Experimental Mathematics Lab at the University of Luxembourg

Magic bodies (show description)

Goal:

Magic squares are famous objects of recreational mathematics: square matrices with (distinct positive) integer entries such that the sum of the numbers in each row, in each column and in the two main diagonals always equals the same number. One can simply obtain magic squares by solving systems of linear equations (but over the integers).

The goal of this project is to define and find 3-dimensional magic objects such as magic balls, magic tori, etc. via triangulations or other tilings. Furthermore, tangible 3d objects shall be created.

The mathematics will mostly be linear algebra (over the integers), but it will likely also involve some basic topology because the Euler characteristic might give some constraints (depending on your definition of being magic).

supervisor: Gabor Wiese

The Kepler Problem (show description)

Goal:

Study the Kepler problem, the relationship with Newton’s laws of motion and use (some) numerical methods to approximate the trajectory of a planet around a single star.

Prerequisites: Course on differential equations, Basic knowledge of Python.

Material: https://www.youtube.com/watch?v=nJ_f1h49jfM https://en.wikipedia.org/wiki/Kepler_problem

supervisors: Luis Maia, Francesca Pistolato

LLN and CLT in action! (show description)

Goal:

Running simulations, we will see the law of large numbers (LLN) and the central limit theorem (CLT) in action, and how they break down when the hypothesis they are based on do not hold.

Prerequisites: Probability 1, Python (Target: 4th semester)

Material: https://intro.quantecon.org/lln_clt.html

supervisors: Luis Maia, Francesca Pistolato

Fast multiplications (show description)

Goal:

What’s the fastest matrix multiplication algorithm? Compare three algorithms in terms of their asymptotic complexity and implement them in Python. What happens when the matrices are sparse?

Prerequisites: any course in linear algebra, Python (Target: 4th semester)

Material: https://en.wikipedia.org/wiki/Computational_complexity_of_matrix_multiplication#Sim ple_algorithms

supervisors: Luis Maia, Francesca Pistolato

Représentations de polyèdres inscrits (show description)

supervisor: Jean-Marc Schlenker

Wythoff’s game (show description)

Goal:

Wythoff’s game is a two-player game played on a large grid of squares, similar to a chessboard. Player A places a figure on any square of the grid, except in the lower-left corner. Then, players A and B take turns moving the figure, with player B starting.

The allowed moves are:

Any number of squares to the left
Any number of squares downwards
Any number of squares diagonally (down-left)

The first player to move the figure into the lower-left corner wins. It turns out that this game has a winning strategy, which is based on certain “losing squares” on the grid: positions from which any move will eventually result in a win for the opponent. The goal of this EML project is to explore variations of the game. What happens if we replace the allowed moves with different ones? Do winning strategies exist in higher dimensions? And what happens if we start removing squares from the grid? These are just some of the questions that could be explored in this project.

supervisors: Anne Fisch, Tim Seuré

Congruence theorems for polygons (show description)

supervisor: Georg Grützner, Antonella Perucca

Isogeny graphs of elliptic curves over finite fields (show description)

supervisor: Alexandre Benoist

Introduction to Statistical Learning (show description)

supervisor: Ujan Gangopadhyay

Colorful fractional Gaussian integers (show description)

Goal:

Fractional Gaussian integers are complex numbers of the form $\frac{a+bi}{q}$, where $a,b,q$ are integers. When mapped to the complex plane, these numbers reveal intriguing geometric patterns and symmetries. The goal of this project is to construct and visualize fractional Gaussian integers by assigning different colors (based on divisibility, modular congruences, density distributions or other properties) in the complex plane. Then to explore and investigate the structures and symmetries within the fractional lattice.

Main reference: https://images-archive.math.cnrs.fr/Dentelles-et-flocons-de-neige-arithmetiques.html?lang=fr

supervisors: Anne Fisch, Guenda Palmirotta (Uni. Paderborn)

Platonic solids and tessellations of the sphere (show description)

Goal:

A Platonic solid is a convex polyhedron in three-dimensional Euclidean space such that all of its faces are congruent regular polygons, and the same number of faces meet at each vertex. The goal of this project is to establish the existence of only five Platonic solids, to construct a model for each of them and to understand their symmetry groups. Additionally, one will explore how tessellations of the sphere are intimately related to Platonic solids.

Prerequisites: Algèbre linéaire, Structures mathématiques

Literature: H. S. M. Coxeter, Regular Polytopes, (1973), 3rd ed.

supervisor: Naomi Bredon

The 17 wallpaper patterns (show description)

Goal:

There are seventeen possible wallpaper patterns to tile a plane. In this project, one studies their classification and the symmetries involved - translations, rotations, reflections, and glide reflections. The goal is to provide a comprehensive description of a few wallpaper patterns that can be found in art and architecture, to understand their symmetries, and to create one’s own tessellations by using an appropriate software (e.g. Geogebra or Inkscape) - for example, inspired by Escher’s work to fit animal-like shapes.

supervisor: Naomi Bredon

The Sierpinski triangle (show description)

Goal:

A fractal is an infinite structure that is self-similar at every scale. This project deals with fractals, focusing on the Sierpin ́ski triangle, a fractal constructed by recursively removing equilateral triangles. The goal is to construct the first few iterations by using a software and to discuss the connections with Pascal’s triangle modulo 2. Several further developments are possible.

supervisor: Naomi Bredon

The Futurama theorem (show description)

Goal:

The Futurama theorem, as established by Ken Keller in an episode of the animated series Futurama, can be stated as follows. Given a group and a mind-swap machine that can only be used once on a pair of people, one can restore each person’s mind to its original body by using at most two additional people. The goal of this project is to write an algorithm to restore order to an arbitrary shuffle, and to understand the minimum number of swaps required. Several related questions about permutations in a finite group could be addressed.

supervisors: Naomi Bredon, Franck Sueur