Currently offered projects

Projects for Summer 2026 (New project added on 18 Feb)

  • Integer polynomials, reductions and symmetry ()
    Goal:

    The objective of the project is to explore various facets of integer polynomials and their symmetries. A toy example is the polynomial f(X) = X^2+1. Of course, we aim at larger degrees.

    Several aspects can be investigated.

    (1) The polynomial f(X) has two roots in the complex plane, namely i and -i, which are symmetric with respect to reflection on the real axis. Other quadratic polynomials have the same kind of symmetry, others show different symmetries. If the degree gets larger, the symmetries become (much!) richer and are not always easy to describe.

    The aim of this part of the project is to understand the symmetries as well as possible and, especially, to visualise them.

    (2) If we reduce f(X) modulo an odd prime p, then sometimes the polynomial will have roots mod p, sometimes it won't. If one does some counting, one will find that both cases appear roughly equally often. This counting pattern is the same for all irreducible integer polynomials of degree 2. However, for degree 3 polynomials or larger, the patterns become much richer.

    This part of the project aims at seeing which patterns occur, at visualising and understanding them.

    (3) One can relate the views in (1) and (2). The reduction modulo p does give rise to a symmetry (in a precise sense).

    The aim of this part is to study this relation and to visualise it as well as possible.

    (4) After being able to do (1), (2) (and (3)) for one polynomial, it becomes interesting to ask what happens if we vary the polynomials. For instance, we can consider many polynomials that have properties in common.

    The aim is to study how the behaviours observed above change and are distributed.

    This project will make you discover hidden properties of polynomial equations and (secretly or not) learn a lot about Galois groups, primes in arithmetic progressions and Chebotarev's density theorem. The experimentation can be done in sagemath (cocalc) or directly in Python, or in other computer algebra systems. Creativity is needed for obtaining nice visualisations.

    Prerequisites: Linear Algebra 1, Structures mathématiques. The course Algèbre helps to understand what is going on theoretically. For carrying out this project, it is not strictly necessary.

    supervisor: Gabor Wiese

  • Kakeya Needle Problem: Numerical Constructions and Experimental Investigations ()
    Description:

    This project focuses on implementing and visualising discrete versions of Kakeya (Besicovitch) sets, exploring their geometric properties, estimating their area and fractal dimensions, and optionally investigating finite-field variants.

    More details here

    supervisor: Thomas Lamby

  • Simulation of Gaussian Random Functions ()

    supervisor: Felix Benning

  • The diameter of totally sum-free sets ()

    supervisor: Mike Daas

  • Rivin’s Rationality Conjecture for Polyhedra ()

    supervisor: Carl Lutz, Jean-Marc Schlenker

  • The sections of the torus ()

    supervisor: Szabi Buzogány

  • Fractal Landscapes ()

    supervisor: Francesco Tognetti

  • Numerical simulation of a harmonic field around a slit-screen with one aperture ()

    supervisor: Franck Sueur

  • Spectrum of the graph Laplacian ()
    Description:

    Imagine an electrical circuit where junctions are nodes and the wires connecting them are edges. If we assign a voltage $v_i$ to each node $i$, Ohm's law states that the current flowing from node $i$ to a connected node $j$ is proportional to their voltage difference, $v_i - v_j$. Assuming every wire has a resistance of one ohm, the current is exactly $v_i - v_j$.

    According to Kirchhoff's current law, the net current leaving any internal node must be zero. Summing the currents to all neighbors of node $i$ gives $d_i v_i - \sum_{j \sim i} v_j$, where $d_i$ is the node's degree. Expressing this system for the entire network yields the graph Laplacian, defined as:

    $$L = D - A$$

    where $D$ is the diagonal degree matrix and $A$ is the adjacency matrix. Similar to how the continuous Laplacian detects the shape of a surface, the eigenvalues of the graph Laplacian—known as its spectrum—reveal the geometric properties of a discrete network.

    Students will compute the eigenvalues of the Laplacian on various graphs numerically. Along with the computational experiments, students will also learn the basic theory of algebraic graph theory.

    Prerequisites: Linear algebra.

    supervisor: Wai Yeung Lam

  • Random scars ()

    More details here

    Prerequisites: Analysis 1, Probability 1, and basic Python knowledge.

    Literature & Links: Azais-Wschebor, Level Sets and Extrema of Random Processes and Fields. Students can also check the "Quantum scar" page on Wikipedia for related introductory information.

    supervisor: Louis Gass

  • Random maze generation algorithm ()

    The goal of this EML is to explore different algorithms to generate a random maze. Several algorithms can be applied, leading to different style and structure for the generated maze. Several points will be studied: complexity of the algorithm, the difficulty to solve, the bias toward a uniform distribution.

    Prerequisites: basic coding skills. Some knowledge in probability and graph theory is appreciated but not mandatory

    supervisor: Louis Gass