Sana Habib’s Masters Presentation

June 29, 2020 @ 2:00 pm – 3:00 pm
Visualizations of Schottky Groups
Often times, to truly understand a mathematical object it must be viewed from several different perspectives, involving several different foundations. In this thesis, I will present eye-catching visualization techniques for something called Schottky groups, which are similar to general linear groups of degree n over the complex numbers. This is built on an understanding of complex numbers to explore the structure of specific kinds of linear groups. My discussions will include the mathematics needed to imagine such an object, as well as the numerics required to compute such an object. Parts of this will include discussion of programming these objects.
This thesis is an exploration of a particular kind of projective linear group,  Schottky groups, and their variations by understanding them geometrically. The general idea here is how we can study fractal-like sets by looking at images of circles representing group elements. This exploration will lead to the discovery and understanding of Schottky groups.
Furthermore, I do this exploration using first and foremost, a background in complex analysis, and in particular, a deep understanding of Mobius maps. This leads to discovery and understanding of anti-Mobius maps, which will be our main tool in understanding reflections. Additionally, I will use Python3 to program examples and experiments of these ideas. The Python code will provide specific examples. as well as mathematical challenges of its own. These challenges will include creating Mobius and anti-Mobius classes, and using these classes to perform all of these operations.