The Reeb Graph of a Smooth Function Encodes the Function Class and Manifold Type

Explore the intricate world of Reeb graphs in this 46-minute lecture presented by Irina Gelbukh for the Applied Algebraic Topology Network. Delve into how the Reeb graph of a smooth function encodes both the class of the function and the type of the manifold. Begin by understanding the concept of Reeb graphs, which are spaces obtained by contracting connected components of a function's level sets to points. Examine the structure of Reeb graphs for Morse functions, which are finite graphs with specific properties, and learn why not every graph can be a Reeb graph of a Morse function. Investigate the structure of Reeb graphs defined by Morse-Bott functions and explore examples of smooth functions whose Reeb graphs are not finite. Discover the relationships between the structure of a Reeb graph, the class of its corresponding smooth function, and the type of manifold it represents. Gain insights into the properties of Reeb graphs that are similar to those of finite graphs, even when they are not finite themselves.