Crooked Planes and Margulis Spacetimes
Mentor(s): Nguyen-Thi Dang and TBA
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 Schlling
Student(s) participating: Yes
Crocheting Adventures with Hyperbolic Planes
Mentor(s): TBA
Student(s) participating: Open!
Details: Coming soon.
Visualizing Seifert Surfaces
Mentor(s): Valentina Disarlo and TBA
Student(s) participating: Open!
Details: Coming soon.
Benoist Cone and Joint Spectrum of Schottky Groups
Mentor(s): Nguyen-Thi Dang and TBA
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 the visualization of 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, Brice Loustau, and Marta Magnani
Student(s) participating: Open!
Details: 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.
Prerequisites: A first course in differential geometry, basic algebra, basic hyperbolic geometry will be needed, but can be learned during the project.
Limit Sets in Spheres
Mentor(s): Maria Beatrice Pozzetti and Brice Loustau
Student(s) participating: Open
Details: The goal of the project is to understand finitely generated groups acting on the real hyperbolic space of dimensions 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.
Prerequisites: A first course in differential geometry, basic algebra. Knowledge of hyperbolic manifolds or geometric group theory could help but is not necessary.
Julia Sets and Kleinian Groups
Mentor(s): Maria Beatrice Pozzetti and Brice Loustau
Student(s) participating: Open!
Details: 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.
Prerequisites: basic complex analysis, basic algebra. A first course in differential geometry and basic hyperbolic geometry could help but are not necessary.
Can You Hear the Shape of a Drum?
Mentor(s): Brice Loustau and TBA
Student(s) participating: Open!
Details: This project 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.