# Currently offered projects

More projects for next semester are coming soon!

• 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.

Tools: SageMath

supervisors: Pieter Belmans, Pietro Sgobba
level: moderate

• Models for cubic surfaces ()
Goal:

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

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

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

Tools: SageMath, D3.js

supervisors: Pieter Belmans, Mélanie Theillière

• 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

supervisor: Pieter Belmans
level: easy

• 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

• Arithmetic billiards (description)

supervisors: Flavio Perissinotto, Antonella Perucca, Sebastiano Tronto
level: any

• Invariants of algebraic curves ()
Goal:

Algebraic curves can be given by an algebraic equation $f(x,y)=0$ with $f(x,y)$ a polynomial in two variables $x$ and $y$. We can take coefficients from a field, usually the complex numbers $\mathbb{C}$. We can vizualize then by plotting a real figure of the solutions in $\mathbb{R}^2$. We will learn that it is better to view these in projective plane.

The idea is to discover experimentally "invariants" of such curves, like the genus. This is a natural number that measures the complexity of the curve. We will discover experimentally (by blowing up the plane) how to arrive at a non-singular model where all points are "smooth" and admit a good tangent line.

We intend to discover experimentally formulas for the genus of the curve and to interpret it in terms of differential forms.

Problems will be given week by week and the students should discover patterns by trying experimentally to solve the problems. It can be seen as a gentle experimental introduction to simple algebraic geometry.

supervisor: Gerard van der Geer

• Counting points on curves of finite fields ()
Goal:

Algebraic curves can be given by an algebraic equation $f(x,y) = 0$ with $f(x,y)$ an irreducible polynomial in two variables $x$ and $y$. When we take coefficients in a finite field of cardinality $q = p^m$ with $p$ a prime, we have only finitely many solutions. We can then try to count this number of solutions and try to find patterns when we vary the cardinality (via $m$) or how it depends on the equation. One can experiment a lot by counting by hand (or mind) or by computer. It will lead to the so-called zeta function of curves over finite fields.

A list of problems will be given week by week and the students should discover patterns by trying to solve the problems.

supervisor: Gerard van der Geer

• 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 exam- ple 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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