Visualizations of Schottky GroupsOften 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.

Join the Hawai‘i Data Science Institute for another Data Science Friday seminar titled “Bayesian Topological Learning for Complex Data Analysis” presented by Assistant Professor of Mathematics Dr. Farzana Nasir on October 16, 2020 at 2 pm on Zoom.

Please find more information below and on the attached flyer.

**Zoom registration:** http://go.hawaii.edu/39f

**Abstract:** Persistent homology is a tool in topological data analysis for learning about the geometrical/topological structures in data by detecting different dimensional holes and summarizing their appearance disappearance scales in persistence diagrams. However, quantifying the uncertainty present in these summaries is challenging. In this talk, I will present a Bayesian framework for persistent homology by relying on the theory of point

processes. This Bayesian model provides an effective, flexible, and noise-resilient scheme to analyze and classify complex datasets. A closed form of the posterior distribution of persistence diagrams based on a family of conjugate priors will be provided. The goal is to introduce a

supervised machine learning algorithm using Bayes factors on the space of persistence diagrams. This framework is applicable to a wide variety of datasets. I will present an application to filament networks data classification of plant cells.**Bio:** Farzana Nasrin graduated from Texas Tech University with a Ph.D. in Applied Mathematics in August 2018. Her research interests span algebraic topology, differential geometry, statistics, and machine learning. Currently, she is holding an assistant professor position at UH Manoa in the Department of Mathematics. Before coming to UHM, she was working as a postdoctoral research associate funded by the ARO in mathematical data science at UTK. She has been working on building novel learning tools that rely on the shape peculiarities of data with application to biology, materials science, neuroscience, and ophthalmology. Her dissertation involves the development of analytical tools for smooth shape reconstruction from noisy data and visualization tools for utilizing information from advanced imaging devices.