Projects

HEGL offers research projects to students mentored by one or more faculty members.

Participating in a project is an opportunity for students to learn fun mathematics, practice one or several programming languages, and use cool technology!

The output of a project consists in a combination of one or more of the following items:

- A talk at the HEGL seminar.
- A blog entry on this website.
- A computer program, web-based and/or hosted on GitHub.
- A piece of mathematical artwork, such as a 3D printed object.
- A written report, especially if the project is part of a Bachelor or Master thesis.

The list of all projects is listed below (click on each project for more details). If you have some ideas for a new project, let us know: we will be happy to discuss them with you!

**To sign up for a project, send us an email**.

List of projects:

- 1. Visualizing subriemanniann billiards
- 2. Planimeters and bicycle tires
- 3. Computing harmonic maps
- 4. Crooked planes and Margulis spacetimes
- 5. Aperiodic tilings of the hyperbolic plane
- 6. Crocheting adventures with hyperbolic planes
- 7. Cayley graphs of right-angled Artin groups
- 8. Visualizing Seifert surfaces
- 9. Benoist cone and joint spectrum of Schottky groups
- 10. Graph embeddings in the hyperbolic plane
- 11. Limit sets in spheres
- 12. Julia sets and Kleinian groups
- 13. Can you hear the shape of a drum?
- 14. Laminations on hyperbolic surfaces in the universal cover
- 15. Numerical nonabelian Hodge correspondence

1. Visualizing subriemanniann billiards

**Mentor:** Dr. Lucas Dahinden and
M.Sc. Lutz Hofmann.

**Student(s) participating:** Open!

**Details:** All the details are contained in this PDF.

2. Planimeters and bicycle tires

**Mentor(s):** Juniorprof. Dr. Gabriele Benedetti.

**Student(s) participating:** Open!

**Details:** Planimeters are devices used to measure the area of a region in the plane by tracing around its boundary. As such, they are a mechanical manifestation of Green's Theorem. Goal of this project is to build models of the polar and of the hatchet planimeter and understand the mathematics behind them. The hatchet planimeter is connected to bicycle tire tracks and the project could further explore a conjecture by Menzin about their geometry using this planimeter.

3. Computing harmonic maps

**Mentor(s):** Dr. Brice Loustau and
Prof. Dr. Jonah Gaster.

**Student(s) participating:** Open!

**Details:** The goal of this project is to compute and visualize harmonic maps between Riemannian manifolds.
The project will involve learning C++, Qt, OpenGL, 3D graphics, and possibly 3D printing. On the theoretical side: Riemannian geometry, hyperbolic geometry,
discrete differential geometry.

4. Crooked planes and Margulis spacetimes

**Mentor(s):** Dr. Nguyen-Thi Dang and *To be announced*.

**Student(s) participating:** Open!

**Details: ***Coming soon*.

5. Aperiodic tilings of the hyperbolic plane

**Mentor(s):** To be announced.

**Student(s) participating:** Open!

**Details: ***Coming soon*.

6. Crocheting adventures with hyperbolic planes

**Mentor(s):** To be announced.

**Student(s) participating:** Open!

**Details: ***Coming soon*.

7. Cayley graphs of right-angled Artin groups

**Mentor(s):** Juniorprof. Dr. Maria Beatrice Pozzetti.

**Student(s) participating:** Jannis Heising

**Details: **The goal of this project is to investigate properties of right-angled Artin groups by visualizing their Cayley graphs.

8. Visualizing Seifert surfaces

**Mentor(s):** Dr. Valentina Disarlo and *To be announced*.

**Student(s) participating:** Open!

**Details: ***Coming soon*.

9. Benoist cone and joint spectrum of Schottky groups

**Mentor(s):** Dr. Nguyen-Thi Dang and *To be announced*.

**Student(s) participating:** Open!

**Details: ***Coming soon*.

10. Graph embeddings in the hyperbolic plane

**Mentor(s):** Juniorprof. Dr. Maria Beatrice Pozzetti and Dr. Brice Loustau.

**Student(s) participating:** Open!

**Details: **
[Prerequisites: A first course in differential geometry, basic algebra, basic hyperbolic
geometry will be needed, but can be learned during the project.]
Recently a lot of interest has been put in visualizing graphs and data
by embedding it in the hyperbolic space. A drawback is that often most
points get pushed to the boundary of the space, making it difficult to
see what is going on. The goal of the project is to use the isometries
of the hyperbolic plane to produce a visualization tool that allows for
a change of perspective, re-centering the model at different points.

11. Limit sets in spheres

**Mentor(s):** Juniorprof. Dr. Maria Beatrice Pozzetti and Dr. Brice Loustau.

**Student(s) participating:** Open!

**Details: **
[Prerequisites: A first course in differential
geometry, basic algebra. Knowledge on hyperbolic manifolds or geometric
group theory could help, but is not necessary.]
The goal of the project is to understand finitely generated groups
acting on the real hyperbolic space of dimension 3 and 4 by visualizing
the minimal invariant subset for the associated action on the boundary,
which is, respectively a 2 and 3 dimensional sphere. This should produce
nice fractals. Time permitting we will also look at the (boundary of
the) 2 dimensional complex hyperbolic space.

12. Julia sets and Kleinian groups

**Mentor(s):** Juniorprof. Dr. Maria Beatrice Pozzetti and Dr. Brice Loustau.

**Student(s) participating:** Open!

**Details: **
[Prerequisites: basic complex analysis,
basic algebra. A first course in differential geometry and basic
hyperbolic geometry could help, but are not necessary.]
The Julia set J(f) is a fractal in the complex plane C associated to the
ratio f(z)=p(z)/q(z) of two polynomials in one complex variable. The
goal of the project is to visualize some of these fractals, and use the
visualizations to guide discovering some of their features. Possibly we
will discuss similarities with limit sets of discrete groups acting on
the three dimensional space, through Sullivan's dictionary.

13. Can you hear the shape of a drum?

**Mentor(s):** Dr. Brice Loustau and *To be announced*.

**Student(s) participating:** Open!

**Details: **
This projects proposes to address the celebrated question "Can you hear the shape of a drum?" using mathematics, computing, and sound equipment.
The idea is to relate, both experimentally and theoretically, the shape of a Riemannian domain and the spectrum of the Laplacian.

14. Laminations on hyperbolic surfaces in the universal cover

**Mentor(s):** Mareike Pfeil and Prof. Dr. Andrea Seppi.

**Student(s) participating:** Farid Diaf

**Details: **
A lamination on a hyperbolic surface is a collection of simple disjoint geodesics whose union is closed. Laminations are very helpful when studying surfaces. For instance, they can be used to deform hyperbolic surfaces and to define coordinates on Teichmüller space. The goal of this project is to visualize lifts of laminations on hyperbolic surfaces in the Poincaré disk model for the hyperbolic plane.

[Prerequisites: A first course in differential geometry, basic algebra, basic hyperbolic geometry will be needed, but can be learned during the project.]

15. Numerical nonabelian Hodge correspondence

**Mentor(s):** PD Dr. Jan Swoboda and
Menelaos Zikidis.

**Student(s) participating:** Open!

**Details: ** All the details are contained in this PDF.