北京雁栖湖应用数学研究院

The path homology theory was created and developed by Grigor'yan, Lin, Muranov, and Yau in the cycle of papers in 2012 - 2015. Afterwards, this theory obtained further development in works of many authors. The path homology theory is a part of algebraic topology methods in graph theory which were developed by the authors of path homology theory. At present time, this theory has the title GLMY-theory by the names of authors. In the talk, we discuss the "universality" of the approach that gives path homology and illustrate this by relations between path homology and the other homology theories.

Graphs are essential tools for modeling pairwise interactions in fields such as biology, material science, and social networks. However, they fall short in representing many-body interactions involving more than two entities. Simplicial complexes and hypergraphs have been developed to address this need but still face challenges, particularly in capturing transitions between different orders of interactions. In this talk, I will present IntComplex, a new framework designed to model these high-order interactions. By incorporating homology theory, IntComplex offers a quantitative topological representation of these interactions. Additionally, I will demonstrate how persistent homology, applied through a filtration process, ensures a stable and robust analysis of the interactions. This framework introduces a new approach to understanding the topological properties of high-order interactions, with applications in complex network analysis.

A digraph $G$ is said to have no multisquares if for any pair of vertices $x, y$ with $d(x, y) = 2$, there are at most two shortest paths from $x$ to $y$. For such a digraph $G$ and a field $\mathbb{F}$, we construct a basis of the vector space of path $n$-chains $\Omega_n(G;\mathbb{F})$ for $n\geq 0$, generalising the basis of $\Omega_3(G;\mathbb{F})$ constructed by Grigor'yan. We consider the $\mathbb{F}$-path Euler characteristic $\chi^\mathbb{F}(G)$ defined as the alternating sum of dimensions of path homology groups with coefficients in $\mathbb{F}.$ If $\Omega_\bullet(G;\mathbb{F})$ is a bounded chain complex, we can use these bases to compute $\chi^\mathbb{F}(G)$. We provide an explicit example of a digraph $\mathcal{G}$ where the $\mathbb{F}$-path Euler characteristic changes depending on whether the characteristic of $\mathbb{F}$ is two. This illustrates differences between GLMY theory and the homology theory of spaces, leading us to conclude that no topological space $X$ can have homology $H_*(X;\mathbb{K})$ isomorphic to path homology ${\rm PH}_*(\mathcal{G};\mathbb{K})$ simultaneously for $\mathbb{K}=\mathbb{Z}$ and $\mathbb{K}=\mathbb{Z}/2\mathbb{Z}$.

High-entropy alloys are attracting ever-increasing attention in the field of catalysis by virtue of the vast chemical space constituted by their diverse tunable active sites. To expedite exploration within this chemical space, a proficient deep-learning model becomes essential. However, in the task of designing crystalline materials through deep learning, three major obstacles arise: the absence of inherent mathematical features contributing to the appealing properties, insufficient training data for the model, and limited confidence in results due to interpretability issues. <br>In light of these challenges, this work proposes a persistent path homology-based semi-supervised prediction and generation framework, empowered by our PathVAEs. This framework aims to predict the adsorption energy and design potential for High-entropy alloy catalysts. It addresses these obstacles by (1) introducing an effective topological approach to extract intrinsic features—ligand and coordination features; (2) utilizing semi-supervised learning to enhance model training; and (3) aligning machine learning operations with catalytic implications. The results are promising: the prediction component attains high accuracy, and the generation aspect yields eight high-performance catalyst designs. This work not only offers an excellent topology-based feature extraction method but also introduces a new research paradigm for the design of crystalline materials.

In this talk, we explore various methods for constructing and analyzing networks, focusing on techniques such as correlation-based network, Bayesian networks, and ODE-based network. Additionally, we introduce the concept of idopNetwork, a specialized approach to network modeling. The practical aspects of network construction are demonstrated using the R programming language, highlighting tools for network visualization and the calculation of key network properties. Through these methods, we provide a comprehensive guide to understanding dynamic and statistical dependencies within complex systems, showcasing their application in real-world data analysis.

A map from a manifold to a Euclidean space is said to be k-regular if the images of any distinct k points are linearly independent. For k-regular maps on manifolds, lower bounds on the dimension of the ambient Euclidean space have been extensively studied. In this talk, we study the lower bounds on the dimension of the ambient Euclidean space for 2-regular maps on Cartesian products of manifolds. As corollaries, we obtain the exact lower bounds on the dimension of the ambient Euclidean space for 2-regular maps and 3-regular maps on spheres as well as on some real projective spaces. Moreover, generalizing the notion of k-regular maps, we study the lower bounds on the dimension of the ambient Euclidean space for maps with certain non-degeneracy conditions from disjoint unions of manifolds into Euclidean spaces.

The extraction of singular patterns is a fundamental problem in detecting the intrinsic characteristics of vector fields. In this study, we propose an approach for extracting singular patterns from discrete planar vector fields. Our method involves converting the planar discrete vector field into a specialized digraph and computing its one-dimensional persistent path homology. By analyzing the persistence diagram, we can determine the location of singularity and segment a region of the singular pattern, which is referred to as a singular polygon. Moreover, the variations of singular patterns can also be analyzed. The experimental results demonstrate the effectiveness of our method in analyzing the centers and impact areas of tropical cyclones, positioning the dip poles from geomagnetic field, and measuring variations of singular patterns between vector fields.

Interactions in complex systems are widely observed across various fields, drawing increased attention from researchers. In mathematics, efforts are made to develop various theories and methods for studying the interactions between spaces. In this talk, we present an algebraic topology framework to explore interactions between spaces. We introduce the concept of interaction spaces and investigate their homotopy, singular homology, and simplicial homology. Furthermore, we demonstrate that interaction singular homology serves as an invariant under interaction homotopy. We believe that the proposed framework holds potential for practical applications.

We study the concept of the $\ell_p$-Vietoris-Rips simplicial set and the $\ell_p$-Vietoris-Rips complex of a metric space, where $1\leq p\leq \infty$. For $p=\infty$ we obtain the classical theory of Vietoris-Rips complexes, and for $p=1$ we obtain the simplicial set, whose homology is the blurred magnitude homology. We prove several results that were known for the Vietoris-Rips complex in the general case: (1) we prove a stability theorem for the interleaving distance between the corresponding persistent modules; (2) we show that for a compact Riemannian manifold and a small enough parameter the homotopy type of all the “$\ell_p$-Vietoris-Rips spaces” coincide with the homotopy type of the manifold; (3) we prove that the $\ell_p$-Vietoris-Rips spaces are invariant (up to homotopy) under taking the metric completion. We also show that the limit of the homology groups of the $\ell_p$-Vietoris-Rips spaces, when the parameter tends to zero, does not depend on $p$; and that the homology groups of the $\ell_p$-Vietoris-Rips spaces commute with filtered colimits of metric spaces.

In [I. Limonchenko et al., Adv. Math., 432 (2023), pp. No. 109274, 2023], the authors construct a cochain complex $CH^*(\mathbb{Z}_K)$ on the cohomology of a moment-angle complex $\mathbb{Z}_K$ and call the resulting cohomology the double cohomology, $HH^*(\mathbb{Z}_K)$. In this talk, we consider the change of rank in double cohomology after gluing an $n$-simplex to a simplicial complex $K$ in certain conditions. As an application, we give a positive answer to an open problem in their paper: For any even integer $r$, there always exists a simplicial complex $K$ such that $\mathrm{Rank} HH^*(\mathbb{Z}_K)=r$.

Since 1970s unstable homotopy decomposition has been a fundamental subject in homotopy theory. It serves as a basic natural way to understand homotopy properties, for instance various global characterizations of homotopy groups. In this talk, I will give an overview of the development of this topic based on our recent comprehensive book ``Unstable Homotopy Decomposition'' joint with Stephen Theriault, including various decomposition methods, some major applications and several open problems.

Interactions among the underlying agents of a complex system are not only limited to dyads but can also occur in larger groups. Currently, no generic model has been developed to capture high-order interactions (HOI), which, along with pairwise interactions, portray a detailed landscape of complex systems. This study proposes a high-order hypernetwork model based on statistical mechanics and topology to analyze HOI in complex systems. While existing models primarily focus on simple pairwise interactions, HOIs play a significant role in complex environments, such as ecosystems. By integrating evolutionary game theory and behavioral ecology, we developed mixed interactive ordinary differential equations (miODE) to dynamically describe the interspecies influences, and applied GLMY homology theory to dissect the topological structure of hypernetworks. Through experimental validation, the model successfully captures interactions involving three or more agents, revealing dynamic changes in interspecies relationships. The results demonstrate that HOIs significantly alter the interaction structure among species, driving community behavior and multi-scale evolutionary processes. This model provides a universal tool for studying complex systems, unveiling hidden dynamic interaction patterns, and understanding HOI across a wide range of physical and biological scenarios.

It is known that each compact connected orientable 3-manifold M admits an $H'$-splitting $H_1\cup_F H_2$, where $F$ is a compact connected orientable surface properly embedded in $M$ and splits $M$ into two handlbodies $H_1$ and $H_2$ (i.e., $H'$-splittings, similar to Heegaard splittings, which are common structures of 3-manifolds). It provides a new way to construct all compact connected orientable 3-manifolds. In this talk, we will discuss the $H'$-splittings of Seifert manifolds (with incompressible $H'$-surface) and handlebodies.

As a major task of natural language processing, speech recognition is one of the essential components of artificial intelligence.<br>We embed speech data into a high-dimensional Euclidean space using a sliding window and then compute persistent homology to topological features, followed by machine learning. This method stands comparison with various mainstream neural network algorithms across multiple datasets.