Topological flow data analysis and its applications to Reeb graphs of Morse functions

Abstract: In this talk, we introduce topological methods to analyze flow data. These methods are based on dynamical systems and Morse theory. So, first, we review the results of generic embeddings of closed surfaces in the three-dimensional Euclidean space and explain the relation between Morse functions and Hamiltonian vector fields on surfaces. In particular, such embeddings are classified by a finite complement invariant, call a molecular. Second, we review topological results in flows on surfaces. Third, we review our complete invariant, called a COT representation, of 2D Hamiltonian flows, its implementation, and the list of all generic transitions of 2D Hamiltonian flow. Moreover, we introduce a complete invariant of 2D flows of finite type and their applications to industrial machines. In addition, as an application of COT representations, we list all generic transitions of Reeb graphs of Morse functions on a sphere. If time allows, we show higher-dimensional results of flows and describe a topological characterization of Morse-Smale flows and a generic transitions between them.

computational geometryalgebraic topologycombinatoricsgeometric topologyprobability

Audience: advanced learners

Asia Pacific Seminar on Applied Topology and Geometry