arXiv - MATH - Algebraic Topology最新文献

Tensor triangular geometry of modules over the mod 2 Steenrod algebra 模 2 Steenrod 代数上模块的张量三角几何

arXiv - MATH - Algebraic Topology Pub Date : 2024-09-16 DOI: arxiv-2409.10731

Collin Litterell

引用次数: 0

Ring operads and symmetric bimonoidal categories 环操作数和对称双元范畴

arXiv - MATH - Algebraic Topology Pub Date : 2024-09-15 DOI: arxiv-2409.09664

Kailin Pan

引用次数: 0

Inferring hyperuniformity from local structures via persistent homology 通过持久同源性从局部结构推断超均匀性

arXiv - MATH - Algebraic Topology Pub Date : 2024-09-13 DOI: arxiv-2409.08899

Abel H. G. Milor, Marco Salvalaglio

{"title":"Inferring hyperuniformity from local structures via persistent homology","authors":"Abel H. G. Milor, Marco Salvalaglio","doi":"arxiv-2409.08899","DOIUrl":"https://doi.org/arxiv-2409.08899","url":null,"abstract":"Hyperuniformity refers to the suppression of density fluctuations at large\u0000scales. Typical for ordered systems, this property also emerges in several\u0000disordered physical and biological systems, where it is particularly relevant\u0000to understand mechanisms of pattern formation and to exploit peculiar\u0000attributes, e.g., interaction with light and transport phenomena. While\u0000hyperuniformity is a global property, it has been shown in [Phys. Rev. Research\u00006, 023107 (2024)] that global hyperuniform characteristics systematically\u0000correlate with topological properties representative of local arrangements. In\u0000this work, building on this information, we explore and assess the inverse\u0000relationship between hyperuniformity and local structures in point\u0000distributions as described by persistent homology. Standard machine learning\u0000algorithms trained on persistence diagrams are shown to detect hyperuniformity\u0000with high accuracy. Therefore, we demonstrate that the information on patterns'\u0000local structure allows for inferring hyperuniformity. Then, addressing more\u0000quantitative aspects, we show that parameters defining hyperuniformity\u0000globally, for instance entering the structure factor, can be reconstructed by\u0000comparing persistence diagrams of targeted patterns with reference ones. We\u0000also explore the generation of patterns entailing given topological properties.\u0000The results of this study pave the way for advanced analysis of hyperuniform\u0000patterns including local information, and introduce basic concepts for their\u0000inverse design.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"206 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Geometric representation of cohomology classes for the Lie groups Spin(7) and Spin(8) Spin(7) 和 Spin(8) 列群同调类的几何表示

arXiv - MATH - Algebraic Topology Pub Date : 2024-09-10 DOI: arxiv-2409.06491

Eiolf Kaspersen, Gereon Quick

引用次数: 0

Computing the homology of universal covers via effective homology and discrete vector fields 通过有效同源性和离散向量场计算普遍盖的同源性

arXiv - MATH - Algebraic Topology Pub Date : 2024-09-10 DOI: arxiv-2409.06357

Miguel Angel Marco-Buzunariz, Ana Romero

{"title":"Computing the homology of universal covers via effective homology and discrete vector fields","authors":"Miguel Angel Marco-Buzunariz, Ana Romero","doi":"arxiv-2409.06357","DOIUrl":"https://doi.org/arxiv-2409.06357","url":null,"abstract":"Effective homology techniques allow us to compute homology groups of a wide\u0000family of topological spaces. By the Whitehead tower method, this can also be\u0000used to compute higher homotopy groups. However, some of these techniques (in\u0000particular, the Whitehead tower) rely on the assumption that the starting space\u0000is simply connected. For some applications, this problem could be circumvented\u0000by replacing the space by its universal cover, which is a simply connected\u0000space that shares the higher homotopy groups of the initial space. In this\u0000paper, we formalize a simplicial construction for the universal cover, and\u0000represent it as a twisted cartesian product. As we show with some examples, the universal cover of a space with effective\u0000homology does not necessarily have effective homology in general. We show two\u0000independent sufficient conditions that can ensure it: one is based on a\u0000nilpotency property of the fundamental group, and the other one on discrete\u0000vector fields. Some examples showing our implementation of these constructions in both\u0000sagemath and kenzo are shown, together with an approach to compute the\u0000homology of the universal cover when the group is abelian even in some cases\u0000where there is no effective homology, using the twisted homology of the space.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Secondary cohomology operations and the loop space cohomology 二级同调运算和环空间同调

arXiv - MATH - Algebraic Topology Pub Date : 2024-09-07 DOI: arxiv-2409.04861

Samson Saneblidze

引用次数: 0

Magnitude homology and homotopy type of metric fibrations 度量纤维的幅同调和同调类型

arXiv - MATH - Algebraic Topology Pub Date : 2024-09-05 DOI: arxiv-2409.03278

Yasuhiko Asao, Yu Tajima, Masahiko Yoshinaga

引用次数: 0

Topological Methods in Machine Learning: A Tutorial for Practitioners 机器学习中的拓扑方法：从业人员教程

arXiv - MATH - Algebraic Topology Pub Date : 2024-09-04 DOI: arxiv-2409.02901

Baris Coskunuzer, Cüneyt Gürcan Akçora

{"title":"Topological Methods in Machine Learning: A Tutorial for Practitioners","authors":"Baris Coskunuzer, Cüneyt Gürcan Akçora","doi":"arxiv-2409.02901","DOIUrl":"https://doi.org/arxiv-2409.02901","url":null,"abstract":"Topological Machine Learning (TML) is an emerging field that leverages\u0000techniques from algebraic topology to analyze complex data structures in ways\u0000that traditional machine learning methods may not capture. This tutorial\u0000provides a comprehensive introduction to two key TML techniques, persistent\u0000homology and the Mapper algorithm, with an emphasis on practical applications.\u0000Persistent homology captures multi-scale topological features such as clusters,\u0000loops, and voids, while the Mapper algorithm creates an interpretable graph\u0000summarizing high-dimensional data. To enhance accessibility, we adopt a\u0000data-centric approach, enabling readers to gain hands-on experience applying\u0000these techniques to relevant tasks. We provide step-by-step explanations,\u0000implementations, hands-on examples, and case studies to demonstrate how these\u0000tools can be applied to real-world problems. The goal is to equip researchers\u0000and practitioners with the knowledge and resources to incorporate TML into\u0000their work, revealing insights often hidden from conventional machine learning\u0000methods. The tutorial code is available at\u0000https://github.com/cakcora/TopologyForML","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The topology of critical processes, III (Computing homotopy) 临界过程拓扑学，III（计算同调）

arXiv - MATH - Algebraic Topology Pub Date : 2024-09-04 DOI: arxiv-2409.02972

Marco Grandis

引用次数: 0

Topological degree as a discrete diagnostic for disentanglement, with applications to the $Δ$VAE 拓扑度作为不纠缠的离散诊断，并应用于 $Δ$VAE

arXiv - MATH - Algebraic Topology Pub Date : 2024-09-02 DOI: arxiv-2409.01303

Mahefa Ratsisetraina Ravelonanosy, Vlado Menkovski, Jacobus W. Portegies

引用次数: 0