Experimental Mathematics Lab at the University of Luxembourg

LASSO (show description)

Goal:

LASSO (Least absolute shrinkage and selection operator) is a method of performing regression and variable selection in a sparse setting i.e., where the number of predictor variables is large but only a few of them are expected to have significant contribution. The goal will be to perform various variants of LASSO and also ordinary regression and compare their performance.

supervisor: Ujan Gangopadhyay

Urn models (show description)

Goal:

Urn Models are one of the fundamental building blocks of probability theory. At the core of the theory of urn models, there are two distinct types of urn models, Polya's urn and Friedman's urn. The goal of the project will be to understand the behavior of these two types of urn models.

supervisor: Ujan Gangopadhyay

Artificial neural networks (show description)

Goal: This project aims to demystify artificial neural networks by studying their mathematical foundations, then applying them to classic machine learning problems, such as handwritten character recognition.

supervisor: Ivan Nourdin

Exploring naive Bayes algorithms (show description)

Goal: The project focuses on the Naive Bayes algorithm, a statistical classification technique. Students will explore its theoretical foundations and implement them in real-world applications, such as spam filtering and sentiment analysis. If time permits, other algorithms or statistical concepts may be studied.

supervisor: Ivan Nourdin

Improving the probability course with computer illustrations (show description)

Goal: The goal of this project is to enrich the semester 3 probability course by integrating computer illustrations of key concepts. Students will review theoretical concepts introduced in the probability course and develop visual and interactive examples to facilitate understanding. Successful illustrations can be incorporated into next year's course materials, contributing to an enhanced learning experience.

supervisor: Ivan Nourdin

Decomposing diamonds (show description)

Goal:

The Hodge diamond of a variety (which is the central type of objects in algebraic geometry) is an integer matrix, with interesting symmetries, and often rotated 45 degrees, to give a diamond shape. We can alternatively use the entries of the matrix to write down a polynomial in $\mathbb{Z}[x,y]$. It gives important information about the variety, but the goal of the project is not to learn all of the required algebraic geometry.

Rather, there are ways in which Hodge diamonds can be decomposed into Hodge diamonds of other (smaller) varieties. This is an important source of conjectures. The goal of this project is to automate the search for such decompositions, so that we can

rediscover some known cases
discover some new decompositions (or even rule out their existence)

This could bring us to some of the frontiers of current research in algebraic geometry, by just manipulating polynomials in $\mathbb{Z}[x,y]$.

supervisors: Pieter Belmans, Sebastián Torres

Distribution of the zeros of random functions: attraction, repulsion and other related properties (show description)

Goal:

The goal of this project is to study the behavior of the zeros set of a random function following a given distribution and comment about the 'shape': do the zeros tend to cluster or to repulse, compared to a Poisson point process. Many models can be studied: zeros of random polynomials with Gaussian coefficients, eigenvalues of random matrices, critical points of random fields, and even optimal transport between roots of a random polynomial and root of its derivative. The goal is to approach some theoretical results from a computational point of view and get nice visual pictures. The literature around this topic is large and the direction of the project will be discussed between the teacher and the student(s).

Prerequisites: we will need some basic probabilities and a bit of Gaussian analysis. Depending on the direction of the project, we might also need a bit of arithmetic, analytic function theory, or even differential geometry. A bit of computational skills will be useful, especially for plotting pictures.

supervisor: Louis Gass

Primitive roots (description)

supervisors: Antonella Perucca, Tim Seuré

On the cardinality of the support of Walsh transforms for functions in a few variables (description)

supervisors: Pierrick Méaux, Gabor Wiese

Formalizing mathematics in Lean (show description)

Goal:

In recent years the idea of formal proof verification has really taken off. It consists of explaining to a computer how a proof goes, and have it check that every step is indeed fully justified. In this project you will learn the basics of Lean, a very user-friendly proof assistant, in which we will formalise proofs of some of the results you have seen in your courses.

Some experience with programming languages will be helpful.

supervisors: Thilo Baumann, Pieter Belmans, Simon Lemal

Drum resonance (show description)

Goal:

A drum's resonance is modeled by a wave equation on a circular membrame with fixed boundary. In playing, it can be varied by the change of the striking position, and by applying pressure to the drum-head. Particularly interesting choices are when the skin is pressed in the nodes (fixed points) of a chosen vibration mode, and when the drum is hit on its atinode (maximally vibrating point). The result seems to be that the choosen mode becomes dominant in the overall drum's sound.

Student(s) will do a computer simulation of the vibrating membrane subject to the above conditions, and compare to the experiment (audio recording).

Literature:

W. Strauss, Partial differential equations: An introduction, John Wiley & Sons, Inc., 1992
T. D. Rossing, Science of Percussion Instruments, Series in popular science – Volume 3, World Scientific Publishing, 2020

supervisor: Damjan Pistalo

Congruence theorems for convex polygons involving heights (show description)

Goal:

In this project we will explore the congruence of convex polygons in the Euclidean plane. The goal is to find geometric data that suffices to determine the polygon. This is a very classical type of problem in geometry, often referred to as a moduli problem.

Let's be a bit more precise. A polygon is a subset of the plane bounded by Euclidean line segments, and it is convex whenever the segment connecting any two points in the polygon is contained in the polygon. We say two polygons are congruent if all of the side lengths and angles are the same. A convex $n$-gon is a convex polygon of $n$ sides. We will distinguish convex $n$-gons up to congruence, that is, consider two convex $n$-gons to be the same if and only if they are congruent to each other. This breaks up the set of convex $n$-gons, into subsets called congruence classes. The goal of this project is to find data that completely determines the congruence class of a convex $n$-gon.

One important piece of data is the set of heights between neighboring sides. Given two sides $AB$ and $BC$ sharing a vertex, the height from $AB$ to $BC$ is the height of the triangle from $A$ to $BC$. For example, it is known that a convex $n$-gon is determined up to congruence if we know all heights as well as the angles between them.

The fundamental question we will be studying is: if we only know the heights, what can we determine about the congruence class of a convex $n$-gon?

We will start by studying the case of triangles. Studying such lower-complexity polygons should illustrate the basic definitions and help us see how the complexity of the situation bumps up once we increase the number of sides.

supervisors: Didac Martinez Granado, Antonella Perucca, Nathanial Sagman

Mathematical and statistical models for game strategy optimization in football sports (show description)

Goal:

In the dynamic and competitive landscape of football sports, optimizing game strategy is crucial for success. This project aims to develop sophisticated mathematical and statistical models to enhance decision-making processes in football coaching, with a focus on strategic optimization. By leveraging advanced analytics and data-driven insights, the project seeks to revolutionize the way teams approach match planning, player positioning, and tactical execution.

Basic experience in Python could be an advantage but is not mandatory. Students can learn some basics from this project.

supervisors: Tianxiao Guo, Senthil Murugan Nagarajan

Visualising systems of linear equations (show description)

Goal:

The aim of this project is to visualise sets of integral solutions of systems of linear equations with integral coefficients, and their reduction modulo an integer. You can study any kind of system of linear equations, but some appear to be particularly intriguing, for example those representing magic squares or similar objects. For magic squares, the variables are arranged in a square and the systems of linear equations are those stating that row, column and diagonal sums are equal.

A way to explore in this project could be the colouring of solutions. As the most basic example, an entry in a solution (vector) which is 0 mod 2 could be represented by black square and 1 mod 2 by a white one. If the entries are arranged in a square (like for a magic square), then you would get a checker board, except that the distribution of black and white squares will be quite a different one.

Of course, this would be only the most simple thing; we could work in the rainbow spectrum, or also in RGB (red-green-blue) or CMY (cyan- magenta-yellow) systems, representing a colour by 3 numbers. Opacity etc. could also be added. There are no limits to your imagination.

In this project we want to understand systems of linear equations better, whilst trying to produce beautiful images.

This project is for students in their 2nd semester.

supervisors: Gabor Wiese, Guenda Palmirotta

Calendars and Friday the 13th (show description)

Goal:

Which day of the week were you born? Which day of the week will be 23 March 4323? There are formulas, using modular arithmetic, for computing the day of the week. The aim of the project is to understand these formulas, implement them, implement similar ones (which you can imagine for other planets) and experiment with them. If done correctly, you will find that it is more likely for a 13th to be on a Friday than on any other day of the week. Can you find other such surprises? Is there any explanation you can give? If you are motivated, you can explore links with pseudo-random numbers.

This project will help you explore modular arithmetic and there will be loads of freedom for you to experiment.

This project is for students in their 2nd semester.

supervisors: Gianni Petrella, Gabor Wiese

Attractive integers (show description)

Goal:

In this project, we would like to make (some) integers (or other numbers) attractive in the following sense. Construct a function $f$ such that iterating it for (m)any start values $x_0$, the sequence defined recursively by $x_n = f(x_{n-1})$ approaches an integer (in a chosen set). An easy way to realise this is Newton's method.

To illustrate it, suppose we want $-1,0,1$ to be attractive. Then we define a polynomial having these integers as zeros: $p(X) := X (X-1) (X+1) = X^3 - X$. Newton's method leads to the function $f(x) := x - p(x)/p'(x) = 2x^3/(3x^2-1)$. Here is an example that shows attraction to 1: starting with $x_0=3$, one finds $x_1=2.07692$, $x_2=1.50057$, $x_3=1.17420$, $x_4=1.03240$, $x_5=1.00146$, $x_6=1.00000$ (up to 5 decimal digits of precision).

One problem (in applications, for instance, in cryptography) with Newton's method is that it involves division. However, one can come up with f that are polynomial, but do their job also fine.

In this project, you are invited to experiment with such functions, find your own ones, and, importantly, visualise them in a way that shows how good they are. It appears to be particularly interesting to generalise this to functions of two (or more) variables, in which cases (certain) pairs of integers could be attractive.

This project can be done by students in their 2nd semester, and the 4th semester.

supervisors: Tim Seuré, Gabor Wiese