De Rham-Hodge Theory Research Articles

Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning

<p>Recently, topological data analysis has become a trending topic in data science and engineering. However, the key technique of topological data analysis, i.e., persistent homology, is defined on point cloud data, which does not work directly for data on manifolds. Although earlier evolutionary de Rham-Hodge theory deals with data on manifolds, it is inconvenient for machine learning applications because of the numerical inconsistency caused by remeshing the involving manifolds in the Lagrangian representation. In this work, we introduced persistent de Rham-Hodge Laplacian, or persistent Hodge Laplacian (PHL), as an abbreviation for manifold topological learning. Our PHLs were constructed in the Eulerian representation via structure-persevering Cartesian grids, avoiding the numerical inconsistency over the multi-scale manifolds. To facilitate the manifold topological learning, we proposed a persistent Hodge Laplacian learning algorithm for data on manifolds or volumetric data. As a proof-of-principle application of the proposed manifold topological learning model, we considered the prediction of protein-ligand binding affinities with two benchmark datasets. Our numerical experiments highlighted the power and promise of the proposed method.</p>

2D Discrete Hodge–Dirac Operator on the Torus

We discuss a discretization of the de Rham–Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge–Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide and prove a discrete version of the Hodge decomposition theorem. The goal of this work is to develop a satisfactory discrete model of the de Rham–Hodge theory on manifolds that are homeomorphic to the torus. Special attention has been paid to discrete models on a combinatorial torus. In this particular case, we also define and calculate the cohomology groups.

EVOLUTIONARY DE RHAM-HODGE METHOD.

The de Rham-Hodge theory is a landmark of the 20th 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 Laplacians 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 potential of the proposed paradigm for data representation and shape analysis of both point cloud data and density maps. To demonstrate the utility of the proposed method, the application is considered to the protein B-factor predictions of a few challenging cases for which existing biophysical models break down.

The de Rham-Hodge Analysis and Modeling of Biomolecules.

Biological macromolecules have intricate structures that underpin their biological functions. Understanding their structure-function relationships remains a challenge due to their structural complexity and functional variability. Although de Rham-Hodge theory, a landmark of twentieth-century mathematics, has had a tremendous impact on mathematics and physics, it has not been devised for macromolecular modeling and analysis. In this work, we introduce de Rham-Hodge theory as a unified paradigm for analyzing the geometry, topology, flexibility, and Hodge mode analysis of biological macromolecules. Geometric characteristics and topological invariants are obtained either from the Helmholtz-Hodge decomposition of the scalar, vector, and/or tensor fields of a macromolecule or from the spectral analysis of various Laplace-de Rham operators defined on the molecular manifolds. We propose Laplace-de Rham spectral-based models for predicting macromolecular flexibility. We further construct a Laplace-de Rham-Helfrich operator for revealing cryo-EM natural frequencies. Extensive experiments are carried out to demonstrate that the proposed de Rham-Hodge paradigm is one of the most versatile tools for the multiscale modeling and analysis of biological macromolecules and subcellular organelles. Accurate, reliable, and topological structure-preserving algorithms for implementing discrete exterior calculus (DEC) have been developed to facilitate the aforementioned modeling and analysis of biological macromolecules. The proposed de Rham-Hodge paradigm has potential applications to subcellular organelles and the structure construction from medium- or low-resolution cryo-EM maps, and functional predictions from massive biomolecular datasets.

The spectral and differential geometric aspects of a generalized De Rham-Hodge theory related with Delsarte transmutation operators in multidimension and its applications to spectral and soliton problems

A review on spectral and differential-geometric properties of Delsarte transmutation operators in multidimension is given. Their differential-geometric and topological structure of Delsarte transmutation operators and associated with them Gelfand–Levitan–Marchenko type equations are studied making use of the generalized de Rham–Hodge–Skrypnik differential complex. The relationships with spectral theory and special Berezansky type congruence properties of Delsarte transmutated operators are stated. Some applications to integrable dynamical systems theory in multidimension are presented.

Generalized de Rham-Hodge Theory of Delsarte Transmutation Operators in Multidimensional Case and Its Applications

A study of spectral and differential-geometric properties of Delsarte transmutation operators is given. Their differential-geometric and topological structure in the multidimensional case is analyzed and the relationships with the generalized de Rham-Hodge theory of generalized differential complexes are stated. Some applications to integrable dynamical systems in the multidimensional case are presented.

De Rham-Hodge theory for Riemannian foliations

De Rham-Hodge theory for Riemannian foliations

De Rham-Hodge theory on a manifold with cone-like singularities

de Rham-Hodge theory on a manifold with cone-like singularities

De Rham-Hodge theory for L2-cohomology of infinite coverings

De Rham-Hodge theory for L2-cohomology of infinite coverings