Summer 2022 Student Projects

Independent Projects

Rolling Knots

Mentors: Anja Randecker, and Maurice Reichert
Team members: Open for 3 members!
Description: In the year 1980 M. H. Freedman conjectured that every generic smooth knot in 3-dimensional euclidean space has a tritangent plane, i.e. a plane tangent to 3 distinct points. Ten years later A. M. Amilibia, J. J. N. Ballesteros and H. R. Morton used trefoil knots to construct a family of counterexamples. Such a tritangentless knot has no stable position, when sitting on a plane or table, and is therefore able to roll in at least 2 directions like a wobbly chair. A. Eget, S. K. Lucas and L. Taalman further optimized the ability to roll on Mortons family of trefoil knots by reducing the transversal deviation of the center of mass.
In this project our goal is to understand the proof by Morton, find new families of tritangentless knots and optimize them with respect to smooth roll-ability. Finally, we also want to 3D print these objects.
Prerequisites: Some knowledge of projective geometry, basic topology, algebra and rudimentary complex analysis skill are needed, but can be learned during the project. Basic programming skills in Python or Matlab are useful for 3D printing and Visualization.

Tilings of the Hyperbolic Plane

Mentor: Anna Schlling
Team members: Yes

Ray Marching in Translation Surfaces

Mentor: Dia Taha
Team members: Fabian Lander, Mara-Eliana Popescu
Description: A translation surface is a surface obtained by gluing together a finite collection of polygons in the Euclidean plane along parallel sides of the same length. In this project, we develop an immersive visualization of the geometry of translation surfaces with raymarching.

SFB Funded Projects

Symplectic Billiards

MentorPeter Albers
Team members: Fabian Lander, Jannik Westermann
Description: The goal of this project is to simulate symplectic billiards on non-convex polygons, and to search for periodic orbits in particular.

Hyperbolic Billiards

Mentor: Dia Taha
Team members: Jannis Heising, Mara-Eliana Popescu
Details: The project aims to study closed billiard trajectories in hyperbolic polygons.