Clifford Research Group

Seminars Academic year 2016-2017

Exploratory data mining: an information-theoretic perspective
Tijl De Bie (UGent)

As the abundance of data grows exponentially, so does the challenge of extracting the full value data holds. Extracting such value is often done by means of machine learning, building models to predict expensive or hard-to-measure target variables. Often however, data analysts have no specific target variables in mind, and only wish to explore the data in order to learn something that is interesting to them. The latter type of task is much less well-defined, as the degree to which a relation found in data is interesting to somebody is inherently subjective. This talk surveys a framework to put it onto a more rigorous footing, and discusses instantiations of this framework to a range of data mining problem types.

Quantum phases of matter and tensor networks
Jutho Haegeman (UGent)

One of the goals of theoretical many body physics is to understand and classify the different phases of matter. Even at zero temperature, different phases of matter can exist due to the presence of quantum fluctuations, and symmetry and topology plays a crucial role in the classification thereof. We will introduce the concept of tensor network states, which provide a convenient framework both for numerical simulations and for a theoretical understanding of the different gapped phases of matter. In particular, we will show how group cohomology emerges in the classification table of quantum phases.

Recent results in discrete Clifford analysis
Hilde De Ridder (CRG)

In this talk i will give an overview of some recent results in discrete Clifford analysis. Discrete Clifford analysis aims to offer a discrete version of Euclidean Clifford analysis which is in itself a refinement of harmonic analysis. We construct an operator theory and consider how typically continuous definitions may be discretized. As examples, I will discuss rotations over general angles and translations over general vectors. I will give some details on the introduction of a (discrete) Spingroup Spin(m) which is similar to the Spingroup in Euclidean Clifford analysis. I will give some preliminary results on integration over the unit sphere.

On the algebra of symmetries of Laplace and Dirac operators
Roy Oste (UGent)

We consider a generalization of the classical Laplace operator in the context of Wigner quantization. For this Laplace-like operator, we determine a class of symmetries commuting with it, which are generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anti-commute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher rank Bannai-Ito algebra.

A model for the higher rank Racah algebra
Wouter van de Vijver (UGent)

We propose a generalization of the Racah algebra by considering the tensor product of n copies of su(1,1). Its role as symmetry algebra for the Z2n Dunkl-Laplacian and its connection to the generalized Bannai-Ito algebra are explained. Bases for Dunkl-harmonics are constructed so that each basis consists of joint eigenfunctions of a maximal abelian subalgebra of the generalized Racah algebra. A method is provided for finding the connection coefficients between these bases. The connection coefficients correspond to the multivariate Racah polynomials as defined by M. V. Tratnik.
We also propose a realization of the higher rank Racah algebra in terms of shift operators. This realization contains the operators found by J. Geronimo and P. Iliev which have the multivariate Racah polynomials as eigenfunctions.
This is joint work with Hendrik De Bie, Vincent Genest and Luc Vinet.

An introduction to thermodynamic time invariance for reversible chemical reactions
Zoë Gromotka (UGent)

For reversible chemical reactions the equilibrium constant is defined as the limit in time of the reaction quotient, a quotient like function of concentrations. The reaction rate of first order and second order reversible reactions can be described using linear and quadratic ODE’s respectively. For these first and second order reversible reactions, quotient like functions of the concentrations (different from the reaction quotient) have been found that equal the equilibrium constant for all time and not just in its limit. In this talk I will give a short synopsis of the derivation for the quotient like functions for the first and second order reversible reactions. Which will be followed with my progress so far to extend this theory to third order reversible reactions.

Discrete analytic functions: from the complex plane to higher dimensions
Stefano Dalbosco: Masterthesis

After we have defined analytic discrete functions, we are going to prove some theorems to come to a special family of functions, the so called expandable discrete analytic functions. In this family we can construct the Cauchy-Kovalevskaya product which allows us to define rational discrete analytic functions. Afterwards, we introduce two extensions: the monogenic one and the harmonic one. We also determine the monogenic extension of the delta function.

Dictionary learning methods in signal and image processing
Srđan Lazendić

In the recent years, dictionary learning techniques and sparse representation models attract the interest of researchers from different fields, both from the theoretical side and the side of the application. The idea is that every signal can be represented as the linear combination of the normalized basis vectors from the (redundant) dictionary D. Moreover, we are interested in adaptive representation which means that the signal can be represented by using just few basis vectors from the dictionary D. Very recent trend in color image processing is introducing quaternions into dictionary construction. The quaternionic representation is ideally suited for representing three color channels as three imaginary units. Also, next to the best-known approach for dictionary learning the so-called K-SVD method, new K-QSVD in quaternionic framework is proposed. New approach already showed remarkable results. Since algebra of quaternions is special case of Clifford algebra, in this talk I will briefly explain already existing results and propose new idea using Clifford algebra methods.

Gelfand-Tsetlin bases in harmonic analysis
Michael Wutzig

In this talk we will give an overview of the Gelfand-Tsetlin construction of bases for spherical harmonics in three different settings.
As an introduction, the Cauchy-Kovalevskaya extension for the classical spherical harmonics is combined with the harmonic Fischer decomposition to derive a recursion formula for basis polynomials. We will present a Maple implementation of this recursive algorithm to visualize these classic results.
When considering harmonic bi-homogeneous polynomials in n complex variables, the construction of basis elements becomes slightly more complicated. Using a complex CK extension, one can again implement a recursive algorithm to create bases for arbitrary (complex) dimensions and orders of homogeneity. Finally there will be an introduction to symplectic polynomials and symplectic harmonics in 2n complex variables. We will present our attempts of using the Gelfand-Tsetlin construction in this setting, as well as the difficulties that are occurring.

Radial derivative of the delta distribution
Fred Brackx

Possibilities for defining the radial derivative of the delta distribution in the setting of spherical co-ordinates are explored. This leads to the introduction of a new class of continuous linear functionals, similar to but different from the standard distributions, to which the radial derivative and some other operations on the delta distribution then belong.

A minimal representation for osp(p,q|2n)
Sigiswald Barbier

Minimal representations are an important class of "small" infinite dimensional unitary representations of Lie groups. They are characterised by the fact that their annihilator ideal is equal to the Joseph ideal. Two prominent examples are the metaplectic representations of Mp(2n) (a double cover of Sp(2n)) and the minimal representation of the indefinite orthogonal group O(p,q).

In this talk I will present a generalisation of the minimal representation of O(p,q) to the Lie superalgebra osp(p,q|2n) using the framework of Jordan (super)algebras. A prominent role is played by the so-called `Bessel operators`. Our representation also has an annihilator ideal equal to a Joseph-like ideal.

Higher spin Laplace operators in several vector variables
Tim Raeymaekers

In this talk, I will introduce higher spin Laplace operators, which generalise the classical Laplace operator to functions taking valuse in arbitrary SO(m)-representations. I will also take a look at a special type of polynomial solutions of these operators, which are believed to generate the entire polynomial kernel of these operators.

Department of Mathematical Analysis Department of Mathematical Analysis

Faculty of Engineering and Architecture