Combinatorial Exploration of Morse–Smale Functions on the Sphere via Interactive Visualization

Youjia Zhou, Janis Lazovskis, Michael J. Catanzaro, Matthew Zabka, Bei Wang

Room: 106

2023-10-22T03:00:00ZGMT-0600Change your timezone on the schedule page
Exemplar figure, described by caption below
We present MSF Designer, a visualization tool that supports the combinatorial exploration of Morse-Smale functions on the sphere. The interface consists of (A) Function and flow visualization panel supports modifying the topology and geometry of the Morse-Smale graph of the function and visualizes the dynamics of its underlying gradient vector field; (B) Elementary moves panel provides a set of elementary moves as fundamental building blocks of a Morse-Smale function; (C) Function adjustment panel allows modifying the function values at singularities; (D) History panel provides undo and redo features; and (E) Barcode panel computes and displays barcodes to guide persistence simplification.
Fast forward

In this paper, we are interested in the characterization and classification of Morse–Smale functions. To that end, we present MSF Designer, an interactive visualization tool that supports the combinatorial exploration of Morse–Smale functions on the sphere. Our tool supports the design and visualization of a Morse–Smale function in a simple way using fundamental moves, which are combinatorial operations introduced by Catanzaro et al. that modify the Morse–Smale graph of the function. It also provides fine-grained control over the geometry and topology of its gradient vector field. The tool is designed to help mathematicians explore the complex configuration spaces of Morse–Smale functions, as well as their associated gradient vector fields and Morse–Smale complexes. Understanding these spaces will help mathematicians expand their applicability in topological data analysis and visualization. In particular, our tool helps topologists, geometers, and combinatorialists explore invariants in the classification of vector fields and characterize Morse functions in the persistent homology setting.