Huitao Feng

As a generalization of graphs and simplicial complexes, hypergraphs have various applications in chemistry, biology, computer science and data science. Recently, the first well-defined hypergraph-based embedded homology and persistent (embedded) homology models have been proposed and achieved great success in drug design. However, the further applications of these models have been significantly hindered by the lack of efficient computational algorithms. In this paper, we propose an algorithm for embedded homology and persistent (embedded) homology of hypergraphs over field coefficient. One of the key issues for hypergraph-based homology models is the absence of proper boundary operators as in traditional simplicial complex-based models. A supremum/infimum chain complex has been proposed in the embedded homology model with well-defined boundary operators. Here we give detailed algorithms for transforming a filtered hypergraph to its filtered infimum chain complex and supremum chain complex. In this way, the general algorithm for chain complex can be used directly on the supremum/infimum chain complexes. Our algorithm has been validated on hypergraph models of protein structures from the protein data bank. It has been found that our algorithm is very robust and efficient.

In this paper, we establish an intrinsic Gauss-Bonnet-Chern formula for Finsler manifolds by using the Mathai-Quillen's superconnection formalism, in which no extra vector field is involved. Furthermore, we prove a more general Lichnerowicz formula in this direction through a geometric localization procedure.

In this short note, we prove that a complex Finsler vector bundle with positive Kobayashi curvature must be ample, which partially solves a problem of S. Kobayashi posed in 1975. As applications, a strongly pseudoconvex complex Finsler manifold with positive Kobayashi curvature must be biholomorphic to the complex projective space; we also show that all Schur polynomials are numerically positive for complex Finsler vector bundles with positive Kobayashi curvature.

In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.

In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.