# Bachelor Project Proposals

Several topics are possible, related to the course Function spaces. These topics typically consider orthogonal polynomials, differential equations, Fourier transforms or (generalizations of) complex analysis. Other topics can be determined upon mutual agreement.

These projects are under the guidance of Prof. Dr. Hendrik De Bie.

A typical example of a topic is given below.

## Bannai-Ito polynomials and association schemes

The Bannai-Ito polynomials are an important class of discrete orthogonal polynomials. The aim of this project is twofold. First the student will study a recent paper about these polynomials, their properties and the algebraic structure that describes them.

Second, we will investigate in what other parts of mathematics these polynomials appear naturally. This is surprisingly diverse: they appear in the study of a generalised 3D Dirac equation, but also appear naturally in a classification problem in algebraic combinatorics, namely in the framework of association schemes.

## The sensitivity conjecture

The sensitivity conjecture was a long standing open problem in computer science and discrete mathematics, solved in July 2019 by Hao Huang.

The aim of this bachelor project is to look at the history of the problem, to study the proof of Huang and compare it with two other versions of the proof. The version of Terence Tao uses convolutions and the version of Daniel Mathews uses Clifford algebras.

## Dunkl operators in dimension 2

Dunkl operators are generalizations of partial derivatives which are a linear combination of differential and difference operators. These operators allow us for example to define a Laplacian which is only invariant under a finite reflection-group and not under the full orthogonal group. The goal of this project is to study these operators in dimension 2, for all dihedral groups. In particular, the Dunkl operators in this situation can be used to define a generalized Cauchy-Riemann operator, of which we plan to determine its kernel explicitly in terms of known orthogonal polynomials. (This project has an analytical, a geometrical and a computational component).