# Past projects

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

The list is not complete yet, and will be extended with projects dating back to 2014.

### 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

• 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

• 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

• 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

• 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 $$\int_a^b e^{-x^2}\mathrm{d}x$$ 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

• 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

### 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 (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

• Arithmetic billiards (description)

supervisors: Flavio Perissinotto, Antonella Perucca, Sebastiano Tronto

students: Francesca Feyereisen, Noah Thielen

• 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: Yann Koehnen, Nicolas Dubreuil

• 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

• 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

students: Maxime Rubio, Yannick Verbeelen

• 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

• 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

### Summer 2021

• 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

• 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 (report)

• 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