EVOLUTIONARY DE RHAM-HODGE METHOD.

Area of Science:

• Topology and Differential Geometry
• Computational Mathematics
• Data Science and Machine Learning

Background:

• De Rham-Hodge theory is a foundational 20th-century mathematical framework with broad applications.
• Analyzing evolving geometric structures, especially those with boundaries, presents significant challenges.
• Existing methods may struggle with complex topological properties of dynamic manifolds.

Purpose of the Study:

• To develop a unified evolutionary de Rham-Hodge method for multiscale geometric and topological analysis.
• To address the complexities of analyzing compact manifolds with 2-manifold boundaries.
• To generate topology-preserving singular spectra for robust manifold evolution analysis.

Main Methods:

• Construction of evolutionary de Rham complexes from manifold filtrations.
• Development of three unique evolutionary Hodge Laplacians for spectral analysis.
• Application of discrete exterior calculus for numerical experiments and data analysis.

Main Results:

• Proposed method generates topology-preserving singular spectra, accurately identifying persistent Betti numbers (dimensions 0, 1, and 2).
• Non-zero eigenvalues reveal topological persistence and geometric progression during manifold evolution.
• Demonstrated effectiveness in data representation and shape analysis for point clouds and density maps.

Conclusions:

• The evolutionary de Rham-Hodge method provides a powerful paradigm for analyzing evolving manifolds, particularly those with boundaries.
• The method shows promise for applications in diverse fields, including biophysics, exemplified by protein B-factor prediction.
• This approach offers a robust framework for understanding complex geometric and topological changes in dynamic data.