北京雁栖湖应用数学研究院

The magnitude homology, introduced by R. Hepworth and S. Willerton, offers a topological invariant that enables the study of graph properties. Hypergraphs, being a generalization of graphs, serve as popular mathematical models for data with higher- order structures. In this talk, we focus on describing the topological characteristics of hypergraphs by considering their magnitude homology. We begin by examining the distances between hyperedges in a hypergraph and establish the magnitude homology of hypergraphs. Additionally, we explore the relationship between the magnitude and the magnitude homology of hypergraphs. Furthermore, we derive several functorial properties of the magnitude homology for hypergraphs. Lastly, we present the Künneth theorem for the simple magnitude homology of hypergraphs.

We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a “bridge” between GLMY theory and group homology theory, which helps to reduce path homology calculations to group homology. We show some cases where this approach allows us to fully express path homology in terms of group homology. To illustrate this method, we provide a path homology computation for the Cayley digraph of the additive group of rational numbers.

Both the Van Kampen theorem and the Mayer-Vietoris sequence are powerful tools in algebraic topology. In GLMY theory, neither of them hold in general, and we would like to discuss the potential connection between their failures.

Human diseases involve metabolic alterations. Metabolomic profiles have served as a biomarker for the early identification of high-risk individuals and disease prevention. However, current approaches can only characterize individual key metabolites, without taking into account their interactions.This work have leveraged a statistical physics model to combine all metabolites into bDSW networks and implement GLMY homology theory to analyze and interpret the topological change of health state from symbiosis to dysbiosis.The application of this model to real data allows us to identify several hub metabolites and their interaction webs, which play a part in the formation of inflammatory bowel diseases.

We construct a coalgebraic structure in path homology (GLMY). It is known that the existence of a coalgebra structure in homology is fundamental to the subject. In order to better understand and describe the properties of GLMY homology, we extend some properties of traditional homology to GLMY homology. Furthermore, we focus on the suspension structure of digraphs. By analogy with the properties of suspension in space, we proved that the suspension of a digraph has a path homology coalgebra structure analogous to that for a suspension of space under certain conditions. This finding provides a strong theoretical support for further study of GLMY homology.