Projects  and Theses

Contact Mireille for possible projects

Availability 2026-2027: Q3+Q4 should be good for bachelor projects; can also handle a student or two in Q2

List of topics, past projects:

The general topics students choose when they have me as a supervisor tend to involve representation theory, algebraic geometry, and/or tensor products. I have covered a few other topics if a student has something specific in mind that involves algebra in general. Projects with me tend to start with the background theory of their relevant topic and small cases in low dimension to get an intuition for the field. A typical goal is to work out what happens in similar situations if the dimension (or some other parameter) increases.

Overview for bachelor students (non-detailed, students are welcome to come talk to me):

Representation theory: A group representation of a group G on a vector space V is a group action of G on V that respects the structure of V, i.e. G acts on V via linear maps. It is very useful in any setting that involves vector spaces with additional symmetries. For a specific group, you can try to find the 'smallest representations' (irreducible representations) that are the building blocks of all representations of that group. Some possible directions are increasing the size of the group, considering families of representations of a specific group, or identifying when representations are 'essentially the same' (isomorphic) by finding their decomposition in irreducible subrepresentations.

Algebraic geometry: A circle can be described in a picture (geometry) and by means of an equation (algebra). The same holds for many more complicated structures. Algebraic geometry covers the relation between these viewpoints, allowing you to translate between the two. In higher dimension, you should expect to deal with polynomial equations in many variables. Some possible directions are finding relevant polynomial equations for given varieties (essentially geometric structures described by the vanishing of polynomial equations), working out the 'best equations' (Gröbner basis) to use from a given starting point, or considering families of varieties (typically ones with lots of symmetries) and comparing their similarities.

Tensor products: Linear algebra is closely related to linear maps. Tensor products are closely related to multi-linear maps. They are used in various real-world settings; a matrix can only relate two variables while a tensor can relate many more. However, tensors are computationally difficult to work with. In projects with me, tensors are often directly used to support the rest of a project, but I've also covered projects on various aspects of tensors, such as the rank of a tensor (very easy in the case of matrices, extremely hard for tensors unless very strong conditions apply).

Some past BFPs outside the above topics (proposed by/tailored to a specific student):

• On circle pairings and limit sets of variations of classical Schottky groups. This title conceals the fact that this is a thesis about fractals. The work was based on a book the student wished to study, and tried to investigate what would happen if certain conditions posed by the book were violated. This resulted in some very nice fractals, some very chaotic pictures, and plenty of results we still haven't explained.

• Techniques for Solving Polyomino Tiling Problems. The basic question for this project was very easy: Can you cover an n by m grid with 5 by 1 rectangles, 1 by 5 rectangles, and 'plus tiles' (a center 1 by 1 tile with four adjacent 1 by one blocks forming the shape of a +) in such a way that at least one plus tile is used (the question is very easy if you don't use plus tiles)? This question is an open problem by David Klarner that has been open for some time. Partial results were obtained for various different tiling patterns, and the student took a capita selecta to find even more relations. The question remains unanswered, but we now have some extra techniques that help look at this problem and that have solved some related problems.

• Structure and Representations of Rubik’s Cube Groups. The rubik's cube is a well-known mathematical puzzle object, and its group has been studied quite a bit. The aim of this project was to understand some recent results about representations of the 3x3x3 group, and to generalize these to NxNxN groups. A negative result showed that these results would not generalize, leaving some open questions as to what is possible in these larger cases. In addition, we got quite close to a full understanding of the NxNxN group, with some part left open for potential future projects.

Sample proposed projects:

I don't typically have any. What are you interested in? As seen above, I'm happy to try to find something that fits your interest, as long as it's algebra-related in a broad sense. Projects with me can be seen as less structured for this reason; we often get started on the theory of a topic a student is vaguely interested in and then figure out an end goal as we go. Expect running into dead ends, going in unexpected directions, suddenly needing to learn or apply extra bits of theory... You'll have both the dangers and the fun of research. If you do have a more concrete end goal in mind, we can go about things in a more structured way, working towards our goal step by step. I've had projects of both types, and am happy to discuss with you what you want.

One topic of recent interest that I haven't done any projects on (and haven't mentioned yet) is localization. How do you determine your position based on known data (like the relative location of nearby satellites)? I'm expecting that there are plenty of opportunities for fun projects here that are far more applied than the topics mentioned above. Feel free to talk to me about this.

Availability: 1-2 students Q2- Q4 2026-2027

Link to file of possible projects

Contact Vahid for possible projects