Algebraic Topology Research Articles

Adversarially Trained Persistent Homology Based Graph Convolutional Network for Disease Identification Using Brain Connectivity.

Brain disease propagation is associated with characteristic alterations in the structural and functional connectivity networks of the brain. To identify disease-specific network representations, graph convolutional networks (GCNs) have been used because of their powerful graph embedding ability to characterize the non-Euclidean structure of brain networks. However, existing GCNs generally focus on learning the discriminative region of interest (ROI) features, often ignoring important topological information that enables the integration of connectome patterns of brain activity. In addition, most methods fail to consider the vulnerability of GCNs to perturbations in network properties of the brain, which considerably degrades the reliability of diagnosis results. In this study, we propose an adversarially trained persistent homology-based graph convolutional network (ATPGCN) to capture disease-specific brain connectome patterns and classify brain diseases. First, the brain functional/structural connectivity is constructed using different neuroimaging modalities. Then, we develop a novel strategy that concatenates the persistent homology features from a brain algebraic topology analysis with readout features of the global pooling layer of a GCN model to collaboratively learn the individual-level representation. Finally, we simulate the adversarial perturbations by targeting the risk ROIs from clinical prior, and incorporate them into a training loop to evaluate the robustness of the model. The experimental results on three independent datasets demonstrate that ATPGCN outperforms existing classification methods in disease identification and is robust to minor perturbations in network architecture. Our code is available at https://github.com/CYB08/ATPGCN.

The algebraic topology of 4-manifold multisections

The algebraic topology of 4-manifold multisections

Uncovering Patterns Beneath the Surface: A Glimpse into the Applications of Algebraic Topology

This paper explores the significant impact of algebraic topology on diverse real-world applications. Starting with an introduction to the historical development and essence of algebraic topology, it delves into its applications in neuroscience, physics, biology, engineering, data analysis, and Geographic Information Systems (GIS). Remarkable applications incorporate the analysis of neural networks, quantum mechanics, materials science, and disaster management, showcasing its boundless significance. Despite computational challenges, this study outlines prospects, emphasizing the requirement for proficient algorithms, noise robustness, multi-scale analysis, machine learning integration, user-friendly tools, and interdisciplinary collaborations. In essence, algebraic topology provides a transformative lens for uncovering stowed-away topological structures in complex data, offering solutions to perplexing problems in science, engineering, and society, with vast potential for future exploration and innovation.

Towards Homological Methods in Graphic Statics

Recent developments in applied algebraic topology can simplify and extend results in graphic statics – the analysis of equilibrium forces, dual diagrams, and more. The techniques introduced here are inspired by recent developments in cellular cosheaves and their homology. While the general theory has a few technical prerequisites (including homology and exact sequences), an elementary introduction based on little more than linear algebra is possible. A few classical results, such as Maxwell's Rule and 2D graphic statics duality, are quickly derived from core ideas in algebraic topology. Contributions include: (1) a reformulation of statics and planar graphic statics in terms of cosheaves and their homology; (2) a new proof of Maxwell's Rule in arbitrary dimensions using Euler characteristic; and (3) derivation of a novel relationship between mechanisms of the form diagram and obstructions to the generation of force diagrams. This last contribution presages deeper results beyond planar graphic statics.

Free Topological Algebra with Separately Continuous Mal’tsev Operation

Free Topological Algebra with Separately Continuous Mal’tsev Operation

PLANE METRICS AND THE UNIFORMIZATION THEOREM ON MINIMAL THE THREE DIMENSIONAL SPHERE

Let (M, g) be a torus with a riemannian metric g. The Uniformization theorem implies the existence of a covering conforma! map <:p : R2 ----> /1.1 such that the group H,t, of diffeo­morphisms p: R2 ---+ R2 such that <Pop= <j,, is a subgroup of the group of translations in R2 and is isomorph to Z2• The classical proof of his theorem uses tools from algebraic topology and PDE's in Sobolev spaces. In this paper we will give an alternative and easier proof of this theorem in the case that M is a minimal immersed torus in the unit three dimensional sphere, S3. We will achieve this theorem by studying the shape operator of minimal torus on the sphere. One of the main ideas of the this paper is to induce the reader to the study of minimal torus in S3• Recall that one important conjecture in differential geometry is Law­son's conjecture. This conjecture states that the only minimal embedded tori in S3 are the Clifford tori.

Various matching keys for asymmetric topology encryption

With the fast development of quantum computer technology, one has to focus on the security of information running in real networks. Topological coding is a technology of basic topological structure, number theory and algebra has potential application value in asymmetric encryption system. The public key and private key of topological graph encryption system designed by topological coding may be able to resist attacks equipped with artificial intelligence technology and quantum computer. In order to further investigate topology encoding, we propose new techniques for designing public and private keys using graph coloring and labeling.

Cutting the Möbius Strip

Summary Using tools from algebraic topology and algebraic geometry (fundamental polygons and blowups, respectively), we prove that cutting along the center of a Möbius strip yields a cylinder.

Some concepts of topology and questions in topological algebra

Some concepts of topology and questions in topological algebra

Multivariate central limit theorems for random clique complexes

Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: the vector of critical simplex counts attained by a lexicographical Morse matching,the vector of simplex counts in the link of a fixed simplex, andthe vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U-statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein’s method.

OVERVIEW ON MODELS IN HOMOTOPICAL ALGEBRA

A covariant functor ∆ → A is referred to as a model object of A, generating a subject matter in A analogous to algebraic topology when A is the category of topological spaces; this work delineates the contexts in which these concepts are developed and outlines the key contributions made by the author concerning model objects.

Review article: Dynamical systems, algebraic topology and the climate sciences

Abstract. The definition of climate itself cannot be given without a proper understanding of the key ideas of long-term behavior of a system, as provided by dynamical systems theory. Hence, it is not surprising that concepts and methods of this theory have percolated into the climate sciences as early as the 1960s. The major increase in public awareness of the socio-economic threats and opportunities of climate change has led more recently to two major developments in the climate sciences: (i) the Intergovernmental Panel on Climate Change's successive Assessment Reports and (ii) an increasing understanding of the interplay between natural climate variability and anthropogenically driven climate change. Both of these developments have benefited from remarkable technological advances in computing resources, relating throughput as well as storage, and in observational capabilities, regarding both platforms and instruments. Starting with the early contributions of nonlinear dynamics to the climate sciences, we review here the more recent contributions of (a) the theory of non-autonomous and random dynamical systems to an understanding of the interplay between natural variability and anthropogenic climate change and (b) the role of algebraic topology in shedding additional light on this interplay. The review is thus a trip leading from the applications of classical bifurcation theory to multiple possible climates to the tipping points associated with transitions from one type of climatic behavior to another in the presence of time-dependent forcing, deterministic as well as stochastic.

PLPCA: Persistent Laplacian-Enhanced PCA for Microarray Data Analysis.

Over the years, Principal Component Analysis (PCA) has served as the baseline approach for dimensionality reduction in gene expression data analysis. Its primary objective is to identify a subset of disease-causing genes from a vast pool of thousands of genes. However, PCA possesses inherent limitations that hinder its interpretability, introduce class ambiguity, and fail to capture complex geometric structures in the data. Although these limitations have been partially addressed in the literature by incorporating various regularizers, such as graph Laplacian regularization, existing PCA based methods still face challenges related to multiscale analysis and capturing higher-order interactions in the data. To address these challenges, we propose a novel approach called Persistent Laplacian-enhanced Principal Component Analysis (PLPCA). PLPCA amalgamates the advantages of earlier regularized PCA methods with persistent spectral graph theory, specifically persistent Laplacians derived from algebraic topology. In contrast to graph Laplacians, persistent Laplacians enable multiscale analysis through filtration and can incorporate higher-order simplicial complexes to capture higher-order interactions in the data. We evaluate and validate the performance of PLPCA using ten benchmark microarray data sets that exhibit a wide range of dimensions and data imbalance ratios. Our extensive studies over these data sets demonstrate that PLPCA provides up to 12% improvement to the current state-of-the-art PCA models on five evaluation metrics for classification tasks after dimensionality reduction.

Persistent Homology as a Heterogeneity Metric for Predicting Pore Size Change in Dissolving Carbonates

AbstractAccurate prediction of physical alterations in carbonate reservoirs under dissolution is critical for development of subsurface energy technologies. The impact of mineral dissolution on flow characteristics depends on the connectivity and tortuosity of the pore network. Persistent homology is a tool from algebraic topology that describes the size and connectivity of topological features. When applied to 3D X‐ray computed tomography (XCT) imagery of rock cores, it provides a novel metric of pore network heterogeneity. Prior works have demonstrated the efficacy of persistent homology in predicting flow properties in numerical simulations of flow through porous media. Its ability to combine size, spatial distribution, and connectivity information make it a promising tool for understanding reactive transport in complex pore networks, yet limited work has been done to apply persistence analysis to experimental studies on natural rocks. In this study, three limestone cores were imaged by XCT before and after acid‐driven dissolution flow‐through experiments. Each XCT scan was analyzed using persistent homology. In all three rocks, permeability increase was driven by the growth of large, connected pore bodies. The two most homogenous samples saw an increased effect nearer to the flow inlet, suggesting emerging preferential flow paths as the reaction front progresses. The most heterogeneous sample showed an increase in along‐core homogeneity during reaction. Variability of persistence showed moderate positive correlation with pore body size increase. Persistence heterogeneity analysis could be used to anticipate where greatest pore size evolution may occur in a reservoir targeted for subsurface development, improving confidence in project viability.

Complexity, Diversity, Homogeneity and Topology Measures Re-Define Biological Tumor Aggressiveness as a Function of Intercellular Connectivity

Similarity between patterns of cellular self-organization in primary tumor and in its normal counterpart underlies the concept of biological tumor aggressiveness. That subjective parameter is supposed to correlate with both dynamics of local tumor growth and the risk of progression. Subjective evaluation during the microscopic examination is inaccurate. The co-application of fractal geometry and algebraic topology offers a novel perspective. A set of prostate carcinomas comprising all patterns of self-organization was classified according to complexity and entropy to investigate how complexity relates to topology of the patterns. Prostate carcinomas evolve from order to disorder along the increasing entropy as many physical dynamic complex systems do. Carcinomas can be classified into the classes of complexity equivalence according to both the values of the global fractal dimensions D0 or D1 and the Shannon entropy H. Topological measures, such as the bottleneck distance or the Wasserstein distance reveal different topology in very low- and low-aggressive prostate carcinomas (class C1, C2, C3) in comparison with high-aggressive carcinomas (class C6, C7). This is contingent on the appearance of infiltrating cancer cells. While the persistent homology entropy Hph remains similar in all classes, the persistent barcode entropy Hpb is significantly decreased in class C1 and increased in class C7 in comparison with the other ones. Patterns of self-organization can now be defined according to changes in intercellular connectivity by the set of complexity, diversity, homogeneity, and topology measures.

The universal property of commutative algebras' internal crossed modules

In this work, we identify subobjects and ideals in the category of internal crossed modules, which provide a deeper understanding of the structure of these objects. Moreover, we provide several propositions through examples, which illustrate the properties and relationships between ideals and subobjects in the category of internal crossed modules. The examples and propositions provided in this work can serve as a foundation for further research in this area and may lead to new insights and discoveries in the study of these complex algebraic structures. Overall, in conclusion, we give a brief overview of the contributions and future research directions of the work presented, highlighting the significance of internal crossed modules in algebraic topology and category theory as well as making suggestions for possible areas of additional research and application.

Heaps of modules and affine spaces

A notion of heaps of modules as an affine version of modules over a ring or, more generally, over a truss, is introduced and studied. Basic properties of heaps of modules are derived. Examples arising from geometry (connections, affine spaces) and algebraic topology (chain contractions) are presented. Relationships between heaps of modules, modules over a ring and affine spaces are revealed and analysed.

IDEALS OF FUNCTION SPACE IN THE LIGHT OF AN EXPONENTIAL ALGEBRA

Exponential algebra is a new algebraic structure consisting of a semigroup structure, a scalar multiplication, an internal multiplication and a partial order [introduced in [4]]. This structure is based on the structure `exponential vector space' which is thoroughly developed by Priti Sharma et. al. in [11] [This structure was actually proposed by S. Ganguly et. al. in [1] with the name `quasi-vector space'] Exponential algebra can be considered as an algebraic ordered extension of the concept of algebra. In the present paper we have shown that the function space $ C^+(\mathbf X) $ of all non-negative continuous functions on a topological space $\mathbf X$ is a topological exponential algebra under the compact open topology. Also we have discussed the ideals and maximal ideals of $ C^+(\mathbf X) $. We find an ideal of $ C^+(\mathbf X)$ which is not a maximal ideal in general; actually maximality of that ideal depends on the topology of $\mathbf{X}$. The concept of ideals of exponential algebra was introduced by us in [4].

Semantical investigations on non-classical logics with recovery operators: negation

Abstract We investigate mathematical structures that provide natural semantics for families of (quantified) non-classical logics featuring special unary connectives, known as recovery operators, that allow us to ‘recover’ the properties of classical logic in a controlled manner. These structures are known as topological Boolean algebras, which are Boolean algebras extended with additional operations subject to specific conditions of a topological nature. In this study, we focus on the paradigmatic case of negation. We demonstrate how these algebras are well-suited to provide a semantics for some families of paraconsistent Logics of Formal Inconsistency and paracomplete Logics of Formal Undeterminedness. These logics feature recovery operators used to earmark propositions that behave ‘classically’ when interacting with non-classical negations. Unlike traditional semantical investigations, which are carried out in natural language (extended with mathematical shorthand), our formal meta-language is a system of higher-order logic (HOL) for which automated reasoning tools exist. In our approach, topological Boolean algebras are encoded as algebras of sets via their Stone-type representation. We use our higher-order meta-logic to define and interrelate several transformations on unary set operations, which naturally give rise to a topological cube of opposition. Additionally, our approach enables a uniform characterization of propositional, first-order and higher-order quantification, including restrictions to constant and varying domains. With this work, we aim to make a case for the utilization of automated theorem proving technology for conducting computer-supported research in non-classical logics. All the results presented in this paper have been formally verified, and in many cases obtained, using the Isabelle/HOL proof assistant.

Tensor Products and Crossed Differential Graded Lie Algebras in the Category of Crossed Complexes

The study of algebraic structures endowed with the concept of symmetry is made possible by the link between Lie algebras and symmetric monoidal categories. This relationship between Lie algebras and symmetric monoidal categories is useful and has resulted in many areas, including algebraic topology, representation theory, and quantum physics. In this paper, we present analogous definitions for Lie algebras within the framework of whiskered structures, bimorphisms, crossed complexes, crossed differential graded algebras, and tensor products. These definitions, given for groupoids in existing literature, have been adapted to establish a direct correspondence between these algebraic structures and Lie algebras. We show that a 2-truncation of the crossed differential graded Lie algebra, obtained from our adapted definitions, gives rise to a braided crossed module of Lie algebras. We also construct a functor to simplicial Lie algebras, enabling a systematic mapping between different Lie algebraic categories, which supports the validity of our adapted definitions and establishes their compatibility with established categories.