# Seminars Academic Year 2019-2020

- 24 January 2020

Hadewijch De Clercq (Ghent University)

Defining relations for quantum symmetric pair coideals of Kac-Moody type - 10 February 2020

Nicolas Crampé (CNRS, Institut Denis-Poisson, Tours)

Missing label for su(3), Gaudin model and Bethe equations - 28 February 2020

Sam Claerebout (Ghent University)

Fock model and Segal-Bargmann transform for the orthosymplectic Lie superalgebra - 6 March 2020

Alexis Langlois-Rémillard (Ghent University)

The symmetry algebra of the three-dimensional dihedral Dunkl-Dirac equation - 8 May 2020

Juliet Cooke (UCLouvain)

Askey-Wilson algebras - 15 May 2020

Hendrik De Bie (Ghent University)

The Dunkl intertwining operator - 12 June 2020

Simon Apers (Inria (France), CWI (the Netherlands))

Quantum Walks and Electric Networks - 19 June 2020

Ren Hu (Ghent University)

Szegö-Radon transform for hypermonogenic functions - 26 June 2020

Srdan Lazendic (Ghent University)

Finite Heisenberg-Weyl group: from theory to applications - 3 July 2020

Frederick Maes (Ghent University)

Halász's theorem for Beurling generalized numbers - 10 July 2020

Alí Guzmán Adán (Ghent University)

Generalized Cauchy-Kovalevskaya extension and plane wave decompositions in superspace - 28 August 2020

Jan Frahm (Aarhus University)

Conformally invariant systems of differential operators on Heisenberg groups and L^{2 }-models for minimal representations

## Defining relations for quantum symmetric pair coideals of Kac-Moody type (Hadewijch De Clerq, Ghent University)

*Abstract:*

Classical symmetric pairs consist of a Kac-Moody algebra g together with its subalgebra of fixed points under an involutive automorphism. Quantum analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras of g by one-sided coideal subalgebras of the quantum group U_q(g). In this talk I will explain this construction and give some motivation and examples arising from quantum integrability. Special attention will be drawn to the presentation of these algebras by generators and relations. I will present a novel set of defining relations of q-Serre or q-Dolan-Grady type for these quantum symmetric pair coideals, based on two different methods. One uses a projection technique established by Letzter and Kolb, the other builds on q-binomial identities.

## Missing label for su(3), Gaudin model and Bethe equations (Nicolas Crampé, CNRS, Institut Denis-Poisson, Tours)

*Abstract:*

I introduce the old problem of the missing label in the direct sum decomposition of two irreducible representations of the Lie algebra su(3). Then, I show how the well-known quantum integrable system called Gaudin model provides this missing label. Finally, I discuss about the resolution via the Bethe ansatz of the Gaudin model and how the associated Bethe equations can be solved to give some explicit examples. This talk is based on an on-going work in collaboration with L. Poulain d'Andecy and L. Vinet.

## Fock model and Segal-Bargmann transform for the orthosymplectic Lie superalgebra (Sam Claerebout, Ghent University)

*Abstract:*

The classical Segal-Bargmann is an integral transform between the Schrödinger space of square-integrable functions and the Fock space of holomorphic functions. The Segal-Bargmann transform was reinterpreted by Hilgert-Kobayashi-Möllers-Ørsted as an intertwining operator between realisations on the Schrödinger and the Fock space of the minimal representation of a Lie group. In this talk I will give a generalisation of this approach to superspaces in order to obtain a Segal-Bargmann transform as an integral transform that intertwines the Schrödinger and Fock model for the orthosymplectic Lie superalgebra osp(m,2|2n).

## The symmetry algebra of the three-dimensional dihedral Dunkl-Dirac equation (Alexis Langlois-Rémillard, Ghent University)

*Abstract:*

The Dunkl-Dirac equation is a deformation of the Dirac equation obtained by changing partial derivatives to Dunkl operators. We consider the cases of three-dimensional Dunkl operators associated to a dihedral root system. The symmetry algebra is considered both abstractly by generators and relations and via the explicit realization by angular momenta operators. The study proceeds by constructing ladder operators on quadratic elements of the algebra of symmetry and by examining the representation theory on the specific example of polynomial monogenics of the Dunkl-Dirac equation obtained in explicit forms by a Cauchy-Kovalevskaya extension theorem.

Work with Hendrik De Bie, Roy Oste and Joris Van der Jeugt

## The Dunkl intertwining operator (Hendrik De Bie, Ghent University)

*Abstract:*

There are two crucial operators in the theory of Dunkl operators. The first is the Dunkl transform, which generalizes the Fourier transform. The second is the intertwining operator, which maps ordinary partial derivatives to Dunkl operators. Although some abstract statements are known about the intertwining operator, the explicit formula for classes of reflection groups is generally not known. In recent work Yuan Xu proposed a formula in the case of dihedral groups and a restricted class of functions. We extend his formula to all functions and give a general strategy on how to obtain similar formulas for other reflections groups. This is based on joint work with Pan Lian, available under arXiv:2002.09065.

## Quantum Walks and Electric Networks (Simon Apers, Inria (France), CWI (the Netherlands))

*Abstract:*

Electric networks are a critical tool in the study of random walks on graphs. In this talk I will discuss the use of electric networks to study *quantum* walks on graphs. While an important building block of many quantum algorithms, quantum walks are much less understood than random walks. I will show how the electric networks analogy helps to interpret and understand the structure and dynamics of quantum walks.

As a running example, I will demonstrate how quantum walks can speed up the estimation of random walk commute times in a graph.

## Szegö-Radon transform for hypermonogenic functions (Ren Hu, Ghent University)

*Abstract:*

In the classical monogenic framework, the Szegö-Radon transform is abstractly defined as an orthogonal projection operator of a Hilbert module of monogenic functions onto a suitable closed submodule of plane waves with parameters in the Stiefel manifold St(m,2). In this talk, we study a refinement of this transform in the hypermonogenic setting. Hypermonogenic functions form a subclass of monogenic functions arising in the study of a modified Dirac operator, which allows for weaker symmetries and also has a strong connection to the hyperbolic metric. In particular, we construct a projection operator from a module of hypermonogenic functions in R^{p+q} onto a suitable submodule of plane waves parametrized now by a vector on the unit sphere of R^{q}. Moreover, we study the interaction of this Szego-Radon transform with the generalized CK extension operator. Finally, we develop a reconstruction (inversion) method for this transform.

## Finite Heisenberg-Weyl group: from theory to applications (Srdan Lazendic, Ghent University)

*Abstract:*

The continuous Heisenberg-Weyl groups have a long history in physics and signal processing. However, their discrete variants haven’t received the deserved attention. When a signal is transformed by an element of the (much larger) discrete symplectic group, autocorrelations, which are mathematical tool for finding repeating patterns in a signal, can be calculated from trace inner products of covariance matrices with signed permutation matrices from the discrete Heisenberg-Weyl group. The mapping between a signal and its autocorrelation coefficients based on the Heisenberg-Weyl group is called the Weyl transform. This instance of the Weyl transform is a special case of a general framework for representation of operators in harmonic analysis. We will make it clear how those beautiful properties of the Weyl transform can be used for particular applications in image processing and show the original results for image classification.

Moreover, we will show that the convolutional neural networks (CNN) can be constructed by using the Heisenberg-Weyl group where elements of the group can be used as the convolutional filters of the network. It has been observed that the incorporation of the Weyl transform into neural networks allows to classify images much more efficiently and accurately than the baseline networks. We will also see that this approach allows better explainability of CNNs and gives a better control of the learning process.

## Halász's theorem for Beurling generalized numbers (Frederick Maes, Ghent University)

*Abstract:*

The Halász theorem is a cornerstone in probabilistic and analytic number theory. It gives insight in the mean-value behaviour of a multiplicative aritmetic function with values in the unit disk, which is desirable in applications. In particular, when applied to the Möbius function, the theorem itself shows the Prime Number Theorem. We recently extended the Halász theorem to hold in the general setting of Beurling numbers. In this talk, we give an introduction to the field of multiplicative analytic number theory by discussing some main results, techniques and examples related to the classical Halász theorem. Next we will discuss Beurling generalized numbers by means of the associated counting functions and zeta function. Finally, we state and comment on our generalized version of the Halász theorem in this context. This last part is based on collaborative work with prof. J. Vindas and dr. G. Debruyne.

## Generalized Cauchy-Kovalevskaya extension and plane wave decompositions in superspace (Alí Guzmán Adán, Ghent University)

*Abstract:*

In this talk, we present a CK-extension theorem that characterizes null-solutions of the Dirac operator in superspace by their restrictions to a sub-supermanifold or arbitrary codimension. In the classical commuting case, this result can be written as a power series of Bessel type of certain differential operators acting on a single initial function. In the superspace setting, novel structures appear in the cases of negative even superdimensions. In these cases, the CK-extension depends on two initial functions on which two power series of differential operators act. These series are not only of Bessel type but they give rise to an additional structure in terms of Appell polynomials. This pattern also is present in the structure of the Pizzetti formula, which describes integration over the supersphere in terms of differential operators. We make this relation explicit by studying the decomposition of this generalized CK-extension into plane waves integrated over the supersphere. Moreover, we apply these results to obtain a decomposition of the super Cauchy kernel into monogenic plane waves, which shall be useful for inverting the super Radon transform.

## Conformally invariant systems of differential operators on Heisenberg groups and L^{2 }-models for minimal representations (Jan Frahm, Aarhus University)

*Abstract:*

On Euclidean space, the Fourier transform intertwines partial derivatives and coordinate multiplications. As a consequence, solutions to a constant coefficient PDE p(D)u=0 are mapped to distributions supported on the variety {p(x)=0}. In the context of unitary representation theory of semisimple Lie groups, so-called minimal representations are often realized on Hilbert spaces of solutions to systems of constant coefficient PDEs whose inner product is difficult to describe. The Euclidean Fourier transform provides a new realization on a space of distributions supported on a variety where the invariant inner product is simply an L^{2 }-inner product on the variety.

Recently, similar systems of differential operators have been constructed on Heisenberg groups. In this talk I will explain how to use the Heisenberg group Fourier transform to obtain a similar picture in this context.