Course description: Topological data analysis (TDA) is a new and fast evolving area of research in pure and applied mathematics. The main idea is that by borrowing tools from algebraic topology, geometry and statistics, one can solve problems related to data analysis. The main goals of this topics course are:
To develop the theory of persistent (co)homology and showcase some of its applications in data science.
To provide an introduction to the theory of fiber/vector bundles, and describe how these ideas have found their way into modern applications.
Some of the topics we will cover are:
Persistence modules and their representation theory
Presheaves and their Cech Cohomology
Principal bundles and their classification (i.e. as Cech cohomology classes and as maps to classifying spaces)
Applications: Data coordinatization with persistent cohomology and classifying spaces; synchronization problems and learning group actions
Oudot, Steve Y. Persistence theory: from quiver representations to data analysis. Vol. 209. Providence, RI: American Mathematical Society, 2015.
Husemoller, Dale. Fibre bundles. Vol. 5. New York: McGraw-Hill, 1966.
Steenrod, Norman. The Topology of Fiber Bundles, volume 14, Princeton Mathematical Series (1951): 16.
Miranda, Rick. Algebraic curves and Riemann surfaces. Vol. 5. American Mathematical Soc., 1995.
Cohen, Ralph. The Topology of Fiber Bundles Lecture Notes, Stanford University, 1998.
Here is a recent online research seminar leveraging the topics from this class.