Analyzing and classifying large and complex datasets is generally challenging. Topology is a field of mathematics that mainly studies “shapes” and allows geometric objects to be deformed, stretched, and bent. Topological data analysis (TDA) uses machinery from algebraic topology and summarizes the shape patterns of the data. However, these topological summaries are not compatible with statistical methods. My recent research interest is in coupling TDA with statistics and machine learning, focusing on complex datasets. In this talk, I will provide a brief introduction to TDA, focusing primarily on persistence diagrams (PDs). The goal is to show how TDA and statistical learning complement each other to provide new insights into complex datasets. A computational Bayesian learning method in the space of PDs will be discussed as a paradigm. This method is underpinned by a point processes model of PDs and a reversible jump Markov chain Monte Carlo (RJ-MCMC) algorithm. This method is applicable to a wide variety of datasets. I will present an application in materials science.

Farzana Nasrin is an assistant professor in the Department of Mathematics at UH Manoa. She received her PhD in applied mathematics from Texas Tech University in 2018. Her research interests span algebraic topology, statistics, and machine learning. 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, ocean science, 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.

Dr. Nasrin will be teaching Math 649K Probability & Statistics in Fall 2024. This course is designed to build theoretical statistics from the principles of probability theory at a graduate level. We will start by covering the basics of probability theory that are necessary for logical developments and proofs of statistical principles.