Beijing Institute of Mathematical Sciences and Applications

In 1994 a certain approach to the construction of cohomologies on digraphs was suggested by Dimakis and -Hoissen. This approach is based on the universal differential calculus on an algebra over a commutative unital ring of functions defined on the set of vertices. However, this approach remained on intuitive level without a precise definition of the corresponding cochain complex. In the paper [Asian J. of Math, 2015] Grigor’yan, Lin, Muranov and Yau constructed the corresponding cochain complex and, afterwards, in a series of works described relations to other cohomology theories, such as simplicial cohomology and Hochchild cohomology. The adequate homotopy theory was constructed by the authors of GLMY theory in 2015. These results initiated the study of relationships between discrete and continuous topology. As a rule, the role of discrete objects is played by objects of various categories of digraphs or graphs and the methods of study are those of classical algebraic topology. In this talk, we discuss the main steps of GLMY theory, open problems, and possible directions of development.

We will discuss a relationship between GLMY-theory and magnitude homology theory.

We develop a generalisation of the path homology theory introduced by Grigor’yan, Lin, Muranov and Yau (GLMY-theory) in a general simplicial setting. We prove the usual properties like homotopy invariance and Kunneth-type formula for box product. Variation of the theory, called square-commutative homology, as well as particular examples like marked groups and marked algebras are considered. Joint work with Sergei O. Ivanov.

First I will give the definition of the path complex of digraph introduced by GLMY. For simplicity, we will focus on the discussion of path complex under the strongly regular condition. We introduce the pair (P, Supp(P)) consisting of minimal path and its supporting digraph. In particular, we will give various examples. Finally, we will talk about the structure theorem and acyclic result. The talk is based on the joint work with S.-T. Yau.

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.

GLMY introduced by Yau and coworkers is mathematically rich and opens new directions in both pure and applied mathematics. As an extension, persistent path homology (PPH) facilitates a comprehensive multiscale analysis of directed graphs (digraphs) and networks. Within the scope of this study, we introduce PPH as a tool to scrutinize and delineate directed structures in the realms of molecular and material sciences. PPH plays a pivotal role in unveiling phenomena such as the Jahn-Teller effect, enabling the differentiation of distinct catalysts even when they share the same conformation, particularly in materials science. In the domain of molecular science, we propose the utilization of angle-based persistent path homology, which aids in discriminating spatial isomers, including Cis-Trans structures and chiral molecules. To further harness the potential of path topology in material science, we introduce the concept of the neighborhood path complex, which facilitates quantitative analyses of the structure and stability of carboranes. Moreover, we extend the application of path topology methods to the field of systems biology, employing them to elucidate the intricate process of blood coagulation formation and revealing its critical stages.

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.

The GLMY homology theory of digraphs introduced by Grigor'yan-Lin-Muranov-Yau in 2012 already successfully applied in various fields. Moreover, they also established the homotopy theory of digraphs. From the perspective of algebraic topology, the homotopy theory of digraphs deserves to be explored deeper to help us understand and dig more potential meaning in digraphs. In this talk, based on the homotopy theory of digraphs established by them, we aim to give an intuitive grids descpition for the homotopy groups of digraphs. Furthermore, we verify the digraph analogue of Puppe sequence. More precisely, we proved that there exists a long exact sequence of homotopy groups of digraphs for any based digraph map.

The utilization of artificial intelligence (AI) in drug design holds considerable promise for a transformative impact on the pharmaceutical industry. Despite its potential, a critical challenge inherent in AI-based drug design models revolves around the efficient representation and featurization of molecules. Recently, the integration of topological data analysis (TDA) with machine learning models has emerged as a noteworthy approach, demonstrating remarkable success in the realm of drug design. In this presentation, I will provide a succinct overview of TDA's application in drug design. Subsequently, I will delve into the introduction of several graph complexes, such as the Dowker Complex, Neighborhood Complex, Hom-Complex, and Hypergraph, as viable solutions for molecular representation and featurization. Through the amalgamation of these mathematical representations with machine learning models, a series of mathematical AI models for the analysis of biomolecular data have been developed. The efficacy of these Mathematical AI models is underscored by their ability to yield state-of-the-art results on widely recognized benchmark datasets.

Molecular descriptors are essential to quantitative structure activity/property relationship (QSAR/QSPR) models and machine learning models. In this talk we will introduce our recently proposed persistent path-spectral (PPS), PPS-based molecular descriptors, and PPS-based machine learning model for the prediction of the protein-ligand binding affinity. For the graph, simplicial complex, and hypergraph representation of molecular structures and interactions, the path-Laplacian can be constructed and the derived pathspectral naturally gives a quantitative description of molecules. Further, by introducing the filtration process of the representation, the persistent path-spectral can be derived, which gives a multiscale characterization of molecules. Molecular descriptors from the persistent path-spectral attributes then are combined with the machine learning model, in particular, the gradient boosting tree, to form our PPS-ML model. We test our model on three most commonly used data sets, i.e., PDBbind-v2007, PDBbind-v2013, and PDBbind-v2016, and our model can achieve competitive results.

In the series of works on path homology and homotopy theory of (di)graphs, Grigor’yan, Lin, Muranov and Yau initiated the study of relationships between discrete and continuous topologies. As a rule, the role of discrete objects is played by objects of various categories of digraphs or graphs and the methods of study are those of classical algebraic topology. The theory of simplicial sets is an efficient method for studying topological spaces. With each topological space X the simplicial set S∆(X) is associated. This set is determined by the singular simplexes f : ∆n → X. The topological realization |K|Top of any simplicial set K is a CW-complex. Hence, the topological realization |S∆(X)|T op of the simplicial set S∆(X) is defined. Moreover, there is a weak homotopy equivalence |S∆(X)|Top ~ X for any CW-complex X. In this talk, we describe shortly the topological situation and describe similar relationships between simplicial sets and their realizations in the categories of quivers and digraphs.

Hypergraphs, as a generalization of simplicial complexes, have long been a subject of interest in their geometric interpretation. The subdivision of simplicial complexes can, to some extent, provide insight into the geometry of simplicial complexes. In this talk, we introduce the concept of hypergraph subdivision. Notably, for simplicial complexes, this subdivision aligns with the barycentric subdivision of simplicial complexes. Moreover, we prove that the subdivision of hypergraphs has the topological invariance of embedded homology.

This report delves into the Extended Persistent Homology, an algorithm initially introduced by Pankaj K. Agarwal, Herbert Edelsbrunner, John Harer, and Yusu Wang in 2006 for application on manifolds. Our study aims to expand the algorithm's applicability to simplicial complexes, enabling its utilization in analyzing point cloud data. By broadening its scope, this adaptation allows for more versatile applications of the algorithm beyond its original manifold-focused design.

In artificial-intelligence-aided signal processing, existing deep learning models often exhibit a black-box structure, and their validity and comprehensibility remain elusive. The integration of topological methods, despite its relatively nascent application, serves the dual purpose of extracting structural information from time-dependent data as well as making models more interpretable. In this talk, I will give an overview of joint work with Pingyao Feng et al., in which we propose a transparent and all-purpose methodology, TopCap, to capture the most salient topological features inherent in time series for machine learning. Rooted in high-dimensional ambient spaces, TopCap is capable of capturing features rarely detected in datasets with low dimensionality. Applying time-delay embedding and persistent homology, we obtain descriptors that encapsulate information such as the vibrations of a time series. This information is then vectorized and fed into multiple machine learning algorithms such as k-nearest neighbors and support vector machine. Notably, in classifying voiced and voiceless consonants, TopCap achieves an accuracy exceeding 96% and is geared towards designing topological convolutional layers for deep learning of speech and audio signals.

Persistent homology is a standard topology data analysis tool for determining the importance of (or ranking of) entities in a connected system that is represented by a simplicial complex. However, many complex systems and datasets have natural multiway interactions that are more faithfully modeled by a hypergraph. Here we introduce a new topological feature, Hˆ homology, on hypergraphs and an efficient algorithm to compute both persistent H inf barcodes and Hˆ barcodes. Our theory is demonstrated by analyzing face-to- face interactions of different populations.

Topology, as a novel method for data analysis, can greatly simplify the complexity of data while retaining critical information and has recently become one of the most promising methods for analyzing the topology of materials in the field of materials science. However, traditional topological methods often struggle with the asymmetric interactions inherent in multi-elemental systems found in crystals and clusters of small inorganic molecules. Addressing this challenge, GLMY theory stands out as an advanced mathematical method specifically designed for oriented systems. Its remarkable capacity to describe unbalanced or asymmetric relationships in data positions this approach as a key player in qualitatively analyzing intrinsic topological features present in materials and small inorganic molecules. This presentation leverages digraph topology and GLMY theory in the analysis of inorganic small molecules and crystal structures to predict the physicochemical properties of materials. The methodology involves constructing digraphs for inorganic small molecules and crystal structures based on their atomic types and chemical bonding relationships. Subsequently, the sustained persistent GLMY homology of each directed graph is computed. The resulting topological features of material structures serve as inputs to a prediction model for both nonlinear and linear predictions of the physicochemical properties of materials. The findings demonstrate a significant improvement in prediction accuracy compared to traditional methods in materials science. This substantiates the superiority of GLMY theory, showcasing its potential to advance physicochemical properties prediction capabilities in materials science.

A cubical set is a discrete object which is based on an union of cubes in various dimensions with a collection of special relations. This set is a natural analog of the simplicial set which is based on a union of simplexes. The notion of a cubical set was introduced by Kan as an algebraic model for the investigation of singular cubical complex S (X) of a topological space X. Similarly to the simplicial case there is a weak homotopy equivalence |S∆(X)|Top~X for any CW-complex X. In this talk, we describe shortly the topological situation and describe similar relationships between cubical sets and their realizations in the categories of quivers and digraphs.