# Currently offered projects

More projects for next semester are coming soon!

• Sharp transitions for percolation in Erdős–Renyi graphs (description)

supervisor: Pierre Perruchaud

• Models for cubic surfaces ()
Goal:

Cubic surfaces have been at the center of algebraic geometry ever since Cayley and Salmon showed all the way back in 1849 that (over an algebraically closed field) they always contain exactly 27 lines (if the surface is smooth). Over the real numbers we can make everything visual, and we can find equations such that all 27 lines are present.

There also exists a classification of singular surfaces, and a count of the number of lines on them.

The goal of the project is to visualise these surfaces using modern interactive online tools, and see what is possible using 3d printing to create physical models.

Tools: SageMath, D3.js

supervisors: Pieter Belmans, Mélanie Theillière
level: moderate to advanced

• Cubic surfaces over finite fields ()
Goal:

Cubic surfaces have been at the center of algebraic geometry ever since Cayley and Salmon showed all the way back in 1849 that (over an algebraically closed field) they always contain exactly 27 lines (if the surface is smooth). Over a finite field we have to modify this count, as the maximum of 27 is not always attained.

Using computer experiments you will determine how many points and lines can lie on a cubic surface in $\mathbb{P}_{\mathbb{F}_2}^3$, and try to visualise this using the tetrahedral model.

As a warm-up we will discuss the Fano plane, and how to visualise curves inside $\mathbb{P}_{\mathbb{F}_2}^2$.

Tools: SageMath

supervisors: Pieter Belmans, Ogier van Garderen
level: easy

• Arithmetic billiards with holes and with other shapes ()
Goal:

Continuing the project of last semester on arithmetic billiards, we want to study some arithmetic/geometric properties of the trajectory of a ball on an arithmetic billiard, which can be a rectangle with a hole or which can have other shapes made of smaller rectangles patched together.

supervisors: Clifford Chan, Flavio Perissinotto

• Triangular arithmetic billiards and the three jugs problem ()
Goal:

This project would be somewhat different from the earlier projects on arithmetic billiards. Instead of considering a ball that is shot with a 45 degrees angle in a rectangular billiard, we consider particular shapes arising from equilateral triangles which can give a geometric solution to the three jugs problem, where the ball is shot either along the sides or with a 60 degrees angle from the sides. We want to investigate which are the shapes that can be found and in what cases, what properties we can find about the trajectory, and maybe try to think of some generalizations.

supervisors: Antigona Pajaziti, Flavio Perissinotto

• 3D models for modular forms ()
Goal:

Modular forms are very important analytic complex-valued functions on the "upper half plane" (the part of the complex plane where the complex numbers have positive imaginary part). They were introduced in the 19th century and studied by mathematicians such as Eisenstein, Jacobi, Poincaré. They were also underlying many of the incredible discoveries of Ramanujan in the beginning of the 20th century, they played a crucial role in the solution of Fermat's Last Theorem and they are still a very active field of research.

Not only that, they are highly symmetric, obey a number of strong transformation laws and are very beautiful: Whereas (colourful) 2D plots of modular forms are (relatively) easy to make with current software and are readily available on the internet, in this project we would like to add one dimension: make 3D models of some modular forms with a 3D printer.

The project consists of understanding the definition of modular forms, studying examples and making 2D images of some modular forms in order to get a feeling of what they look like, before creating files for printing some modular forms on a 3D printer.

As mathematical prerequisites, the courses of the first year are sufficient. The 2D plotting can be done, for instance, with SageMath, but other software can also be used. Experience with 3D printing is not necessary, but some time will be necessary to learn about it.

supervisor: Gabor Wiese

• Visualising roots of algebraic numbers ()
Goal:

This picture shows the zeros of the polynomials of degree at most 24 with coefficients either 1 or -1. A discussion around this fascinating topic of the (visualisation of the) distribution of roots of integral polynomials can be found here.

The connected question of the proportion of irreducible (i.e. one cannot write them as a product of two integral polynomials in a non-trivial way, as you will learn (resp. learned) during your Algebra class) among all such polynomials is still being studied intensively, see e.g. here.

A systematic study (which is partly accessible to students) of the distribution of the roots (from a hyperbolic perspective) has recently been undertaken by the three mathematicians Edmund Harriss, Katherine E. Stange and Steve Trettel and led, among others, to the website Algebraic Starscapes.

The project will be a computer experimentation with integral polynomials and their roots. Students might start by reproducing (some simple version of) some pictures found on the mentioned websites, and then create their own ones as variations (e.g. other sets of polynomials, other colourings, other normalisations, etc.).

Along the way, observations shall be made, illustrated by pictures and, if possible, explained.

supervisor: Gabor Wiese

• Rational points on quadratic varieties ()
Goal:

A quadric in $n$-dimensional projective space is the zero set of a homogeneous quadratic equation in $n+1$ variables. Whereas the real points (in 2- or 3-dimensional projective space) lead to nice pictures, we will be interested in rational points, i.e. points with rational coordinates. Since we work in projective space, we can arrange the coordinates to be integers.

A simple example is the unit circle (with centre the origin), given by the polynomial $X^2 + Y^2 - Z^2$. Its rational points correspond to Pythagorean triples.

Further, we will be interested in intersections of quadratics, leading to quadratic surfaces. This is actually quite general because a theorem of Mumford asserts that every projective variety can be written as an intersection of quadrics (if a good embedding into a projective space is chosen). In general, varieties are expected to have few rational points.

A motivation, which can also be studied in this project, comes from the question if, given an integral $m\times n$ matrix $M$, there are integers $a_1,\ldots, a_n$ such that $M$ times the column vector $(a_1^2,\ldots,a_n^2)$ is a vector whose coordinates are all squares. This question can be recast as a question on rational points on the intersection of quadrics.

This project can go into several different directions (which can also be done together):

• define and illustrate quadrics and rational points on them;
• illustrate intersections of quadrics and, if possible, rational points on them;
• make experiments and observations in which situations one can expect the existence of a rational point;
• make experiments on the motivation described above.

supervisor: Gabor Wiese

• Centrifuges (description)

supervisors: Vincent Wolff, Tim Seuré

• Drum resonance ()
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

• Fractals and iterated function systems (description)

supervisor: Laurent Loosveldt

• Printable planetarium (description)

supervisor: Laurent Loosveldt

• Problème du cavalier ()
Goal:

In chess (an $8\times 8$ grid), the knight can move in directions $2x1$. It is easy to see that he can reach all the squares of the board. Let us generalize the problem: given a grid of size $N*N$ (with $N$ an integer, or infinity) and a knight who can move in directions of type $axb$, which squares are reachable? The result is partially describable in terms of $N$, $a$ and $b$. Conversely, one can fully describe the configurations $N$, $a$ and $b$ such that the whole chessboard is reachable. The proof I had imagined is very visual, and not totally trivial. It is made of easy arithmetic and a little imagination. On can then study various questions: the most difficult square to reach, the access time, optimal paths, etc. One could also imagine adding probabilities, generalizing to higher dimensions, on other types of tilings. Most of these questions can be intuited with the help of the computer tool and can give rise to nice visuals of chessboard filling. It can be a nice team work because there are a lot of tasks, both from computer science and mathematical. The problem is very open and it can also encourages students to ask themselves questions and explore them.

supervisor: Louis Gass