# Master Thesis Proposals

Subjects will be proposed customized to the student. A number of examples can be found below.

## Subject 1: Generalized Fourier transforms: discrete versus continuous

The classical Fourier transform (FT) can be related to a realization in terms of differential operators of the Lie algebra sl(2). This way of rewriting allows us to get far-reaching generalizations of the FT., by realizations of sl(2) (or even more complicated Lie (super) algebras) through new differential operators. During the last years, this is an active area of research, where both continuous and discrete transformations were introduced. In many cases, there is moreover an interesting quantum mechanical interpretation.

The purpose of this thesis is to study a number of such transformations in detail and make a comparing study of the discrete and continuous case.

**Supervisors**: Hendrik De Bie (S8, office 130.062) and Joris Van der Jeugt

## Subject 2: Study of the symplectic Dirac operator

The goal of this master thesis subject is to go through a recent book of Habermann and Habermann, which deals with the symplectic Dirac operator. This differential operator is, in comparison to the classical orthogonal invariant Dirac operator, invariant under the symplectic group. The consequence is that one has to use symplectic Clifford algebras, which act on the infinite dimensional space of symplectic spinors. Knowledge of differential geometry is necessary, since this operator has to be constructed very generally.

**Supervisor**: Hendrik De Bie (S8, office 130.062)

## Subject 3: Tensor product representations for Lie (super) algebras

The goal of this thesis is on the one hand a study of some known techniques from representation theory of Lie algebras, and on the other hand an application on open questions for Lie super algebras. First, the technique of Littlewood-Richardson and Young tableaus will be applied to tensor powers of the fundamental representations of the linear and orthogonal Lie algebra. Together with taking trace-representation, this offers a new framework to study the complete reducibility of representations of the orthosymplectic Lie super algebra.

**Supervisor**: Hendrik De Bie (S8, office 130.062)

## Subject 4: Hermitian Clifford analysis

Hermitian Clifford analysis is a refinement of orthogonal Clifford analysis, where the classical Dirac operator is rewritten as the sum of two complex differential operators. The consequence of this approach is that it leads to a much richer function theory, which consists of complex analysis in several vector variables.

The goal of this subject is to investigate a number of recent papers thoroughly. After a study of the basic frame work, there are a number of different possibilities, namely a study of the polynomial null solutions (the so-called h-monogenic system), fundamental solutions, Cauchy-Kowalevskaya extensions, etc.

**Supervisors**: Hendrik De Bie (S8, office 130.062) and Hennie De Schepper (S8, office 130.064)