• Vor- und Aufbereitung (unter Anleitung) von Forschungsfragen und -ergebnissen im Bereich der Geometrie und Dynamik
• Bearbeiten (unter Anleitung) von Forschungsfragen, z.T. mit Programmier- und Visualisierungsaspekten

• Sie haben Interesse an mathematischen Fragestellungen in Geometrie und Dynamik.
• Sie haben eine hohe Motivation und Bereitschaft, neue mathematische Konzepte zu erlernen.
• Sie sind bereit, kleinere Forschungsprojekte unter Anleitung zu bearbeiten, weiter zu entwickeln und später zu präsentieren.
• Sie kennen Programmier- / Visualisierungstechniken oder sind bereit, diese zu erlernen und einzusetzen (je nach Projekt).
• Sie habe die Vorlesungen Analysis 1 und Lineare Algebra 1 erfolgreich abgeschlossen.

• Eine offene und freundliche Arbeitsatmosphäre und einen regelmäßigen Austausch mit erfahrenen Wissenschaftler*innen
• Einblicke in den Forschungsalltag im Bereich Geometrie und Dynamik und in aktuelle Forschungsthemen
• Teilnahme an den Aktivitäten der Forschungsstelle Geometrie & Dynamik und dem Heidelberg Experimental Geometry Lab (HEGL)
• Die Möglichkeit aus dieser Tätigkeit heraus im Anschluss eine Bachelor- oder Masterarbeit anzufertigen
• Flexible Arbeitszeiten in Absprache mit den Gruppenleiter*innen

Supervisor:

Peter Albers

Prerequisites: 

Grundlagen der Geometrie und Topologie

Abstract: 

A simple model of bicycle kinematics is the following: A bicycle is represented by an oriented line segment of constant length \(\ell\), which can move in such a way that the velocity of its rear wheel always points in the direction of the segment (the rear wheel is rigidly attached to the bicycle frame). The motion takes place in the Euclidean plane, but it can equally well - although with less physical motivation - be considered in \(\mathbb{R}^n\).

The goal is to understand this (discrete and/or continuous) transformation and to visualize and simulate it through examples.

Literature and further reading:

• S. Tabachnikov: On the bicycle transformation and the filament equation: results and conjectures, arXiv:1602.06455
• S. Tabachnikov, E. Tsukerman: On the discrete bicycle transformation, arXiv:1211.2345

Supervisor:

Raphael Appenzeller

Prerequisites:

At least one of the following:

• Attended the lecture "Coxeter Groups and buildings".
• Familiar with (or highly motivated to learn about) 3D visualization tools such as the Unity engine and others. 

Abstract:

Spherical and affine buildings come up in various fields of mathematics. For a long time, the only concrete models/visualizations of buildings were certain graphs. Recently, Bram Bekker [1,2] has created a stunning visualization of some 2-dimensional buildings, namely some of type $A_3$ and $\widetilde{A}_2$, which are accessible on the webpage https://buildings.gallery. With Prof. Stephan Witzel we have created further visualizations of buildings of type $\widetilde{A}_2$ which could be added to the website (as part of this project). We would also like to create models for other types of buildings, especially $C_3$ and $\widetilde{C}_2$. Some 3D-printable files are available [2].

In this project, your tasks could include:

• the generation of more new buildings (of different type or for different fields) 
• the visualization of these using Unity, or some other suitable program
• making the data available for other people to use
• creating 3D-prints of more models.

Literature and further reading:

[1] Bram Bekker, The buldings gallery: https://buildings.gallery.
    
[2] Bram Bekker, Maarten Solleveld, The buildings gallery: visualizing buildings. Journal of Mathematics and the Arts, vol  16, 2022.
    
[3] Raphael Appenzeler, Affine building of type $\widetilde{A}_2$, https://www.thingiverse.com/thing:6301564, 2023. 
    
[4] Kenneth S. Brown, Buildings, Springer Monographs in Mathematics, 1989.

Supervisor:

Florent Schaffhauser

Prerequisites:

• Adequate for 3rd semester Bachelor students and beyond.
• An interest in learning to use 3D printing software.

Abstract:

The Heidelberg Experimental Geomeytry Lab has an extensive collection of 3D printed models, whose many interesting properties are nonetheless not immediately visible! In this survey project, we “reverse engineer” the construction of the model and establish a protocol for the creation of future 3D models. Questions to be answered are:

• What is this object?
• How was it discovered in the first place?
• What are its mathematical properties?
• How is the model built in practice?

The student(s) working on this project will explore and enrich the HEGL gallery of 3D printed objects while documenting their process on the HEGL website. The expected output is a new 3D model, which will be accompanied by a blog post and added to the Lab’s collection!

Literature and further reading:

Check out our cool gallery of 3D printed models on the HEGL website!

Supervisor:

Christoph Schnörr

Prerequisites:

• LA/Ana 
• programming skills 
• complex analysis would be beneficial

Abstract:

Conformal maps including the Riemann mapping theorem have beautiful general mathematical properties. Yet naive discretizations of closed-form relations a plagued by difficulties, like the crowding phenomenon. This project examines recent progress towards reliable numerical realizations and approximation. The long-term motivation for this project concerns mathematical shape representations which support machine-learning based shape recognition.

Literature and further reading:

• S. R. Bell. The Cauchy Transform, Potential Theory and Conformal Mapping. CRC Press, 2nd edition, 2016.
• N. Dym, R. Slutsky, and Y. Lipman. Linear Variational Principle for Riemann Mappings and Discrete Conformality. PNAS, 116(3):732--737, 2019.
• A. Gopal and L. N. Trefethen. Representation of Conformal Maps by Rational Functions. Numerische Mathematik, 142:359--382, 2019.
• Q. Chang, C. Gotsman, and K. Hormann. Real-Time Conformal Maps and Parametrizations. Computer Graphics Forum, (e70310), 2026.

Supervisor:

Petra Schwer

Prerequisites: 

Students interested in this project should ideally bring at least one of the following: 

• Solid knowledge on Coxeter groups e.g. as covered in the course "Coxeter groups and buildings" taught in the summer term 2026, or
• Solid knowledge about and experience in programming in java-script and HTML.

Abstract:

 We host a web-app showing Bruhat intervals and various kinds of other shadows in Coxeter groups. In small dimensions these objects can be visualized as combinatorially defined sub-patterns of the tilings of the hyperbolic plane or euclidean 2-space.  

You can find and play with the applet under the link listed below in the literature section. The main goal of the project is to improve and enrich the functionality of the applet which is currently in use for ongoing research. Depending on time and students interest and prior knowledge we may also think about research questions related to these objects.

Literature and further reading:

• https://www.mathi.uni-heidelberg.de/~reflectiongroupsnetwork/shadows/shadows.html
• Björner, Brenti. Combinatorics of Coxeter groups, Springer 2005. 
• Brown. Buildings, Springer, 1989. (See also here: https://pi.math.cornell.edu/~kbrown/buildings/)