Projects

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

Participating in a project is an opportunity to learn fun mathematics, practice one's programming skills, and use cool technology!

The output of a project can be one or more of the following:

- A talk at the HEGL Community Seminar.
- A blog entry on this website.
- A computer application, 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 ideas for new projects, 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:

- Visualizing subriemanniann billiards
- Planimeters and bicycle tires
- Crooked planes and Margulis spacetimes
- Tilings of the hyperbolic plane
- Crocheting adventures with hyperbolic planes
- Cayley graphs of right-angled Artin groups
- Visualizing Seifert surfaces
- Benoist cone and joint spectrum of Schottky groups
- Graph embeddings in the hyperbolic plane
- Limit sets in spheres
- Julia sets and Kleinian groups
- Can you hear the shape of a drum?
- Laminations on hyperbolic surfaces in the universal cover
- Geometric constructions with the MIRA

Visualizing subriemanniann billiards

**Mentor:** Lucas Dahinden and
Maksim Schreck.

**Student(s) participating:** Jakob Niessner

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

Planimeters and bicycle tires

**Mentor(s):** Lucas Dahinden.

**Student(s) participating:** Aaron Osburg

**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.

Crooked planes and Margulis spacetimes

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

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

**Details:** The goal of this project is to draw or print crooked planes: fundamental
domains in three-dimensional space for discrete groups of affine transformations.
This will allow to visualize their quotient space called Margulis
spacetimes.
This project involves learning 3D printing. Theoretical side: hyperbolic
geometry, affine and projective geometry.

Tilings of the hyperbolic plane

**Mentor(s):** Anna Schilling.

**Student(s) participating:** Yes

Crocheting adventures with hyperbolic planes

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

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

**Details: ***Coming soon*.

Cayley graphs of right-angled Artin groups

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

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

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

Visualizing Seifert surfaces

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

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

**Details: ***Coming soon*.

Benoist cone and joint spectrum of Schottky groups

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

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

**Details:** The goal of this project is to draw numerically the joint spectrum of a
finite set of square matrices of determinant one.
The joint spectrum spans the Benoist cone, which means that the project
enables to visualize both objects, especially their boundaries.
This project involves random walks and numerical libraries on
eigenvalues and singular values of matrices. Theoretical side: linear
algebra.

Graph embeddings in the hyperbolic plane

**Mentor(s):** Maria Beatrice Pozzetti, Dr. Brice Loustau, and M.Sc. Marta Magnani.

**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.

Limit sets in spheres

**Mentor(s):** 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.

Julia sets and Kleinian groups

**Mentor(s):** 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.

Can you hear the shape of a drum?

**Mentor(s):** 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.

Laminations on hyperbolic surfaces in the universal cover

**Mentor(s):** Mareike Pfeil and 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.]

Geometric constructions with the MIRA

**Mentor(s):** Denis Vogel

**Student(s) participating:** Silas Gramlich

**Details: **
The MIRA is a semi-transparent reflective device used for constructing geometric objects. The goal of this project is to build such a device and understand the mathematics behind it.