On the Inverse Problem Between Discrete Morse Functions and Merge Trees

Explore the inverse problem connecting discrete Morse functions on graphs to merge trees in this 46-minute conference talk from the Applied Algebraic Topology Network. Delve into discrete Morse theory as a combinatorial algebraic topology tool that employs well-behaved functions from face posets of regular CW complexes to real numbers, providing sublevel-filtrations and theoretical guarantees for various algorithms. Learn about merge trees as combinatorial descriptors that track path component development across filtered space levels, originally introduced for visualization and contour tree computation. Examine the central inverse problem: given a specific merge tree, discover all discrete Morse functions on graphs that produce that merge tree and develop structured methods to find them systematically. The presentation synthesizes research findings on merge trees and discrete Morse functions on paths and trees, as well as investigations into cycles and merge trees, offering insights into the mathematical relationships between these fundamental topological data analysis concepts.