract:*

The Deligne category is Deligne's answer to the following question: "If we were to define the symmetric group outside the non-negative integer, what would its representation theory look like?" Following this idea, he constructed a category that "interpolates" between the categories of finite-dimensional representations of all the symmetric groups for any element of an arbitrary characteristic 0 field. When specialized to non-negative integer, it is equivalent, up to quotient, to the category of finite-dimensional representations of the symmetric group. In general, the Deligne category admits a diagrammatic construction via partition diagrams, which was used extensively by Comes and Ostrik to study its blocks. Thickening partition diagrams gives an equivalence between the partition category generated by partition diagrams and the 2D cobordisms up to two additional relations. In a recent work, Khovanov and Sazdanovic have given a multi-parameters generalization of the Deligne category starting from cobordism categories by using the extra information given by the genus of the surfaces involved. This talk presents their interesting construction image by image.

## A Schrödinger model, Fock model and intertwining Segal-Bargmann transform for the exceptional Lie superalgebra D(2,1;ɑ) (Sam Claerebout, Ghent University)

*Abstract:*

We construct two infinite-dimensional irreducible representations for D(2,1;ɑ): a Schrödinger model and a Fock model. Further, we also introduce an intertwining isomorphism. These representations are similar to the minimal representations constructed for the orthosymplectic Lie supergroup and for Hermitian Lie groups of tube type. The intertwining isomorphism is the analogue of the Segal-Bargmann transform for the orthosymplectic Lie supergroup and for Hermitian Lie groups of tube type.

## Inverse Source Problems in Fractional Dual-Phase-Lag heat conduction (Frederick Maes, Ghent University)

*Abstract:*

Non-classical thermal models based on a non-Fourier type law have attracted a lot of interest in the past few decades. In this talk I will discuss the fractional dual-phase-lag heat equation and uniqueness in an associated inverse source problem. First an introduction to the modeling part and fractional calculus will be given. Next I will state and discuss our main uniqueness results of determining a space dependent source given the final time observation. Finally, a possible relaxation of the assumptions will be investigated in two modified models.

This talk is based on joint work with Marian Slodicka.

## Clifford invariants of symmetric Lie algebra pairs (Kieran Calvert, University of Manchester)

*Abstract:*

Kostant proved that if g is a semisimple Lie algebra then the Clifford invariants Cl(g)^{g} is isomorphic to a Clifford algebra on the primitive subspace related to g. We prove the analog of Kostants theorem for symmetric pairs (g; k), where g decomposes as p + k. The Clifford invariants Cl(p)^{k} are used in Dirac cohomology. In this talk we describe the structure of Cl(p)^{k }as a graded algebra.

## PBW-type bases for simple osp(1|2n)-modules (Asmus Bisbo, Ghent University)

*Abstract:*

The Lie superalgebra osp(1|2n) describes the interactions of parabosonic particles, relevant in several areas of theoretical physics. We construct polynomial PBW-type bases for an important class of simple osp(1|2n)-modules know as parabosonic Fock spaces. We present two complementary expressions for the basis elements. Firstly, they are written as polynomials in the odd generators of osp(1|2n) acting on the lowest weight vector. Secondly, they are written as monomials in the generators of the u(n) subalgebra of osp(1|2n) acting on the u(n)-highest weight vectors. Matrix elements corresponding to this basis will also be discussed.

## Distributed order fractional wave equations with irregular coefficients (Srđan Lazendić, Ghent University)

*Abstract:*

In this talk, we derive and analyze fractional wave equations describing wave propagation in one-dimension viscoelastic media, modeled by distributed-order fractional constitutive stress-strain relation. More precisely, we consider the system of

equations which consists of the equation of motion of the one-dimensional deformable-body, the constitutive equation of distributed order fractional type, which describes the mechanical properties of the linear viscoelastic body, and the strain for small local deformations. The system is equivalent to the integrodifferential wave-type equation. First, we show that that the fundamental solution to the generalized Cauchy problem for the distributed order wave equation exists and it is unique. In particular, the fundamental solutions corresponding to four thermodynamically acceptable classes of linear fractional constitutive models and power-type distributed-order models will be discussed. Those results regarding homogeneous materials are obtained in [1].

Further, we are interested in obtaining similar results for the cases when viscoelastic material/media are heterogeneous. This implies that the coefficients appearing in equations become non-smooth functions depending on space (or even time) which could also be irregular, such as Dirac delta distribution. We shall concentrate on existence and uniqueness results of weak and very weak solutions. Finally, since the fractional order derivatives are difficult to handle numerically in their original form, we will motivate why the obtained mathematical model, which combines the Laplace transform together with the variational methods, is perfectly suitable for further practical applications.

Joint work with: Sanja Konjik and Ljubica Oparnica.

[1] S. Konjik, S. Lazendic and Lj. Oparnica, Distributed-order fractional constitutive stress-strain relation with irregular coefficients in wave propagation modeling, 2021, Submitted.