Currently offered projects

Projects for Winter 2026 (Updated on 15 Sep 2026) - More be added

  • Experimenting with AI to study open questions in geometry (show description)

    supervisor: Jean-Marc Schlenker

  • Word Complexity of Representations of Integer Sequences ()

    Consider the sequence $3^n$ in binary representation:

    1
11
1001
11011
1010001
11110011
1011011001
100010001011
1100110100001
100110011100011
1110011010101001
101011001111111011
10000001101111110001
110000101001111010011
10010001111101101111001
110110101111001001101011
10100100001101011101000001
111101100101000010111000011
10111000101111001000101001001
1000101010001101011001111011011
11001111110101000001101110010001

    How “complex” is this representation asymptotically? Will there be a pattern? How are the digits or blocks of digits distributed? What about other sequences? What about non-binary representations, e.g. decimal representations?

    This project is triggered by a very recent article on the arXiv (Yann Bugeaud: On the binary representation of powers of 3, https://arxiv.org/abs/2608.23017), which proves that in a precise way the binary representations of $3^n$ are “not too simple”. It also contains a very nice theorem that views this kind of question from a different perspective, namely starting from an infinite sequence $a_0,a_1,a_2,\dots$ of 0's and 1's, one can consider the sequence of natural numbers with binary representations $a_0, a_0a_1, a_0a_1a_2,\dots$
    The theorem says that if the sequence of 0's and 1's is “relatively simple” (expressed in a precise way), then the sequence of natural numbers has infinitely many prime divisors (in particular, it does not only consist of powers of 3).

    Goal:
    The aim of the project is to study these questions, variations and similar ones experimentally and to make observations, to form hypotheses, to test them, and to report on and illustrate the results.

    Methods:
    The main tool will be computer experimentation, e.g. in Python or SageMath. The use of AI tools is allowed. However, the students must be able to justify each line they write. Their own thoughts are highly appreciated a lot, whilst copy&pasted AI output is forbidden.

    Prerequisites:
    The project is suitable for all students.

    supervisor: Gabor Wiese

  • Simulating random functions (show description)

    supervisor: Felix Benning

  • Cyclotomic polynomials (show description)

    supervisors: Alexandre Benoist, Antonella Perucca

  • No three in a line: the geometry of SET (show description, Video on moodle)

    supervisor: Javier Fernández Píriz

  • Combinatorics & LLM (show description)

    supervisors: Szabi Buzogány, Antonella Perucca

  • Union of Subgroups (show description)

    supervisors: Mike Daas, Antonella Perucca

  • Free bands (show description, Video on moodle)

    supervisor: Michael A. Daas

  • Bold or cautious (show description)

    supervisors: Luís Maia, Théo Niemann

  • 3D-Printing Hyperbolic Polyhedra (show description)

    supervisors: Quirijn Boeren, Carl Lutz

  • Counting spanning trees via the graph Laplacian ()

    Description:

    A graph is a set of nodes joined by edges. Two matrices describe it: the adjacency matrix \(A\), which records which nodes are joined, and the diagonal degree matrix \(D\), which records how many edges meet at each node. Their difference

    \[ L = D - A \]

    is the graph Laplacian.

    A spanning tree is a choice of edges that connects every node without forming a loop, the cheapest way to keep a network in one piece. A graph usually has many, and their number \(\tau(G)\) says how robustly connected it is. Counting them by hand is hopeless beyond a few nodes.

    Kirchhoff's matrix-tree theorem does it with linear algebra. If \(\lambda_1 = 0 \le \lambda_2 \le \cdots \le \lambda_n\) are the eigenvalues of \(L\), then

    \[ \tau(G) = \frac{1}{n}\,\lambda_2 \lambda_3 \cdots \lambda_n. \]

    An eigenvalue computation replaces an enormous combinatorial search.

    Students will:

    1. verify the theorem on small graphs by listing all spanning trees by hand and comparing with the eigenvalues;
    2. compute \(\tau\) for the \(m \times n\) grid;
    3. study how \(\tau\) grows as the grid gets large, and find the constant governing that growth;
    4. repeat the computation for other lattices such as triangular, honeycomb, cylinders and tori, and compare.

    Prerequisites: Linear algebra.

    supervisor: Wai Yeung Lam