Course Description:

Topological Data Analysis (TDA) is a field in algebraic topology concerned with analyzing data by extracting information about its shape. While the field of algebraic topology enjoys a long history within mathematics, the recent development of software to compute topological invariants in data sets has drastically increased the ability of researchers to apply these sophisticated tools in a variety of ways. Topological measurements such as Betti numbers, which count holes of different dimensions, and persistent homology that records and tracks topological features in an appropriate filtration of the data, have shown promise in extracting structure from data. Researchers have applied persistent homology to application areas such as Biology, Material Science, Climatology, Neuroscience, and Medical Imaging. In this short course, we introduce mathematical foundations of TDA, provide hands-on tutorial on TDA software, and present some recent work on applications to various datasets from different scientific disciplines. Background knowledge on algebraic topology is not required, but programming experiences with any languages are necessary. By the end of this short course, students and researchers will be able to apply tools in TDA to their own datasets.

Tentative Schedule:

5/28 Lecture 1: Introduction to Computational Topology

5/30 Lecture 2: Persistent Homology: a Multi-scale Analysis

6/4 Lecture 3: Hands-on Tutorial on Persistent Homology

6/6 Lecture 4: Applications to Natural Sciences