  The de Rham-Hodge theory is a landmark of the 20$^\text{th}$ Century's
mathematics and has had a great impact on mathematics, physics, computer
science, and engineering. This work introduces an evolutionary de Rham-Hodge
method to provide a unified paradigm for the multiscale geometric and
topological analysis of evolving manifolds constructed from a filtration, which
induces a family of evolutionary de Rham complexes. While the present method
can be easily applied to close manifolds, the emphasis is given to more
challenging compact manifolds with 2-manifold boundaries, which require
appropriate analysis and treatment of boundary conditions on differential forms
to maintain proper topological properties. Three sets of unique evolutionary
Hodge Laplacian operators are proposed to generate three sets of
topology-preserving singular spectra, for which the multiplicities of zero
eigenvalues correspond to exactly the persistent Betti numbers of dimensions 0,
1, and 2. Additionally, three sets of non-zero eigenvalues further reveal both
topological persistence and geometric progression during the manifold
evolution. Extensive numerical experiments are carried out via the discrete
exterior calculus to demonstrate the utility and usefulness of the proposed
method for data representation and shape analysis.
