Past projects

The following is a list of all projects that were part of the Experimental Mathematics Lab.

Summer 2024

  • Urn models ()
    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

    students: Christophe Rafael Ferreira Simoes, Diana Couto Melo

  • Artificial neural networks ()
    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

    students: Nicolas Lebon, Maxime Mary, Sergej Soules

  • Exploring naive Bayes algorithms ()
    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

    student: Eden Carbonell

  • Improving the probability course with computer illustrations ()
    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

    students: Max Brimeyer, Luca Desquiotz, Jules Nies

  • Decomposing diamonds ()
    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

    students: Inês Paiva Sirgado, Natalija Postolovska, Mahima Kumar

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

    supervisors: Pierrick Méaux, Gabor Wiese

    student: Jean-Philippe Marichal

  • Formalizing mathematics in Lean ()
    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

    students: Lina Metzger, Alex Mille

  • 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

    students: Kin Shing Yeung, Elza Hoti, Ruben Soares Monteiro

  • Congruence theorems for convex polygons involving heights ()
    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

    students: Enrico Ganassin, Guillaume Sylvain Verfaille, Shi-Yuan Zhou

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

    students: Maxime Laruccia, Andrea Valentini, Raphael Araujo

  • Visualising systems of linear equations ()
    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

    students: Ian Beck, Sevan Chrétien, Amelia Gonzalez Saura

  • Calendars and Friday the 13th ()
    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

    student: Kim Lazzeri

  • Attractive integers ()
    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

    students: Alexandre Wilmes, Abdourrahmane Makni

Winter 2023

  • Spotting patterns in infinity norms (description)

    supervisors: Tim Seuré, Vincent Wolff

    students: Lola Husson, Tom Sachsen, Noa Reichert (report)

  • Celebrating the 20th anniversary of our university with Fourier series ()
    Goal:

    One can create many two-dimensional "one-line drawings" with complex Fourier series. The aim of this project is to do this for a concrete, interesting logo, namely our university logo and the EML logo. This video gives a nice insight: youtube.com/watch?v=r6sGWTCMz2k

    Basic knowledge in analysis and linear algebra is recommended, particularly a certain familiarity with Fourier series, and some experience with programming.

    supervisor: Guenda Palmirotta

    students: Charles Chattou, Gil Schroeder

  • Random walks: elephants and sharks ()
    Goal:

    In a regular (Markovian) random walk, each step is independent of the past. However, in the elephant random walk (ERW) and shark random swim (SRS) models, the next step is either a fresh step with some probability $1-p$, or a repetition of some randomly selected past step with probability $p$. Generally, these random walks are called ERW if the distribution of the steps are light-tailed, and they are called SRS if the steps are heavy-tailed. One can consider many different mechanisms for selecting the past step. Also, instead of the whole past, one can consider just the recent past, or one can consider using only the memory near the beginning of the process.

    Only in some cases, theoretical results are known. Almost all of the results are about some phase transition. Because if $p$ is close to 0, we will see behavior similar to a standard random walk, whereas if $p$ is close to 1, the trajectories are close to straight lines. It is easy to see these happening via simulations (although it is not easy to guess the critical $p$ from simulation).

    The project will involve simulating various models of ERW and SRS and observing the phase-transitions.

    supervisor: Ujan Gangopadhyay

    students: Silvia Ferreira, Inês Sofia Texeira Noguiera

  • Imaginary integers ()
    Goal:

    Let $d$ be a positive integer. The subset of the complex plane formed by elements of the form $a+b\sqrt{-d}$ for $a,b$ rational numbers is called an imaginary quadratic field. In it, one has the so-called ring of integers, which generalises the usual integers. The most famous example is that of the Gaussian integers, that is, the points $n+m\mathrm{i}$, where $n,m$ are integers and $\mathrm{i}$ is the imaginary unit, i.e. $\mathrm{i}^2=-1$.

    The aim of the project is to study properties of such integer rings by means of visualisation, concretely, by pictures drawn on the complex plane.

    Visualisations include:

    • the ring of integers for varying d
    • existence of a division with remainder (Euclidean division)
    • "ideals"
    • class number

    This project can help to illustrate notions taught in the Algebra class of the BMATH.

    In the case that these are not enough questions, there exist three-dimensional analogs, which could also be investigated.

    Basic Python and R will be used.

    supervisors: Thierry Meyrath, Gabor Wiese

    students: Andreia Correia, Eva Scardillo (report)

  • Statistical approach for data manipulation and analysis in healthcare ()
    Goal:

    In this project, we will do broad research on statistical approaches such as descriptive analysis, inferential analysis, predictive analysis, prescriptive analysis, and exploratory data analysis. The main goal is to work with openly available data collected from healthcare industries and use statistical approaches to find important insights and do some predictions.

    • Descriptive analytics helps understand historical patient data to compare and discover trends across various aspects of healthcare.
    • Inferential statistics are widely used in healthcare to analyze the effectiveness of new drugs, medical treatments, and interventions.
    • Predictive analytics in healthcare helps medical institutions identify people who are at risk of developing chronic conditions and give them preventive care before the disease progresses.
    • Prescriptive analytics enables healthcare decision-makers optimize business outcomes by recommending the best course of action for patients or providers.
    • Exploratory data analysis helps to recognize natural patterns hidden in the data.

    Based on above description, the students will work each statistical approaches by applying openly available healthcare dataset.

    supervisor: Senthil Murugan Nagarajan

    students: Nadine Haas, Naomi Picard

  • Areas of convex hulls and Viterbo's conjecture (description)

    supervisors: Alexey Balitskiy, Mingkun Liu

    students: Lou Meylender, Thomas Thalmaier

  • Caustic curves and 3d prints (description)

    supervisors: Mélanie Theillière, Ogier van Garderen

    students: Inês Paiva, Natalija Postolovskaija

    student: Irma Skrjijelj

  • 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.

    In this edition of the project, we would like to focus more on new interesting features:

    1. "Zoom" into some of the interesting parts of the picture of zeros, by, for instance, going higher and higher in the degree (maybe only even degrees) and adding points to the picture while taking more and more polynomials into account.
    2. Separate the set of polynomials into subsets: only those polynomials with one 1, only those with two 1, etc.; alternatively, take one polynomial and permute its coefficients in any possible way (or only even permutations), etc.
    3. Investigate the similarity between the inner part of the picture and dragons

    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.).

    supervisors: Gabor Wiese, Guenda Palmirotta

    students: Claudia Marichal, Catia Alves Piro (report)

Summer 2023

  • 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

    students: Mahima Kumar, Ben Welter

  • 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

    students: Catia Alves, Andrea Cunha, Eva Scardillo

  • 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.

    supervisors: Gabor Wiese, Thierry Meyrath, Guenda Palmirotta

    students: Lynn Hellenbrand, Tom Kasel, Christophe Thelen (report)

    students: Salma Belmir, Mikala Eisen, Melissa Genoud (report)

  • Transitive subgroup art ()
    Goal:

    It is possible to visualise elements of a symmetric group as explicit permutations, similar to how braid groups are represented. There are various choices of generating sets one can use, and they will give different visualisations of the elements of the symmetric group.

    You will experiment with visualising group elements using certain generating sets. There are also interesting subgroups of the symmetric group that one can consider, with other interesting visualisations.

    supervisor: Pieter Belmans

    students: Ines Paiva, Natalija Postovolovska

    students: Anne-Julie Bertinchamps, Tia De Waha, Sara Zouabi

  • Left-right reversal illusions ()
    Goal:

    It is almost unbelievable, but true: there exist 3d objects that, when seen in a mirror, are exactly reversed. For example, you see an arrow pointing right, but its mirror image points left.

    If you see the photos in this article, you will be impressed.

    However, this illusion only works when looking at the object from the correct angle, which makes the phenomenon a little more believable.

    The aim of the project is to understand the underlying mathematics (elementary geometry), to design such objects and to 3d print some of them. Maybe we can even have a small collection in the end?

    supervisors: Gianni Petrella, Gabor Wiese

    students: Andreia Correia Martiniano, Claudia Marichal (report)

  • Frobenius problem: Solving special cases (description)

    supervisors: Tim Seuré, Vincent Wolff

    students: Lou Meylender, Thomas Thalmaier (report)

  • Fractals and iterated function systems (description)

    supervisor: Laurent Loosveldt

    students: André Rodrigues Costa, Jean-Philippe Marichal, Sara Zouabi

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

    supervisor: Pierre Perruchaud

    students: Mathieu Amasio, Nicky Soares Monteiro

Winter 2022

  • Printable planetarium (description)

    supervisor: Laurent Loosveldt

    students: Anne-Julie Bertinchamps, Tia de Waha, Gabriele Terenziani (report)

  • 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

    students: Diana Couto Melo, Christophe Rafael Ferreira Simoes, Camal Tahireddine (report)

  • Centrifuges (description)

    supervisors: Vincent Wolff, Tim Seuré

    students: Jean-Philippe Marichal, André Rodrigues Costa (report)

  • 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

    students: Jie Chen, Leonid Gnutov (report)

  • 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

    students: Salma Belmir, Luke Hastert, Tom Kasel (report)

  • 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

    students: Matthieu Amasio, Nicky Soares Monteiro

Summer 2022

  • Distribution of Galois groups ()
    Goal:

    Associated to an irreducible monic polynomial with rational coefficients we have its Galois group, which measures the symmetries of the complex solutions of this polynomial.

    If instead of rational coefficients we use integral coefficients, we can consider those polynomials for which the absolute value of the coefficients is bounded above by a small constant, so that we have a finite set. If we compute all the Galois groups that appear for such polynomials, and let the bound grow higher and higher, we will observe some patterns.

    The goal is to understand which pattern appears, and understand some of the group theory that goes into these patterns.

    supervisors: Pieter Belmans, Pietro Sgobba

    students: Vicky Huiqi Zheng, Dewei Zou

  • Exploring Runge's phenomenon ()
    Goal:

    Consider a continuous function $f$ on an interval $[a,b]$. Then, by a classic approximation theorem due to Weierstrass, $f$ can be uniformly approximated on $[a,b]$ to any degree of accuracy by polynomials. On the other hand, given $N+1$ distinct points $x_i$ in $[a,b]$, it is well known that there exists a unique polynomial $P_N$ of degree at most $N$ that interpolates the function $f$ in these points, i.e. that fulfills $P_N(x_i) = f(x_i)$.

    It seems promising to use these interpolation polynomials to approximate $f$ on $[a,b]$, in particular, one might believe that if the number of interpolation points $x_i$ in $[a,b]$ increases, the corresponding interpolation polynomials will approximate the function $f$ better and better. While this might indeed be the case, it was first observed by Carl Runge that depending on the function $f$ and the location of the interpolation points in $[a,b],$ the opposite might actually occur, meaning that increasing the number of interpolation points makes the accuracy of the approximation worse. This somewhat surprising result, known as "Runge's phenomenon", shows that interpolation polynomials can not in general be used to approximate functions.

    The goal of this project is to understand and explore Runge's phenomenon, and to illustrate it with numerical calculations. Moreover, different approaches to circumvent the phenomenon, such as interpolating in the so-called Chebyshev nodes, can also be investigated.

    supervisor: Thierry Meyrath

    students: Aylin Cosgun, Jules Nies (report)

  • Arithmetic billiards (description)

    supervisors: Flavio Perissinotto, Antonella Perucca, Sebastiano Tronto

    students: André Manuel Rodrigues Costa, Enzo Fantino, Jean-Philippe Marichal

  • Divisibility graphs ()
    Goal:

    Divisibility graphs are a funny tool to visualize divisibility criteria for positive integers. This means for example that there is a graph with 7 nodes and various arrows to understand whether a number is divisible by 7. According to the number, one walks on the divisibility graph and if the walk ends in one specific node, then the number is divisible by the chosen starting number. The first aim of the project is understanding how the divisibility graphs work, namely what is their precise definition and how they can be constructed. Then we would like to compare divisibility graphs corresponding to different numbers. For example, such graphs differ by visible properties, if one compares the distribution of the arrows (see below). Do these properties reflect for example that the given divisor is squarefree, or a prime power, or that it is related to the base number 10? Moreover, we can try to think how the divisibility graphs can be combined. Indeed, we can try to see how the 12-divisibility graph can be recovered by its two ”shadows” the 3-divisibility graph and the 4-divisibility graph. Notice that there are already programs that can draw these graphs.

    supervisors: Antonella Perucca, Emiliano Torti, Vincent Wolff

    students: Anne-Julie Bertinchamps, Tia De Waha (report)

  • Queens ()
    Goal:

    There are a number of famous chess inspired mathematical puzzles, for example that of the Eight Queens. You can consult the corresponding Wikipedia page. It is possible to make many variations of this puzzle, for instance, other sizes and shapes of the board, including boards with holes, or even boards on the surface of a ball or a torus.

    The goal is to find interesting such variations, programme solutions, visualise them and also analyse them theoretically. Ideally, the simulations will lead to observations that can be recorded and possibly proved.

    Tools:

    Any programming language or any computer algebra system, and image processing software

    supervisors: Thierry Meyrath, Gabor Wiese

    students: Nicolas Dubreuil, Yann Koehnen (report)

  • Magic polyhedra ()
    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 cubes, magic footballs, and similar. Furthermore, tangible 3d objects shall be created (via a 3d printer or by buying simple footballs/models or in similar ways).

    Some inspiration and mathematical background on polyhedra can, for instance, be found here.

    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).

    Tools:

    Any programming language or any computer algebra system; potentially software for creating 3d prints.

    supervisor: Gabor Wiese

    students: Jie Chen, Leonid Gnutov (report)

  • Patterns in primes ()
    Goal:

    Prime numbers are full of surprises. The famous prime number theorem proved at the end of the 19th century describes the prime counting function (i.e. the number of primes up to bound $x$) asymptotically. However, given any integer $x$, there is no formula, depending on $x$, that produces the prime closest to $x$.

    When visualising the distribution of prime numbers, such as with the Ulam spiral and the Klauber triangle, one can observe surprising phenomena that have not yet been fully explained.

    The goal of this project is to visualise the distribution of primes in various ways, inspired by the Ulam spiral, make observations, formulate them precisely and, if possible, relate them to theorems and conjectures.

    Tools:

    Any programming language or any computer algebra system, and image processing software

    supervisor: Gabor Wiese

    students: Matthieu Amasio, Arkin Mustafic, Daniel Sok

  • Numerical integration methods ()
    Goal:

    It is well-known that, if $-\infty<a<b<+\infty$, it is impossible to give an explicit value for \begin{equation} \int_a^b e^{-x^2}\mathrm{d}x \end{equation} while this integral plays a huge role in various fields of mathematics, especially probability. It is therefore very important to have efficient numerical methods to give a nice approximation of the value of such integrals. Many such methods exist, using different approximation tools. For instance, one can estimate the value of an integral by summing the areas of different geometric shapes (mainly rectangles or trapezoids) using the interpretation of the integral as the area of the surface that is under the graph of the function. One can also first approximate the function by a polynomial and then compute the integral of this polynomial, which is easier.

    Another strategy is to pick $N$ points $(x_j)_{j=1}^N$ randomly in $[a,b]$ and compare the average value $\frac{1}{N}\sum_{j=1}^N f(x_j)$ with $b-a$. The raison d'être of all these methods of course relies on different important theorems in analysis and probability.

    The aim of this project is to write a program which computes different numerical integration methods and to compare the efficiency of these methods on various examples. One can for instance try to find criteria under which a method is better than an other one. As an application, one can also give numerical estimations of well-known numbers such as $\pi$ or $e$.

    supervisor: Laurent Loosveldt

    students: Luke Hastert, Tom Kasel, Luca Kremer

  • Numerical estimation of areas ()
    Goal:

    In this project, we aim at judging the efficiency of a numerical method to estimate an area.

    The method consists in drawing a surface, delimited by lines to make it simpler, into the square $[0,1]^2$. Then we pick $N$ random points uniformly in $[0,1]^2$ and count the number $M$ of points that are in the surface. If $N$ is large, $\frac{M}{N}$ should be a good approximation of the area.

    Depending on how well the project is going, one can consider some extensions of the method, for instance to compute areas of more complicated surfaces or a multidimensional extension.

    supervisor: Laurent Loosveldt

    student: Eden Carbonell

  • Random tilings and the Arctic circle theorem ()
    Goal:

    Consider a $2n\times 2n$ box that we want to fill with $2\times 1$ dominoes. For most of the possible tilings, the human eye will not see any kind of emerging tiling. On the other hand, if one considers instead the shape pictured below (the so-called Aztec diamond), then one can see that for most tilings, the dominoes tend to form a brick-wall pattern in all four corners. In fact, one can show that for very large diamonds, almost all of the tilings will exhibit this pattern, and the central region will get closer and closer to the inscribed circle. This is called the Arctic circle theorem.

    The first objective of this project would be to find how to generate a "typical" tiling of the Aztec diamond, so as to illustrate the Arctic circle theorem. If one wants to push the exploration further, one could wonder what happens if we try a different shape, or tilings with different tiles.

    See also for a short introduction in French, by V. Kleptsyn for the 5 minutes Lebesgue.

    supervisor: Pierre Perruchaud

    students: Julien Pezzi, Sébastien Plaasch (report)

  • Turmites ()
    Goal:

    Imagine an insect sitting on an infinite grid of coloured cells. There are m colours, the insect can be in n states of mind and it can face into 4 different directions. According to some fix set of rules, the insect will now move from cell to cell and change their colours. This is called a turmite. The goal of this project is to investigate the behavior of turmites under different rules. For example, can we generate spiral growth by tweaking the rule matrices? Your own questions for investigation are also welcome.

    supervisor: Tara Trauthwein

    students: Oliver Philip Jack, Francesca Feyereisen, Noah Thielen (report)

Winter 2021

  • Rubik's snakes on a plane ()
    Goal:

    Rubik's Snake is a toy, and we investigate its planar configurations.

    supervisors: Francesco Grotto, Antonella Perucca

    student: Tatjana Van Steenbergen Bergeron (report) (external link)

  • Exploring Runge's phenomenon ()
    Goal:

    Consider a continuous function $f$ on an interval $[a,b]$. Then, by a classic approximation theorem due to Weierstrass, $f$ can be uniformly approximated on $[a,b]$ to any degree of accuracy by polynomials. On the other hand, given $N+1$ distinct points $x_i$ in $[a,b]$, it is well known that there exists a unique polynomial $P_N$ of degree at most $N$ that interpolates the function $f$ in these points, i.e. that fulfills $P_N(x_i) = f(x_i)$.

    It seems promising to use these interpolation polynomials to approximate $f$ on $[a,b]$, in particular, one might believe that if the number of interpolation points $x_i$ in $[a,b]$ increases, the corresponding interpolation polynomials will approximate the function $f$ better and better. While this might indeed be the case, it was first observed by Carl Runge that depending on the function $f$ and the location of the interpolation points in $[a,b]$, the opposite might actually occur, meaning that increasing the number of interpolation points makes the accuracy of the approximation worse. This somewhat surprising result, known as "Runge's phenomenon", shows that interpolation polynomials can not in general be used to approximate functions.

    The goal of this project is to understand and explore Runge's phenomenon, and to illustrate it with numerical calculations. Moreover, different approaches to circumvent the phenomenon, such as interpolating in the so-called Chebyshev nodes, can also be investigated.

    supervisor: Thierry Meyrath

    students: Beatriz Freire Sousa, Selma Jusufovic, Steve Schmol

  • Rigidity of weakly convex polyhedra ()
    Goal:

    A classical results of Legendre and Cauchy states that convex polyhedra are rigid: they are uniquely determined by the "shape" of their faces. They are also infinitesimally rigid, that is, any non-trivial first-order deformation induces a non-zero first-order deformation of the shape of a face.

    More recently, it was conjectured that this infinitesimal rigidity property extends to weakly convex polyhedra, that is, polyhedra which are not convex but have the same vertex set as a convex polyhedron, under an additional condition of decomposability: a polyhedron is decomposable if it can be cut into convex pieces without introducing any additional vertex.

    This infinitesimal rigidity conjecture turns out to be true (see Izmestiev, Ivan and Schlenker, Jean-Marc, Infinitesimal rigidity of polyhedra with vertices in convex position. Pacific J. Math. 248 (2010), no. 1, 171–190) under an additional condition of codecomposability (basically, the complement of a polyhedron in its convex hull is decomposable). The main goal of the project would be to explore whether this additional condition is really necessary.

    An intermediate step, or tool, would be to compute a certain symmetric matrix associated to decomposable polyhedron, and explore the index of this matrix.

    supervisor: Jean-Marc Schlenker

    student: Jilly Kevo (external link)

  • Arithmetic billiards (description)

    supervisors: Flavio Perissinotto, Antonella Perucca, Sebastiano Tronto

    students: Francesca Feyereisen, Noah Thielen (report)

  • Magic squares of squares: the modular approach ()
    Goal:

    A magic square is a square of distinct (usually positive) integers such that the sum of each row, each column and each diagonal is the same. In previous projects in the Experimental Mathematics Lab, students have created very interesting and beautiful "magic objects".

    A magic square of squares is a magic square such that each of its entries is a square. It is an open problem to decide if a 3x3 magic square of squares exists. For the 4x4 case, Euler gave a solution, which in modern algebra terms stems from the multiplicativity of the norm of quaternions.

    Many variations are possible, for instance, magic tetrahedra of squares. Using some linear algebra combined with a "brute-force" search, I produced an example.

    The goal this term is to push the "modular method" to obtain "magic objects of squares" or to compute lower bounds for the size of the entries in potential magic objects of squares. More precisely, the "modular method" will produce necessary conditions that entries in a magic object need to satisfy (usually in forms of congruences). One can then run through all tuples of integers (up to a bound given by the available computing time) satisfying these necessary conditions and either produce magic objects of squares or prove that any magic object of squares must have entries greater than the bound of the computations.

    Tools:

    Any programming language or any computer algebra system, and image processing software

    Literature:
    • Christian Boyer: Some Notes on the Magic Squares of Squares Problem, The Mathematical Intelligencer, 2005
    • O. M. Cain: Gaussian Integers, Rings, Finite Fields and the Magic Square of Squares, arXiv:1908.03236

    supervisors: Thierry Meyrath, Gabor Wiese

    students: Nicolas Dubreuil, Yann Koehnen

  • Continued fractions on Hecke triangle surfaces ()
    Goal:

    The group of 2x2-integer matrices with determinant 1, i.e. $\mathrm{SL}_2(\mathbb{Z})$, is of fundamental importance in number theory. It is quite elementary to show that it is generated by the matrices $T=\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$ and $S=\left(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right)$. In fact, if one mods out scalar matrices, any matrix can be written in a unique (subject to some obvious condition) way as a "word" in $S$ and $T$.

    Such a word can be interpreted as a continued fraction, and vice-versa, a continued fraction (with a certain sign condition) gives rise to an element of $\mathrm{SL}_2(\mathbb{Z})$.

    Essentially the Euclidean algorithm can be used to compute the continued fraction expansion of a rational number. This algorithm hence also gives a way to compute the expression as a word in $S$ and $T$ of any matrix in $\mathrm{SL}_2(\mathbb{Z})$. This algorithm also has a geometic interpretation.

    In this project, we may also explore generalisations, such as (other) Hecke triangle surfaces, or surfaces arising from quaternion algebras. The aim is to implement and visualise the interplay between continued fractions and geometry.

    Literature:
    • David Rosen: A class of continued fractions associated with certain properly discontinuous groups
    • Stefan Müller-Stach, Jens Piontkowski: Elementare und algebraische Zahlentheorie, Vieweg+Teubner

    supervisors: Lassina Dembelé, Gabor Wiese

    students: Béatrice Bach, Lara Suys, Léa Micard (report)

  • Distribution of Galois groups ()
    Goal:

    Associated to an irreducible monic polynomial with rational coefficients we have its Galois group, which measures the symmetries of the complex solutions of this polynomial.

    If instead of rational coefficients we use integral coefficients, we can consider those polynomials for which the absolute value of the coefficients is bounded above by a small constant, so that we have a finite set. If we compute all the Galois groups that appear for such polynomials, and let the bound grow higher and higher, we will observe some patterns.

    The goal is to understand which pattern appears, and understand some of the group theory that goes into these patterns.

    supervisor: Pieter Belmans

    students: Nicolas Capesius, Tania Da Cruz Rebelo, Sara Da Silva Carregueira (report)

    students: Maxime Rubio, Yannick Verbeelen (report)

  • Solving the rainbow ball ()
    Goal:

    The rainbow ball is a quite recent logical toy for children from the age of 6. It is quite simple to solve intuitively and even small children usually manage very well. The aim of this project is to understand the ball, and use the understanding to give solution strategies and implement them on the computer.

    The following steps could be done:

    • Find a formal notation system for describing the current state of the ball and also for describing moves, for computer use.
    • Represent the ball on the computer: either in text notation (simple) or graphically (more advanced).
    • Simulate the ball on the computer (allow the user to play the ball on the computer: either in text mode, or graphically).
    • Analyse the possible moves: find and describe basic (sequences of) moves, such as, exchanging two neighbouring colours without changing the other colours.
    • Use the language of permutations (e.g. cycles, transpositions) that you know from "Structures mathématiques" or "Algèbre linéaire" to make and prove theorems about the ball; in particular, answer the question: "Can any position of the colours be achieved?"
    • Use the basic moves and your mathematical analysis to write a solution algorithm.
    • (If time permits) Analyse the running time of the algorithm and try to find improvements.

    supervisors: Tara Trauthwein, Gabor Wiese

    students: José Eduardo Castro Da Silva, Hugo Miguel Fernandes Rodrigues, Jennifer Mendes Moreira (report)

  • Tuning your piano and why you are bound to fail doing so ()
    Goal:

    A piano (or any other instrument in Western music) has 12 notes in an octave. Why is that the case, and why is this only an approximation of what might be an "ideal" (but impossible) tuning system?

    This problem has kept musicians and mathematicians busy since Pythagoras. In this project you will learn why it is an impossible problem, which approximations have been used in history, and how we can compare them.

    The goal is to visualise the comparison of different tuning systems, and produce sound files that illustrate the problem and the various solutions.

    supervisors: Thilo Baumann, Pieter Belmans

    students: Clara Duchossois, Arjanita Dingu, Karine Fontaine (report)

Summer 2021

  • Arithmetic billiards ()
    Goal:
    n/a

    supervisors: Antonella Perucca, Sebastiano Tronto

    student: Joe Reguengo de Sousa (external link)

  • Three-dimensional arithmetic billiards ()
    Goal:
    n/a

    supervisors: Antonella Perucca, Flavio Perissinotto

    student: Steve Mendeleev (external link)

  • A functional equation for Riemann's zeta function ()
    Goal:

    The Mellin transform, just as any other analytic function of one complex variable, can be conveniently visualized by color-encoding its phase and mapping it onto a plane, creating a so-called phase plot, with all the important functional information perceivable at a glance. Since this kind of integral transform plays a crucial role in the theory of the Riemann zeta function, it becomes possible to create a bridge between the functional equation of the latter and the properties of Mellin integrals—a bridge constructed upon illustrations. Towards the end, this functional equation morphs into the key for unraveling the mystery behind the value of $\zeta(0)$, while concomitantly performing an empirical residue calculation, again based on visual representations of $\zeta$.

    supervisor: Thierry Meyrath

    student: Jilly Kevo (report)

  • Arithmetic billiards ()
    Goal:

    Arithmetic billiards are a geometric construction which allow an interesting visualisation for the greatest common divisor and the least common multiple of two positive integers. For an introduction to the subject, see this article. For the convenience of the general public we plan to make a website where it is possible to insert the side lengths of the billiard tables (as variables) and the starting point, and then one automatically has the nice image of the billiard path. This is surely doable and easy in dimension 2, and we will also see what we can do in dimension 3. We will also consider adding more information, for example the set of boundary points.

    supervisors: Antonella Perucca, Sebastiano Tronto

    student: Steve Schmol

  • Lët'z box-counting! (description)

    supervisors: Lara Daw, Guenda Palmirotta

    student: Kim Da Cruz (report)

  • Counting points over finite fields ()
    Goal:

    We look at solutions of polynomial equations $f(x,y)=0$ over finite fields and try to discover patterns in the number of such solutions. In fact, we are counting points on curves over finite fields. With little experimenting very quickly you will see some patterns arising. Prerequisites are that you know what a finite field is and that you are willing to experiment to find the number of solutions with a computer (or even by hand if you have enough perseverance). Once you see some patterns appear the successive experiments will present themselves. The project file suggests step by step what experiment to do. At the end you know what to do.

    Tools:

    Any programming language or any computer algebra system

    supervisors: Bryan Advocaat, Gerard van der Geer

    students: Kim Da Cruz, Dylan Mota, Clara Popescu (report)

  • Behind the secrets of Hadamard matrices and their applications (description)

    supervisor: Guenda Palmirotta

    students: Lea Micard, Eva Ragazzini, Lara Suys (report)

  • Knights and queens ()
    Goal:

    Among the famous chess inspired mathematical puzzles are those of the Knight's Tour and the Eight Queens. You can consult the corresponding Wikipedia pages on the Knights and the Queens. It is possible to make many variations of these puzzles. A previous project has explored some variations of the Knights problem. This term, you can explore other variations, or focus on variations of the Queens problem.

    Tools:

    Any programming language or any computer algebra system, and image processing software

    supervisors: Thierry Meyrath, Gabor Wiese

    students: Fátima De Jesus Dionisio Campos Gomes, Selma Hamzaj

  • Voter models and influencers in social networks ()
    Goal:

    Imagine a graph, i.e. a set of edges and vertices, in which each vertex is in one of two possible states. For example, the vertices could represent Facebook users, edges are friend relationships (two vertices are connected by an edge if the corresponding users are friends on Facebook) and the states represent whether a user is willing to buy some product (vote for a specific candidate in an election, holds some political view etc.). A voter model is such a graph, together with a set of random rules which describe how the states evolve over time. A simple rule for the Facebook example could be:

    1. Randomly choose a person.

    2. Randomly choose a friend of that person with a different opinion. This friend then switches his opinion.

    If you apply this rule ad infinitum, will the persons eventually reach a consensus, i.e. will each vertex be in the same state, no matter with which initial states you start? How about different rules? What happens if instead of two, there are n possible opinions?

    Now we change the model a bit: At the beginning, all persons have the same opinion. We are allowed to choose k persons (so-called influencers), which then switch their opinion. Afterwards, the above rule (where step 2 is only executed if a person with the same opinion as the influencers has been chosen) is applied again ad infinitum. How can we choose the influencers so that in the short/medium/long run the number of persons with our preferred opinion is maximized on average? What happens if there are also "negative" influencers, i.e. persons who propagate the opposite opinion?

    For obvious reasons, solutions to such questions are of very high interest to companies, political parties, lobbyists and governments, which is why quite a lot of research has been done in this direction. In this project, you will heuristically discover some known results and also test some conjectures by simulating models as described above.

    supervisor: Simon Campese

    students: Francesca Feyereisen, Oliver Philip Jack, Noah Thielen (report)

  • Langton and his ant (description)

    supervisor: Tara Trauthwein

    students: Vicky Zheng, Dewei Zou (report)

  • Pseudo-random numbers ()
    Goal:

    To generate chance, computers use pseudo-random numbers. These are numbers that are supposed to be close to a statistically perfect hazard, as one would for instance obtain by successively throwing a fair coin. Because they are usually created by an algorithm, pseudo-random numbers cannot be considered completely random. This is why different methods have been developed over the years to test the quality and properties of a pseudo-random number generator, which play a central role in many applications, including simulations (Monte Carlo method), electronic games (initialization), and cryptography. The aim of this project is to test the good randomness of two families of pseudo-random numbers. The first family will be classical, and generated by a linear congruence. The second family to be tested will be that invented in 2006 by Alain Schumacher (under patent), who will act as an external expert.

    supervisors: George Kerchev, Ivan Nourdin, Gabor Wiese

    students: Anne Fisch, Maxime Rubio, Yannick Verbeelen (report)

  • Magic squares of squares: the modular approach ()
    Goal:

    A magic square is a square of distinct (usually positive) integers such that the sum of each row, each column and each diagonal is the same. In previous projects in the Experimental Mathematics Lab, students have created very interesting and beautiful "magic objects".

    A magic square of squares is a magic square such that each of its entries is a square. It is an open problem to decide if a 3x3 magic square of squares exists. For the 4x4 case, Euler gave a solution, which in modern algebra terms stems from the multiplicativity of the norm of quaternions.

    Many variations are possible, for instance, magic tetrahedra of squares. Using some linear algebra combined with a "brute-force" search, I produced an example.

    The goal this term is to push the "modular method" to obtain "magic objects of squares" or to compute lower bounds for the size of the entries in potential magic objects of squares. More precisely, the "modular method" will produce necessary conditions that entries in a magic object need to satisfy (usually in forms of congruences). One can then run through all tuples of integers (up to a bound given by the available computing time) satisfying these necessary conditions and either produce magic objects of squares or prove that any magic object of squares must have entries greater than the bound of the computations.

    Tools:

    Any programming language or any computer algebra system, and image processing software

    Literature:
    • Christian Boyer: Some Notes on the Magic Squares of Squares Problem, The Mathematical Intelligencer, 2005
    • O. M. Cain: Gaussian Integers, Rings, Finite Fields and the Magic Square of Squares, arXiv:1908.03236

    supervisor: Gabor Wiese

    students: Eden Carbonell, Abigail Verholen

Winter 2020

  • Solving polynomial equations ()
    Goal:

    The goal of this project is to implement, to explore and to illustrate various methods for solving polynomial equations in one variable.

    You will study equations:

    • over the real numbers using known formulas for small degrees (see Wikipedia),
    • over the real numbers using Newton approximation (see Wikipedia),
    • over finite fields, using Berlekamp's algorithm and/or other algorithms (see Wikipedia),
    • modulo $p^n$: starting from a solution modulo a prime number $p$ (obtained for instance by Berlekamp's algorithm), one can "improve" it to a solution mod $p^2$, and then further mod $p^3$, etc. using a variant of Newton approximation, called Hensel's lemma (see Wikipedia).

    supervisors: Guenda Palmirotta, Gabor Wiese

    students: Kim Da Cruz, Dylan Mota, Clara Popescu (report)

  • Eigenvalues of random matrices ()
    Goal:

    In school, many of you have done the classical dice throw experiment: You repeatedly throw a die, note which number comes up and then compute the running averages of all numbers thrown so far. If you throw long enough, this average will converge to to 3.5 (the arithmetic mean of all numbers on the die, i.e. one to six). This phenomenon is known as the law of large numbers. In this project, instead of throwing a die, we will do something much more interesting by generating random matrices in several ways and investigating their eigenvalues. If you want, you can stick to dice or even coin flips but your computer can easily sample from more complicated distributions as well) and then fill a square matrix with $n$ rows and columns in a symmetric way (i.e. the upper diagonal entries are filled randomly and then copied to the lower diagonal entries) by repeated die throw/coin flips. Afterwards, compute the $n$ eigenvalues of the obtained matrix. If you generate enough matrices in this way, choose $n$ sufficiently large and then compare the obtained eigenvalues, you will see a pattern emerge in the eigenvalues. The same is true if you only record the largest eigenvalue for each matrix. Questions you will answer are:

    • Do the emerging patterns depend on the source of randomness (die throws, coin flips, sampling from a Gaussian distribution etc.)?
    • Do the emerging patterns depend on the types of matrices (in the example above symmetric) we generate?
    • How do the patterns depend on the dimension $n$ of the matrix?

    All these experiments are connected to results from the fascinating field of random matrix theory, and by the end of the project (or earlier, depending on your curiosity) you will get the theoretical "solutions" (in the form of mathematical theorems) to the conjectures obtained by your simulations.

    supervisor: Simon Campese

    students: Tim Seuré, Max Thill (report)

  • Stock trading with machine learning models ()
    Goal:

    The opportunities for applying algorithmic and probabilistic techniques in finance are vast. A whole industry called FinTech has emerged in recent years in parallel to the ever-increasing usage of computers to help and model financial activities.

    Historically, applications of computer systems for trading can be traced to the 70s. In particular, in 1971 the Nasdaq stock market began operations as the world's first electronic market. In the 80s synthetic products (a type of contract) on a stock portfolio have been created making use of the Black-Scholes formula (a theoretical estimate for the solution of stochastic differential equation). Around the same time Renessaince technologies was founded - one of the most profitable hedge funds that specializes in systematic trading using quantitative models derived from mathematical and statistical analysis.

    The purpose of this project is to let students explore possible applications of probabilistic models for prediction of economic regimes and stock prices. Students will learn about a probabilistic model, how to apply it to a data set and how to evaluate its performance. Here is a possible list of models to be considered:

    • Linear regression
    • Hidden Markov models
    • Long-short term memory (recurrent neural networks)

    supervisor: George Kerchev

    students: Gil Moes, Mirza Muharemovic (report)

    students: Beatrice Bach, Anne Fisch (report)

  • The secretary problem ()
    Goal:

    Imagine an administrator wants to hire a secretary out of 10000 applicants, who are interviewed one by one. A decision about each particular applicant is to be made immediately after the interview. If rejected, the applicant cannot be recalled. Naturally, there are many possible policies for hiring the candidates, each which will have an associated "probability of choosing the best candidate". For instance, if we decide to take the policy of "always hiring the first person in the queue", the probability of choosing the best candidate equals the probability that the first candidate in the queue is the best, which is 0.0001. However, as one might suspect, this is not the best course of action. The goal of this project is to find the strategy that maximizes the probability of selecting the best candidate.

    supervisor: Arturo Jaramillo Gil

    students: Sebastien Plaasch, Maxime Rubio, Yannick Verbeelen (report)

  • Illustrating number theory ()
    Goal:

    Number theory can generate beautiful and intriguing pictures. Samples can be found in a nice expository article by Kate Stange. The aim of this project is to re-create some of the pictures from the article, to make variations of them, and possibly to find (elsewhere or by oneself) similar sources.

    supervisor:

    students: Jan Huberty, Ibrahim Hukic, Maurice Desquiotz

  • Buffon's needle problem ()
    Goal:

    Suppose that we drop many needles in a floor made out of strips as in the picture

    One of the earliest problems in the area of geometric probability is the so called "Buffon's needle problem", which poses the question of determining the probability that a needle intersects one of such lines. Direct computations allow us to derive (in a reasonably simple way), very clean expressions for such probability. Simulations for this experiment can be easily implemented. The reader is referred to this website for a very nice visual explanation of the problem. The purpose of this project consists of determining explicit solutions for this problem and validate the findings with the implementation of a suitable simulation.

    supervisor: Arturo Jaramillo Gil

    students: Dany Da Silva Brandao, João Gil Rocha Figueiredo (report)

  • Graphs and puzzles ()
    Goal:

    The goal of the project is to explore complexity of some combinatorial puzzles using tools from graph theory. Some puzzles admit a graph model in which possible states of the games are represented by vertices, and an edge is added between two vertices if the corresponding states can be reached from each other by application of the rules of the puzzle. By using these graphs models, it is sometimes possible to apply graph theoretic algorithms to find optimal solutions to the puzzles, or to quantify their complexity.

    supervisors: Hugo Parlier, Bruno Teheux

    students: Inas Bosch, Yanis Bosch (report)

  • Counting points over finite fields ()
    Goal:

    We look at solutions of polynomial equations $f(x,y)=0$ over finite fields and try to discover patterns in the number of such solutions. In fact, we are counting points on curves over finite fields. With little experimenting very quickly you will see some patterns arising. Prerequisites are that you know what a finite field is and that you are willing to experiment to find the number of solutions with a computer (or even by hand if you have enough perseverance). Once you see some patterns appear the successive experiments will present themselves. The project file suggests step by step what experiment to do. At the end you know what to do.

    Tools:

    Any programming language or any computer algebra system

    supervisors: Bryan Advocaat, Gerard van der Geer

    students: Selma Jusufovic, Irma Skrijelj (report)

  • The Collatz conjecture (description)

    supervisors: Luca Notarnicola, Massimo Notarnicola

    students: Belmin Adrovic, Luca Di Cino, Raphael Proenca

    students: Clara Duchossois, Antonios Pantelis, Julien Pezzi

Summer 2020

  • Music in Fibonacci numbers ()
    Goal:

    The idea is to compare, in terms of mathematical and musical interest, different ways in which the division algorithm produces melodies, starting from the Fibonacci series. This will require the student to understand some mathematical properties of this particular recursive number series; understand some connections between cyclic groups and representations of musical theory. the output will be a number of different melodies, along with their rhythmic patterns (also extracted from the Fibonacci series), and if there is enough time, along with harmonic progressions obtained from such series.

    supervisor: Eduardo Ibarguengoytia

    students: Beatrice Bach, Loris Chenneaux (report)

  • Knights (and queens) ()
    Goal:

    Among the famous chess inspired mathematical puzzles are those of the Knight's Tour and the Eight Queens. You can consult the corresponding Wikipedia pages on the Knights and the Queens. It is possible to make many variations of these puzzles. For instance, we will ask to find the shortest path for a knight to jump from one square to another one. Of course, we shall not do this only on the usual $8\times8$ board, but on arbitrary chess boards, not necessarily square ones. We shall actually go to three dimensions (or even $n$ if we are daring); we could allow a three-dimensional knight to jump a distance of 3 in one direction (e.g. along the $x$-axis), a distance of 2 in another direction (e.g. the $y$-axis) and a distance of 1 in the last direction. The goal is to program these (and other) variations, illustrate them and make observations (for instance, on the (im)possibility for certain paths, the minimal length of paths, etc.).

    supervisors: Guenda Palmirotta, Gabor Wiese

    students: Lea Micard, Eva Ragazzini, Lara Suys (report)

  • Magic squares of squares ()
    Goal:

    A magic square is a square of distinct (usually positive) integers such that the sum of each row, each column and each diagonal is the same. A magic square of squares is a magic square such that each of its entries is a square. It is an open problem to decide if a $3\times3$ magic square of squares exists. For the $4\times4$ case, Euler gave a solution, which in modern algebra terms stems from the multiplicativity of the norm of quaternions.

    Many variations are possible. One can, for instance, ask for bimagic squares. Those are magic squares that are also magic when one squares their entries. Another variation are magic cubes.

    Christian Boyer has created an interesting website.

    One goal is to study known constructions and examples, and to use them to make a nice collection of interesting magic squares and cubes and to illustrate them in an attractive way (this requires some effort especially for cubes).

    Another goal is to create "congruence magic squares of squares" using modular arithmetic.

    supervisors: Guenda Palmirotta, Gabor Wiese

    students: Arjanita Dingu, Clara Duchossois (report)

    students: Sabrina Cunha, Anne Fisch, Yannick Verbeelen

  • Primes versus random sets ()
    Goal:

    The famous prime number theorem states that the number of primes up to a bound $x$ behaves asymptotically as $x/\log(x)$. This can be translated into saying that the $n$th prime is roughly $n\log(n)$. If one believes in the Riemann hypothesis, the arguably most famous open problem in mathematics, then one also has very good bounds on the "error", that is, on the discrepancy between "reality" and the asymptotic prediction.

    These results about the distribution of primes may suggest that primes are very "regular" and that most things are known. But this is not the case. The world of prime numbers has many mysteries and many famous (and even more not well-known) conjectures and questions about primes are open, such as the twin prime conjecture.

    In this project you will first make a collection of known facts about the distribution of primes, as well as a collection of open (and maybe also some solved) questions about primes. You will then create many sets of random integers (which will be hopefully huge) sharing the same distribution properties as the set of primes. Then you will test the conjectures you found on your random sets (as well as on the primes). In some cases, this may give a hint if the question or conjecture you are looking at are of an "arithmetic" nature or are "just" related to the distribution properties.

    supervisors: Luca Notarnicola, Gabor Wiese

    students: Irma Skrijelj, Karine Fontaine, Eric Wagener

    students: Sebastien Plaasch, Maxime Rubio, Lucas Villière

  • Perturbations of the Lorenz system ()
    Goal:

    The Lorenz system is a system of ordinary differential equations originally studied by Edward Lorenz in 1963 as a simple model of atmospheric convection. The system admits chaotic solutions for certain values of the parameters and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. In this project, you will investigate the Lorenz system and in particular, check the behaviour for various values of the parameters. After this, you can similarly investigate the behaviour of related systems including a periodically perturbed Lorenz system. One can search the vast literature for other examples or indeed propose your own modification of the system.

    supervisor: Andrew Bruce

    student: Ibrahim Hukic (report)

  • Fractal limit sets and Schottky groups ()
    Goal:

    The Möbius maps are transformations of the plane sending circles and straight lines to circles and straight lines, and preserving the angles between curves. Schottky groups are some particular groups consisting of Möbius transformations, and constructed by pairing couples of circles. Whenever such a group is given, there is an associated subset of the plane, called its "limit set". These sets have lots of symmetries related to the Shottky group, and are very often beautiful fractal sets.

    The goal of this project is to be able to draw and explore these beautiful subsets of the plane. It will be also a nice opportunity to understand Möbius transformations and their link with $2\times 2$ matrices, as well as some topics around groups.

    supervisor: Miguel Acosta

    students: Gil Armand Arsène Moes, Mirza Muharemovic (report)

    students: Inas Bosch, Yanis Bosch (report)

  • Percolation ()
    Goal:

    Consider a $n\times n$ grid of sites. Each site is either blocked or open. Initially open sites are empty. A full site is an open site that can be connected to an open site in the top row via a chain of neighboring (left, right, up, down) open sites. If there is a full site in the bottom row, then we say that the system percolates (see picture below). If sites are independently set to be open with vacancy probability $p$, what is the probability that the system percolates? It turns out that there is a value of $p$ above which the system percolates almost always. This threshold value of $p$ can be estimated via Monte Carlo simulation. One can study further related concepts like bond percolation and passage time.

    supervisor: George Kerchev

    student: Tim Seuré (report)

  • Machine learning for trading ()
    Goal:

    The opportunities for applying algorithmic and probabilistic techniques in finance are vast. A whole industry called FinTech has emerged in recent years in parallel to the ever-increasing usage of computers to help and model financial activities.

    Historically, applications of computer systems for trading can be traced to the 70s. In particular, in 1971 the Nasdaq stock market began operations as the world's first electronic market. In the 80s synthetic products (a type of contract) on a stock portfolio have been created making use of the Black-Scholes formula (a theoretical estimate for the solution of stochastic differential equation). Around the same time Renessaince technologies was founded - one of the most profitable hedge funds that specializes in systematic trading using quantitative models derived from mathematical and statistical analysis.

    The purpose of this project is to let students explore possible applications of probabilistic models for prediction of economic regimes and stock prices. Students will learn about a probabilistic model, how to apply it to a data set and how to evaluate its performance. Here is a possible list of models to be considered:

    • Linear regression
    • Hidden Markov models
    • Long-short term memory (recurrent neural networks)

    supervisor: George Kerchev

    students: Anthony Pantelis, Julien Pezzi (report)

    students: Belmin Adrovic, Luca di Cino, Raphael Jose Proenca Carvalho (report)

    students: Dylan da Silva Moreira, Tiago Dias (report)

Winter 2019

  • Arithmetic billiards ()
    Goal:

    Arithmetic billiards are a geometric construction which allow an interesting visualisation for the greatest common divisor and the least common multiple of two positive integers. For an introduction to the subject, see this article. At least the following two generalizations can be addressed:

    • Allowing the starting point not to be a corner.

    • Considering three (or even more) integers i.e. the trajectory of a ball bouncing inside a parallelepiped rather than a rectangle.

    Collecting experimental data will allow to make conjectures concerning the shape of the path, which then one can try to prove. It is possible that the key ideas used for the base case can be extended to prove the conjectures.

    supervisor: Antonella Perucca

    student: Bruno Carvalho da Veiga (report)

  • Magic squares of squares ()
    Goal:

    A magic square is a square of distinct (usually positive) integers such that the sum of each row, each column and each diagonal is the same. A magic square of squares is a magic square such that each of its entries is a square. It is an open problem to decide if a $3\times3$ magic square of squares exists. For the $4\times4$ case, Euler gave a solution, which in modern algebra terms stems from the multiplicativity of the norm of quaternions.

    Many variations are possible. One can, for instance, ask for bimagic squares. Those are magic squares that are also magic when one squares their entries. Another variation are magic cubes.

    Christian Boyer has created an interesting website.

    One goal is to study known constructions and examples, and to use them to make a nice collection of interesting magic squares and cubes and to illustrate them in an attractive way (this requires some effort especially for cubes).

    Another goal is to create "congruence magic squares of squares" using modular arithmetic.

    supervisors: Guenda Palmirotta, Gabor Wiese

    students: Selma Jusofovic, Manal Kaaouane, Mirza Muharemovic, Gil Moes

  • Betting games ()
    Goal:

    Two players play the following game. Flip a coin until either HHH or THH occurs in the sequence of flips. In the first case, Player 1 wins, otherwise, it's Player 2. What is the probability of Player 1 winning? What if both players can choose a triplet to observe? The game can be modeled by a Markov chain giving precise answers while computer simulations can give estimates. How do the two compare?

    supervisor: George Kerchev

    students: Belmin Adrovic, Raphael Jose Proenca Carvalho, Luca di Cino (report)

  • Betting on Catalan ()
    Goal:

    A fair coin is flipped 100 times yielding a sequence of heads and tails. Player A observes that in the end there are 50 heads and 50 tails and furthermore, after each flip there have been at least as many heads as tails. Player B thinks that such a sequence is extremely unlikely and offers a bet to A with $100:1$ odds that this will not happen if they throw again the coin 100 times. Should A accept the bet? What are the true odds for such an event? Is there a good graphical representation for sequences with the properties observed by A? What happens if there are $2n$ coin flips? Biased coin?

    supervisor: George Kerchev

    students: Dylan da Silva Moreira, Ibrahim Hukic (report)

  • Runs in random sequences ()
    Goal:

    Imagine you are observing a random binary sequence, i.e. at each step a (biased) coin is flipped, yielding a zero or one with some given probabilities $p$ and $1-p$. This is a good model for many repetitive processes, for example the production of some product in a factory (product with/without a defect), the failure of a hard drive (bit is saved correctly/incorrectly) etc. A run of length $k$ is the occurence of $k$ ones in a row. In this project, your task is to verify and visualize several central limit theorems and asymptotics regarding such runs by simulation. Among other things, you will find answers to the following questions: After $n$ steps, how many runs of length $k$ can be observed on average? How likely is it that no run of length $k$ has occured?

    supervisor: Simon Campese

    students: Yanis Bosch, Tim Seuré (report)

  • Perturbations of the Lorenz system ()
    Goal:

    The Lorenz system is a system of ordinary differential equations originally studied by Edward Lorenz in 1963 as a simple model of atmospheric convection. The system admits chaotic solutions for certain values of the parameters and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. In this project, you will investigate the Lorenz system and in particular, check the behaviour for various values of the parameters. After this, you can similarly investigate the behaviour of related systems including a periodically perturbed Lorenz system. One can search the vast literature for other examples or indeed propose your own modification of the system.

    supervisor: Andrew Bruce

    students: Sven Federspiel, Alex Schammo

Summer 2019

  • Non-standard fair dice ()
    Goal:

    A dice is a polyhedron with different numbers on its faces. When you throw a dice, the result of the throw will be the number facing the ground. A dice is fair if the probability law on the set of numbers written on the dice resulting from the throw is uniform.

    A common way to create fair die is to use the most symetric polyhedrons a.k.a. Plato's polyhedrons.

    But there are many other ways to create fair die! For instance in an article by K. Robin McLean (Dungeons, Dragons and Dice The Mathematical Gazette, Vol. 74, No. 469 (Oct., 1990), pp. 243-256) the author draws another polyhedron which leads to a fair dice.

    The above example is still very symetric in a specific sense: it is isohedral, i.e., every face can be obtained from the other by applying a rotation.

    The goal is to conceive many different kinds of fair die, especially non-isohedral ones. Ideally these die would be 3D printed and have their fairness studied.

    supervisor: Clément Guérin

    students: Mjremea Aganovic, Ibrahim Hukic (report)

  • Benford's law ()
    Goal:

    Imagine you would keep track of the first digit after the comma on your next electricity bills (3 for a total of 81.34 Euros, 1 for a total of 53.12 Euros etc.). What would you find in the long run? Does every digit between zero and nine occur with the same frequency? Surprisingly, the answer is no! Instead, the digits follow the so called Benford probability law. Your task in this project is to test for this Benford law using various datasets from STATEC and EUROSTAT (the statistical databases of Luxembourg and the European Union)

    supervisor: Simon Campese

    students: Dylan Alves Ferreira, Marko Peric, Joe Reguengo de Sousa

  • Visualisation of Bianchi fundamental polyhedra ()
    Goal:

    The Bianchi groups are $2x2$ matrix groups, and each of them acts on 3-dimensional hyperbolic space by pushing it forward and backwards in some directions, rotating it, and combining these movements. A fundamental polyhedron for this action is the selection of an area of hyperbolic space such that its copies under the movements by the Bianchi group together cover up all of hyperbolic space.

    We can require this area to be a polyhedron; and in order for it not to become too big, we can require that it touches its copies only at its facets. The shape of the Bianchi fundamental polyhedra is computed by the project supervisor's software Bianchi.gp in terms of hyperbolic coordinates, and has been visualized in the upper-half space model, where planes get distorted to hemispheres.

    The students will collect screenshots from Geomview and organise them in animation files (or develop a more practical export procedure).

    supervisor: Alexander Rahm

    student: Kelly Jost (report)

  • Experimental tests on the $abc$-conjecture (description)

    supervisor: Alexander Rahm

    students: Marie Baroni, Darina Zakharova

    student: Arno Geimer (report)

Winter 2018

  • Lattice reduction (description)

    supervisors: Luca Notarnicola, Gabor Wiese

    students: Joris Limonier, Dany Alves Marques, Joe Reguengo de Sousa

Summer 2018

  • Spectrum of random matrices ()
    Goal:

    Check numerically that the empirical measure of the spectrum of any symmetric matrix with random entries converges, as the dimension of the matrix tend to infinity, to the semicircular law. Study other related examples.

    supervisor: Massimo Notarnicola

    students: Fabio Carvalho dos Santos, Sven Federspiel, Alex Schammo (report)

  • Monte Carlo simulation ()
    Goal:

    Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Their essential idea is using randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches.

    The goal of this project would be to vizualise and study several examples of situations where Monte Carlo estimation can be used.

    supervisor: Guenda Palmirotta

    students: Philippe Karst, Bruni Carvalho, Arno Geimer

Winter 2017

  • Modular value distribution ()
    Goal:

    In many cases one is interested in the values taken by a function. In this project we study the values of integer polynomials modulo integers. For instance, the values of $f(x)=x^2$ at natural numbers are precisely the squares: $0,1,4,16,25,36,\ldots$

    Considering them modulo 8, one finds $0,1,4,0,1,4,\ldots$ In fact, one easily proves that 0,1,4 are the only residues modulo 8 that appear for squares. One could consider other polynomials like $f(x)=x^3$ or $f(x)=x^2+x+1$. The modulus can be a prime, a prime power or any positive integer. In fact, the latter case can usually be reduced to the previous ones by the Chinese Remainder Theorem.

    The aim is to make computer experimentation on this kind of question, to nicely present and illustrate the results, and to see what can be theoretically explained by the Chinese Remainder Theorem.

    This project is motivated by a concrete question from a research project. The questions and the details will be explained if the project is chosen.

    supervisors: Luca Notarnicola, Gabor Wiese

    students: Bruno Carvalho, Alex Schammo, Sven Federspiel

  • Convex hulls of finite packs of spheres ()
    Goal:
    The linear pack of spheres of length $n$ in dimension $d$ is the convex hull of $n$ non-overlapping balls of radius one, with centers arranged on a line segment and without gaps between the balls. The volume of this linear pack shall be compared against the volume of the convex hull of other non-overlapping arrangements of $d$-dimensional balls, in the quest of finding the arrangement of minimal volume. It is obvious that in dimension $d=2$, the triangular arrangement is more efficient (in terms of lower volume of its convex hull) than the linear pack.

    The task of the project group will be to fix $d > 2$, and to find an arrangement with $n$ as small as possible which is more efficient than the linear pack of length $n$.

    Already in dimension $d=3$, this requires some efforts, but also dimension $d=4$ should be tackled. Arrangements constructed in low dimensions should be generalised for arbitrary dimension, and the volume of their convex hull computed on the machine for several values of $d$, in order to compare against the linear pack.

    supervisor: Alexander Rahm

    student: Fabio Marcelo Carvalho dos Santos (report)

  • Enveloppes convexes d'ensembles limites ()
    Goal:
    n/a

    supervisor: Jean-Marc Schlenker

    students: Arno Geimer, Philippe Karst, Vincent Wolff (report)

Summer 2017

  • Primality testing ()
    Goal:

    For cryptography, coding theory, and for experiments in number theory, it is of utmost importance to be able to decide if a given positive integer is prime or not. This is called a primality test. Note that a primality test does NOT give the factorisation of a given integer into primes, it outputs only yes or no. In fact, it is believed that factorising a given integer into its prime factors is a hard, and often practically intractable problem: the very important cryptographic system RSA relies on this hardness.

    One distinguishes between deterministic and probabilistic primality tests: the latter ones usually run faster, but if they answer "yes", then this only means that the number is prime with a high probability. In the project, some primality tests shall be described, implemented, tested and compared.

    supervisor: Gabor Wiese

    students: Robert Contignon, Paulo Portinha (report)

  • Arithmetic densities ()
    Goal:

    A basic result in analytic number theory states that the "likelihood" for an integer to be squarefree is $6/\pi^2$, i.e. roughly $61\%$.

    The aim of this project is to explore this and similar situations and to illustrate them.

    supervisor: Gabor Wiese

    students: Bruno Carvalho, Alex Schammo, Philippe Karst, Arno Geimer (report)

Winter 2016

  • Arcsine law ()
    Goal:
    Vizualize the arcsine distribution followed by the last visit to 0 (resp. the time above 0) of the simple random walk on $\mathbb{Z}$.

    supervisor: Ivan Nourdin

    students: Lia Malato Leite, Robert Contignon (report)

Summer 2016

  • Magic squares ()
    Goal:

    Magic squares are integral square matrices such that the sums of the entries in every column, in every row and in both main diagonals are all the same. One speaks of a diabolic magic square if in addition the sums of the entries in broken diagonals are also equal to the same number.

    Magic squares appear more magic than they really are; there are many known ways for constructing them. But, there are also many questions that are still open.

    The goal is to construct magic squares on the computer by using some of the many constructions that are available in the literature. The examples obtained can be used to study some open questions.

    supervisors: Panagiotis Tsaknias, Gabor Wiese

    students: Lia Malato Leite, Victoria Jacquemin, Noemie Boillot (report)

  • Visualising modular forms ()
    Goal:

    Modular forms are analytic functions on the upper half-plane of the complex plane satisfying certain simple transformations and/or symmetries. They play a prominent role in number theory.

    In this project, modular forms shall be visualised. For instance, plots like those on the wikipedia page of Klein's j-invariant shall be produced for other modular forms. Optionally, one could also think about 3D-plots of modular forms.

    supervisors: Panagiotis Tsaknias, Gabor Wiese

    students: Joel Costa da Rocha, Loqman Salamatian, Alex Ferreira Costa (report)

  • Pythagorean triples and rational points ()
    Goal:

    Pythagorean triples are integer triples $(a,b,c)$ occurring as side lengths of rectangular triangles. By Pythagoras' theorem, they are hence the triples such that $a^2 + b^2 = c^2$. Setting $x=a/c$ and $y=b/c$ we obtain $x^2 + y^2 = 1$. Thus, our Pythagorean triple has given rise to a rational point on the unit circle; the word "rational" refers to the fact that the coordinates $(x,y)$ are rational numbers. Conversely, any rational point on the unit circle gives rise to a "primitive" Pythagorean triple (that is, a triple such that the gcd of $a,b,c$ is 1). Any Pythagorean triple is proportional to a primitive one, so it is no loss just to consider primitive triples.

    As a generalisation, one can consider rational points on other (smooth) plane curves. The circle is a special case of a plane conic section. Rational points on plane conic sections (assumed to be defined over $\mathbb{Q}$) can be parametrised in a simple way: fix a point on it and draw a line of rational slope through it; it will intersect the conic section in one point.

    Pythagorean triples can be enumerated; one way to obtain an enumeration is by using the geometric description. Another one uses certain matrix groups.

    The goal is to illustrate rational points on different curves, mostly plane conic sections, and to give them a number-theoretic meaning. The subject can be developed into many different ways; this will be subject to a discussion with interested participants.

    supervisors: Panagiotis Tsaknias, Gabor Wiese

    student: Jerry Hilgert (report)

Winter 2015

  • Galois groups in generalisations of Maeda's conjecture ()
    Goal:
    n/a

    supervisors: Panagiotis Tsaknias, Gabor Wiese

    students: Lucija Calic, Valnea Skansi

  • Simulation of probability distributions ()
    Goal:

    The goal of this project is to review several methods for simulating the most popular and widely used probability distributions. Then, effective implementations of those techniques will be done and will be used to empirically compute some features (such as median, moments, etc.) of the considered distributions.

    supervisor: Ivan Nourdin

    students: Lynn Van der Weken, Fa Zhu (report)

Summer 2015

  • Benford's law ()
    Goal:

    Imagine you would keep track of the first digit after the comma on your next electricity bills (3 for a total of 81.34 Euros, 1 for a total of 53.12 Euros etc.). What would you find in the long run? Does every digit between zero and nine occur with the same frequency? Surprisingly, the answer is no! Instead, the digits follow the so called Benford probability law. Your task in this project is to test for this Benford law using various datasets from STATEC and EUROSTAT (the statistical databases of Luxembourg and the European Union).

    supervisor: Ivan Nourdin

    students: Lynn Van der Weken, Julie Kersch, Christina Haack (report)

  • Triangulation de Delaunay ()
    Goal:

    Given a set of points in the plane (resp. in Euclidean 3-dimensional space, or in higher dimensions) one can define a "Delaunay triangulation" of their convex hull, as defined for instance in this Wikipedia page. It is a basic tool in computational geometry. The definition involves the positions of the points with respects of circles (resp. spheres).

    More recently, more exotic types of Delaunay-like decompositions have been proved to exist, where the circles (resp. spheres) are replaced by special kinds of hyperboloids or paraboloids. The main goal of the project is to compute those "exotic" Delaunay triangulations.

    To achieve this goal, the students involved will be lead to understand relatively simple notions of projective geometry as well as the basic geometry of spaces of constant curvature, and in particular their projective and conformal models.

    supervisor: Jean-Marc Schlenker

    student: Guenda Palmirotta (report)

  • Décomposition de Delaunay ()
    Goal:

    Given a set of points in the plane (resp. in Euclidean 3-dimensional space, or in higher dimensions) one can define a "Delaunay triangulation" of their convex hull, as defined for instance in this Wikipedia page. It is a basic tool in computational geometry. The definition involves the positions of the points with respects of circles (resp. spheres).

    More recently, more exotic types of Delaunay-like decompositions have been proved to exist, where the circles (resp. spheres) are replaced by special kinds of hyperboloids or paraboloids. The main goal of the project is to compute those "exotic" Delaunay triangulations.

    To achieve this goal, the students involved will be lead to understand relatively simple notions of projective geometry as well as the basic geometry of spaces of constant curvature, and in particular their projective and conformal models.

    supervisor: Jean-Marc Schlenker

    student: Guenda Palmirotta (report)

  • How many zeroes does a random polynomial have? ()
    Goal:

    The number (and the distribution) of the zeros of different ensembles of random polynomials have attracted the attention of physicists and mathematicians for many years. Besides their implicit interest, they have a vast range of applications: algorithm complexity, quantum physics, etc.

    The first class of studied random polynomials were the usual ones \[ p_K(t)=a_0+a_1x+\ldots+a_Kx^K \]

    In this case it is known that, as $K$ tends to infinity, the average and the variance of the number of zeros on the whole real line are asymptotic to $2/\pi\log(K)$ and $4/\pi(1 - 2\pi)\log(K)$ respectively, for a large range of choices of the random coefficients $a_n$ (centered and with variances equal to one). Nevertheless, with a convenient choice of the coefficients (centered but with different variances of a very concrete form) it is known that the asymptotic number of zeros of such a polynomial is $\sqrt{K}$ that is really much more than the preceeding value.

    Besides, there are results that state that in case of systems of complex homogeneized polynomials, the roots are surprisingly isotropically distributed on the sphere.

    Other classes of studied random polynomial include the trigonometric polynomials and the hypergeometric polynomials. For trigonometric polynomials, central limit theorems for their number of zeros have been established recently in the situation of independent standard Gaussian coefficients. In the case of the hyperbolic polynomials there are known asymptotics for the mean number of zeros under different choices of the random coefficients (all Gaussian for the moment).

    The aim of this mémoire is (at least) threefold:

    • to review the existing bibliography on the subject (with a special emphasis on trigono- metric polynomials).
    • to study the techniques that have been used so far.
    • to make some numerical simulations in order to "confirm" some of the already known results and to "discover" some new ones.

    supervisor: Federico Dalmao

    student: Massimo Notarnicola (report)

  • Testing conjectures on primes ()
    Goal:

    Prime numbers are very mysterious objects and they have always fascinated people. Many basic questions about prime numbers are still open, for instance:

    • (Goldbach conjecture) Is every even positive integer the sum of two primes?
    • (Twin prime conjecture) Are there infinitely many prime numbers $p$ such that $p+2$ is also prime?
    • (Riemann conjecture, equivalent formulation) Let $\Pi(x)$ be the number of primes less than $x$. Is the absolute value of the difference $\Pi(x) - x/\log(x)$ bounded by the square root of $x$ divided by $\log(x)$ for all large enough $x$?

    The goal is to test some conjectures on prime numbers on a computer.

    Possible conjectures to test are next to the many famous conjectures (including those mentioned above) also those by the Chinese mathematician Zhi-Wei Sun who is currently making an enormous number of new ones (see for instance this article).

    supervisors: Panagiotis Tsaknias, Gabor Wiese

    students: Joel Costa da Rocha, Alex Ferreira Costa (report)

  • Visualising the distribution of primes ()
    Goal:

    Prime numbers appear to be sporadic and hard to predict. If, however, one counts the number of primes less than $x$, then one can prove (that's the famous prime number theorem) that this number is asymptotically equal to $x/\log(x)$.

    This means that the $n$th prime, call it $p_n$, is asymptotically of size $n\log(n)$, i.e. the distance between primes gets bigger and bigger (on average).

    The goal is to visualise the distribution of primes. One can do this by

    • plotting the prime counting function and comparing it with x/log(x),
    • plotting the integer nearest to p_n/log(n),
    • plotting the Klauber triangle or the Ulam spiral (see e.g. on wikipedia),
    • other ways that you may devise.

    supervisors: Panagiotis Tsaknias, Gabor Wiese

    students: Jim Barthel, Pietro Sgobba, Fa Zhu (report)

  • Primes is in P ()
    Goal:

    For many purposes in cryptography it is essential to have "big" prime numbers; for instance, in the very widely used cryptosystem RSA one uses prime numbers having 2048 binary digits (i.e. of size $2^2048$, having 617 decimal digits). The question thus arises of how to find such big prime numbers. The answer is that one takes random numbers and tests if they are prime. One hence needs a fast test whether a given integer n is prime; such a test is called a primality test.

    It is important to remark that a primality test only tests if an integer $n$ is prime, it does not yield its decomposition into prime factors. This latter question is computationally very hard to solve (the security of RSA depends on this hardness).

    It has been known for some time that probabilistic primality testing is possible in time polynomial in the size of $n$ (this means that the run time is bounded from above by a polynomial in $\log(n)$, i.e. it can be bounded by $\log(n)^m$ for some $m \in \mathbb{N}$); for instance, the Miller-Rabin test achieves this. By probabilitistic primality testing one means that the result is true with a very high chance, but is not proved to be true.

    It was a great surprise when in 2002 Agrawal, Kayal and Saxena found a primality test (called AKS-test) that runs in time polynomial in $\log(n)$ (like Miller-Rabin) and yields proved results.

    supervisor: Gabor Wiese

    student: Luca Notarnicola (report)

  • Elliptic curve cryptography and the Weil pairing ()
    Goal:

    An elliptic curve is an algebraic curve equipped with an abelian group law. Elliptic curves are very important objects in current number theory. Apart from this, they are also used in cryptography (e.g. in the German identity card and the German passport). For this application, one uses elliptic curves defined over "big" finite fields. The abelian group is then a "big" finite abelian group. If one chooses the curve well, this group can be cyclic, generated by a point $P$. Any other point $Q$ is then of the form $Q = mP$ for some $m \in \mathbb{Z}$.

    Given $Q$ and $P$, it seems to be very hard to compute $m$. This is called the Discrete Logarithm Problem for elliptic curves.

    Let us denote the elliptic curve by $E$ and the neutral element $O$ (it is also natural to call it $\infty$) and let $n \in\mathbb{N}$. The $n$-torsion points are defined to be the set of points of order a divisor of n. André Weil defined a pairing on $E[n]$. This pairing has many very nice features.

    supervisor: Gabor Wiese

    student: Steve Dias da Cruz (report)

Winter 2014

  • Simple closed curves on surfaces ()
    Goal:

    Compute the maximum number of non-homotopic simple closed curves that can fit on a surface of small genus so that two curves intersect at most once (or at most $N$ times, for $N$ small). Focus on the special case where this family of curves contains a maximal family of non-intersecting curves (a pants decomposition).

    supervisor: Jean-Marc Schlenker

    student: Léonard Cadilhac (report)

  • Computing Galois groups via Chebotarev density (description)

    supervisors: Panagiotis Tsaknias, Gabor Wiese

    student: David Gasperini (report)

  • Factorisation of polynomials modulo $p^n$ (description)

    supervisors: Panagiotis Tsaknias, Gabor Wiese

    student: Salima Lamhar (report)

  • Compressed sensing ()
    Goal:

    Compressed sensing (CS) is a signal processing technique for efficiently acquiring and reconstructing a sparse signal, by finding solutions to underdetermined linear systems. In 2004, Emmanuel Candès and Terence Tao, and David Donoho independently, proved that given knowledge about a signal's sparsity, the signal may be reconstructed with even fewer samples than the sampling theorem requires. Since then, CS has seen a impressive number of spectacular applications, in areas including photography, holography, facial recognition, astronomy, magnetic resonance imaging, shortwave-infrared cameras, etc.

    The goals of this project is to

    • Understand the proof of a version of the Candès–Tao theorem
    • Propose a program (in Matlab) to find the exact number of samples that are required to reconstruct a sparse signal exactly.

    supervisor: Ivan Nourdin

    student: Lucien May (report)