2: Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers

Abstract: Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form $x+\tau y$ for $x,y \in \mathbb{R}$, and $\tau^2=1$ but $\tau \not= \pm 1$, create the natural number system in which to encode geometric properties of the Minkowski space $\mathbb{R}^{1,1}$. We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

1: Spherical Geometry and the Least Symmetric Triangle

Abstract: We study the problem of determining the least symmetric triangle, which arises both from pure geometry and from the study of molecular chirality in chemistry. Using the correspondence between planar n-gons and points in the Grassmannian of 2-planes in real n-space introduced by Hausmann and Knutson, this corresponds to finding the point in the fundamental domain of the hyperoctahedral group action on the Grassmannian which is furthest from the boundary, which we compute exactly. We also determine the least symmetric obtuse and acute triangles. These calculations provide prototypes for computations on polygon and shape spaces.