Department of Electronics and Information Systems

Seminars Academic Year 2022-2023

Graphical calculus for quantum vertex operators (Hadewijch De Clercq, Ghent University)

Abstract:

Graphical calculus provides a diagrammatic framework for performing topological computations with morphisms in strict ribbon categories. This amounts to a functorial identification of such morphisms with oriented diagrams colored by a ribbon category, such as the category of finite-dimensional representations of a quantum group. In this talk I will explain how the graphical calculus can be extended to a larger category of quantum group representations, encompassing the q-analog of the BGG category O. In particular, this extended framework allows to graphically represent quantum vertex operators on Verma modules, as well as morphisms depending on a dynamical parameter, such as dynamical R-matrices. I will demonstrate the potential of this approach by graphically deriving q-difference equations for twisted trace functions of N-point quantum vertex operators. These include the dual q-KZB and dual Macdonald equations first obtained by Etingof and Varchenko, as well as some generalizations.

I will conclude the talk by a short demo of the IPE drawing editor, which I use as a tool to create graphical calculus diagrams.

This talk is based on joint work with Jasper Stokman (University of Amsterdam) and Nicolai Reshetikhin (UC Berkeley and Tsinghua University).

Almansi and Fueter’s theorems for slice regular functions of several quaternionic variables (Giulio Binosi, Università di Trento)

Abstract:

Classical Almansi Theorem (1899) states that any p-harmonic real function can be expressed as a p-tuple of harmonic functions. Moreover, since any slice regular function f : H → H is bi-harmonic, Almansi Theorem has been generalized for slice regular functions of one quaternionic variable. Motivated by the reasonable expectation that slice regular functions in several quaternionic variables were (separately) bi-harmonic w.r.t. each variable, we provide Almansi-type decompositions in the several variables setting, too. Namely, if f : Hn → H is slice
regular, we can decompose f in 2n different ways, for any H ∈ P({1, ..., n}) as 

f(x) = \sum_{K⊂H}(−1)|H\K|xH\K ⊙ SHK(f),
where every component SHK(f) is a slice function, separately harmonic and constant over the sets Sα,β ={α + SHβ}, w.r.t. xh, ∀h ∈ H. As a consequence, we are able to prove that Δ2hf = 0, for any h = 1, ..., n. Moreover, if we consider decompositions related to any complete ordered set {1, ...,m − 1}, we find integral mean value formulas, as well as a generalization of Fueter’s theorem for several quaternionic variables: for every K ⊂ {1, ...,m − 1}, ΔmS{1,...,m−1}K (f) is an xm-monogenic function, i.e. ΔmS{1,....,m−1}K (f) ∈ ker(∂xm), where ∂xm := 1/2 (∂αm + i∂βm + j∂γm + k∂δm) is the Cauchy-Riemann-Fueter (or Dirac) operator w.r.t. xm. Finally, it has been proved that for any odd n = 2m+ 1, any slice regular function f : Rn+1 → Rn over the Clifford Algebra Rn satisfies Δm+1f = 0 and an Almansi-type decomposition has been found. Next step will be provide a similar decomposition for slice regular functions of several Clifford variables.

Null-solutions, CK-extensions and symmetry operators of the parabolic Dirac operator (Teppo Mertens, Ghent University)

Abstract:

In classical Clifford analysis, one uses Clifford algebras to factorize the Laplacian as minus the square of the Dirac operator.

In this talk, I will discuss how one can use similar techniques to factorize the heat operator as the square of the so-called parabolic Dirac operator.

Furthermore, I will discuss a special class of null-solutions of the parabolic Dirac operator generated as a series expansion.
This will result in the study of CK-extensions in this setting.
Finally I will discuss generalized symmetry operators of the parabolic Dirac operator.
The key to finding such operators is to study all the possible null-solutions of the parabolic Dirac operator.

A hermitian refinement of symplectic Clifford analysis (Güner Muarem, University of Antwerp)

Abstract:

In this talk I will start by introducing the symplectic Dirac operator on the canonical symplectic manifold R^2n. The operator is then obtained as a contraction between Weyl algebra (symplectic Clifford algebra) generators and derivatives by using the symplectic form. This is in complete analogy with the classical Dirac operator, where one makes the contraction of Clifford algebra generators and derivatives using the Riemannian (flat) metric. Further analogy between the orthogonal and symplectic case is however broken due to the fact that the symplectic spinor space is infinite dimensional, making the representation theory of the symplectic spinors more involved. In particular, we will consider a symmetry reduction of sp(2n) to u(n) by introducing a compatible complex structure J. This will lead to a couple of symplectic Dirac operators with unitary invariance leading to interesting Fischer decompositions.

Uncoiled algebras, their Wenzl–Jones idempotents and how to construct them (Alexis Langlois-Rémillard, Ghent University)

Abstract:

The affine Temperley–Lieb algebras are a family of infinite-dimensional algebras with a diagrammatic presentation. They have been studied in the last 30 years, mostly for their physical applications in statistical mechanics, where the diagrammatic presentation encodes the connectivity property of the models. Most of the relevant representations for physics are finite-dimensional. In this work, we define finite-dimensional quotients of the affine Temperley–Lieb algebras that we name uncoiled algebras in reference to the diagrammatic interpretation of the quotient and construct a family of Wenzl–Jones idempotents, each of which projects onto one of the one-dimensional modules these algebras admit. We also prove that the uncoiled algebras are sandwich cellular and sketch some of the applications of the objects we defined.

This is joint work with Alexi Morin-Duchesne.

Uncertainty relations for multiple observables (Pan Lian, Tianjin Normal University)

Abstract:

The uncertainty relations for multiple observables can be traced back to Robertson's work in 1934.  In this talk, I will introduce some interactions between Clifford analysis and uncertainty relations for multiple operators. Firstly, we will prove the sum uncertainty relation proposed by Dodonov in 2018 using an inequality of Clifford numbers. Then based on the Wigner-Yanase skew information and the Kirillov-Kostant-Souriau Kahler structure on the quantum phase space, we will introduce new geometric uncertainty relation for multiple observables, which further refines Dodonov's uncertainty relation. (Joint work with Bin Chen, Tianjin Normal University.)

A general approach to constructing minimal representations of Lie supergroups (Sam Claerebout, Ghent University)

Abstract:

I describe an approach to generalise minimal representations to the super setting for Lie superalgebras obtained from Jordan superalgebras using the TKK construction. This approach was used successfully to construct a Fock model, a Schrödinger model and intertwining Segal-Bargmann transform for the orthosymplectic Lie supergroup OSp(p, q|2n) and the exceptional Lie supergroup D(2, 1; α).

The Segal-Bargmann transform (David Avtandilov, Master student, Ghent University)

Abstract:

In this talk, we discuss the analytic and the classical approach to the construction of the Segal-Bargmann transform. We highlight its quantum mechanical relevance and focus on its main properties.  We also cover one of many generalizations of the Segal-Bargmann transform, specifically in the context of finite Coxeter groups.

Global Harmonic Analysis and the Unramified Eisenstein Spectrum (Marcelo De Martino, Ghent University)

Abstract:

In this talk, I wish to give a motivated introduction to the harmonic analytic aspects of the global Langlands programme. I will also report on a joint work with E. Opdam and V. Heiermann on the parametrization of the space square-integrable functions on the automorphic quotient of a split semisimple group which occur as residues of the unramified Borel Eisenstein series.

Point clouds real-time generation (Nicolas Vercheval, Ghent University)

Abstract:

Point clouds are a simple way of describing a three-dimensional object. They consist of an unordered collection of coordinates easily renderable for 3D graphics. Unsupervised learning of these objects allows for rapid image generation with virtual and augmented reality applications. This seminar will show how a machine-learning model called VQVAE can be the base of a generative point cloud model that produces quality random samples in real time.

Fourier kernels associated with the Clifford-Helmholtz system (Ze Yang, Ghent University)

Abstract:

The Clifford-Helmholtz system is a system of partial differential equations in a Clifford algebra that refines the classical Helmholtz equation. In this talk I will introduce a family of solutions of the Clifford-Helmholtz system. All these solutions can be used as integral kernels of generalized Fourier transforms in hypercomplex analysis. In the Laplace domain they have interesting expressions in terms of terminating hypergeometric functions. This allows us to compute recursion relations between the different kernels and the generating functions for two special cases. Also, the solutions were used to construct a new class of Fourier transforms.  Based on that we obtain several versions of uncertainty principles and real Paley-Wiener theorems for this novel class of Fourier transforms. Since the majority  of these integral transforms are not unitary, the results deviate slightly with the classical ones.

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

Semisimplifying Lie Rep(alpha_3)-algebras (Michiel Smet, Ghent University)

Abstract:

First, we introduce the semisimplification functor on the category Rep(alpha_3) and explain how we can construct Lie superalgebras that only exist over fields of characteristic 3 using this functor. Second, we will apply this functor to a class of Lie Rep(alpha_3)-algebras, namely those that correspond to J-ternary algebras, and, as such, construct some Lie superalgebras unique to characteristic 3.

Interpretable Deep Learning Models for Multichannel and Multimodal Data Processing using Hypercomplex Algebra and Convolutional Sparse Coding (Srđan Lazendić, Ghent University)

Abstract:

In this presentation, I will showcase the results of my PhD studies, focusing on the utilization of representation learning, sparse representation learning, and deep learning as potent tools for high-dimensional data analysis. The primary objective of this research is to enhance the interpretability and explainability of these representation learning tools. To demonstrate their effectiveness, I will present the results achieved in the domains of multichannel image restoration and multimodal image segmentation. By improving interpretability and explainability, these findings contribute to the broader understanding and application of representation learning techniques in complex image analysis tasks.

Diagram categories of Brauer type (Sigiswald Barbier, Ghent University)

Abstract:

Diagram categories are a special kind of tensor categories that can be represented using diagrams. In this talk I will give an introduction to categories represented using Brauer diagrams. In particular I will explain the relation with the Brauer algebra and how the categorical framework can be applied to representation theory of the corresponding algebra.