We compute the Balmer spectrum of a certain tensor triangulated category of comodules over the mod 2 dual Steenrod algebra. This computation effectively classifies the thick subcategories, resolving a conjecture of Palmieri.
{"title":"Tensor triangular geometry of modules over the mod 2 Steenrod algebra","authors":"Collin Litterell","doi":"arxiv-2409.10731","DOIUrl":"https://doi.org/arxiv-2409.10731","url":null,"abstract":"We compute the Balmer spectrum of a certain tensor triangulated category of\u0000comodules over the mod 2 dual Steenrod algebra. This computation effectively\u0000classifies the thick subcategories, resolving a conjecture of Palmieri.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258517","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}
We generalize the classical operad pair theory to a new model for $E_infty$ ring spaces, which we call ring operad theory, and establish a connection with the classical operad pair theory, allowing the classical multiplicative infinite loop machine to be applied to algebras over any $E_infty$ ring operad. As an application, we show that classifying spaces of symmetric bimonoidal categories are directly homeomorphic to certain $E_infty$ ring spaces in the ring operad sense. Consequently, this provides an alternative construction from symmetric bimonoidal categories to classical $E_infty$ ring spaces. We also present a comparison between this construction and the classical approach.
{"title":"Ring operads and symmetric bimonoidal categories","authors":"Kailin Pan","doi":"arxiv-2409.09664","DOIUrl":"https://doi.org/arxiv-2409.09664","url":null,"abstract":"We generalize the classical operad pair theory to a new model for $E_infty$\u0000ring spaces, which we call ring operad theory, and establish a connection with\u0000the classical operad pair theory, allowing the classical multiplicative\u0000infinite loop machine to be applied to algebras over any $E_infty$ ring\u0000operad. As an application, we show that classifying spaces of symmetric\u0000bimonoidal categories are directly homeomorphic to certain $E_infty$ ring\u0000spaces in the ring operad sense. Consequently, this provides an alternative\u0000construction from symmetric bimonoidal categories to classical $E_infty$ ring\u0000spaces. We also present a comparison between this construction and the\u0000classical approach.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258518","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}
Hyperuniformity refers to the suppression of density fluctuations at large scales. Typical for ordered systems, this property also emerges in several disordered physical and biological systems, where it is particularly relevant to understand mechanisms of pattern formation and to exploit peculiar attributes, e.g., interaction with light and transport phenomena. While hyperuniformity is a global property, it has been shown in [Phys. Rev. Research 6, 023107 (2024)] that global hyperuniform characteristics systematically correlate with topological properties representative of local arrangements. In this work, building on this information, we explore and assess the inverse relationship between hyperuniformity and local structures in point distributions as described by persistent homology. Standard machine learning algorithms trained on persistence diagrams are shown to detect hyperuniformity with high accuracy. Therefore, we demonstrate that the information on patterns' local structure allows for inferring hyperuniformity. Then, addressing more quantitative aspects, we show that parameters defining hyperuniformity globally, for instance entering the structure factor, can be reconstructed by comparing persistence diagrams of targeted patterns with reference ones. We also explore the generation of patterns entailing given topological properties. The results of this study pave the way for advanced analysis of hyperuniform patterns including local information, and introduce basic concepts for their inverse design.
{"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}
By constructing concrete complex-oriented maps we show that the eight-fold of the generator of the third integral cohomology of the spin groups Spin(7) and Spin(8) is in the image of the Thom morphism from complex cobordism to singular cohomology, while the generator itself is not in the image. We thereby give a geometric construction for a nontrivial class in the kernel of the differential Thom morphism of Hopkins and Singer for the Lie groups Spin(7) and Spin(8). The construction exploits the special symmetries of the octonions.
{"title":"Geometric representation of cohomology classes for the Lie groups Spin(7) and Spin(8)","authors":"Eiolf Kaspersen, Gereon Quick","doi":"arxiv-2409.06491","DOIUrl":"https://doi.org/arxiv-2409.06491","url":null,"abstract":"By constructing concrete complex-oriented maps we show that the eight-fold of\u0000the generator of the third integral cohomology of the spin groups Spin(7) and\u0000Spin(8) is in the image of the Thom morphism from complex cobordism to singular\u0000cohomology, while the generator itself is not in the image. We thereby give a\u0000geometric construction for a nontrivial class in the kernel of the differential\u0000Thom morphism of Hopkins and Singer for the Lie groups Spin(7) and Spin(8). The\u0000construction exploits the special symmetries of the octonions.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"178 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195565","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}
Effective homology techniques allow us to compute homology groups of a wide family of topological spaces. By the Whitehead tower method, this can also be used to compute higher homotopy groups. However, some of these techniques (in particular, the Whitehead tower) rely on the assumption that the starting space is simply connected. For some applications, this problem could be circumvented by replacing the space by its universal cover, which is a simply connected space that shares the higher homotopy groups of the initial space. In this paper, we formalize a simplicial construction for the universal cover, and represent it as a twisted cartesian product. As we show with some examples, the universal cover of a space with effective homology does not necessarily have effective homology in general. We show two independent sufficient conditions that can ensure it: one is based on a nilpotency property of the fundamental group, and the other one on discrete vector fields. Some examples showing our implementation of these constructions in both sagemath and kenzo are shown, together with an approach to compute the homology of the universal cover when the group is abelian even in some cases where there is no effective homology, using the twisted homology of the space.
{"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}
Motivated by the loop space cohomology we construct the secondary operations on the cohomology $H^*(X; mathbb{Z}_p)$ to be a Hopf algebra for a simply connected space $X.$ The loop space cohomology ring $H^*(Omega X; mathbb{Z}_p)$ is calculated in terms of generators and relations. This answers to A. Borel's decomposition of a Hopf algebra into a tensor product of the monogenic ones in which the heights of generators are determined by means of the action of the primary and secondary cohomology operations on $H^*(X;mathbb{Z}_p).$ An application for calculating of the loop space cohomology of the exceptional group $F_4$ is given.
在循环空间同调的激励下，我们构建了同调$H^*(X; mathbb{Z}_p)$的二次运算，使之成为简单连接空间$X的霍普夫代数。这回答了 A. Borel 将霍普夫代数分解为它们的张量乘积的问题，在张量乘积中，生成器的高度是通过对$H^*(X;mathbb{Z}_p)$ 的一级和二级同调运算的作用来确定的。
{"title":"Secondary cohomology operations and the loop space cohomology","authors":"Samson Saneblidze","doi":"arxiv-2409.04861","DOIUrl":"https://doi.org/arxiv-2409.04861","url":null,"abstract":"Motivated by the loop space cohomology we construct the secondary operations\u0000on the cohomology $H^*(X; mathbb{Z}_p)$ to be a Hopf algebra for a simply\u0000connected space $X.$ The loop space cohomology ring $H^*(Omega X;\u0000mathbb{Z}_p)$ is calculated in terms of generators and relations. This answers\u0000to A. Borel's decomposition of a Hopf algebra into a tensor product of the\u0000monogenic ones in which the heights of generators are determined by means of\u0000the action of the primary and secondary cohomology operations on\u0000$H^*(X;mathbb{Z}_p).$ An application for calculating of the loop space\u0000cohomology of the exceptional group $F_4$ is given.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195563","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}
In this article, we show that each two metric fibrations with a common base and a common fiber have isomorphic magnitude homology, and even more, the same magnitude homotopy type. That can be considered as a generalization of a fact proved by T. Leinster that the magnitude of a metric fibration with finitely many points is a product of those of the base and the fiber. We also show that the definition of the magnitude homotopy type due to the second and the third authors is equivalent to the geometric realization of Hepworth and Willerton's pointed simplicial set.
在本文中，我们证明了具有共同基点和共同纤维的两个度量纤度具有同构的幅同调，甚至具有相同的幅同调类型。这可以看作是 T. Leinster 所证明的一个事实的一般化，即具有有限多点的度量纤度的幅是基点和纤维的幅的乘积。我们还证明了第二位和第三位作者关于幅同调类型的定义等同于赫普沃思和威勒顿的点简集的几何实现。
{"title":"Magnitude homology and homotopy type of metric fibrations","authors":"Yasuhiko Asao, Yu Tajima, Masahiko Yoshinaga","doi":"arxiv-2409.03278","DOIUrl":"https://doi.org/arxiv-2409.03278","url":null,"abstract":"In this article, we show that each two metric fibrations with a common base\u0000and a common fiber have isomorphic magnitude homology, and even more, the same\u0000magnitude homotopy type. That can be considered as a generalization of a fact\u0000proved by T. Leinster that the magnitude of a metric fibration with finitely\u0000many points is a product of those of the base and the fiber. We also show that\u0000the definition of the magnitude homotopy type due to the second and the third\u0000authors is equivalent to the geometric realization of Hepworth and Willerton's\u0000pointed simplicial set.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"87 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195569","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}
Topological Machine Learning (TML) is an emerging field that leverages techniques from algebraic topology to analyze complex data structures in ways that traditional machine learning methods may not capture. This tutorial provides a comprehensive introduction to two key TML techniques, persistent homology and the Mapper algorithm, with an emphasis on practical applications. Persistent homology captures multi-scale topological features such as clusters, loops, and voids, while the Mapper algorithm creates an interpretable graph summarizing high-dimensional data. To enhance accessibility, we adopt a data-centric approach, enabling readers to gain hands-on experience applying these techniques to relevant tasks. We provide step-by-step explanations, implementations, hands-on examples, and case studies to demonstrate how these tools can be applied to real-world problems. The goal is to equip researchers and practitioners with the knowledge and resources to incorporate TML into their work, revealing insights often hidden from conventional machine learning methods. The tutorial code is available at https://github.com/cakcora/TopologyForML
{"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}
Directed Algebraic Topology studies spaces equipped with a form of direction, to include models of non-reversible processes. In the present extension we also want to cover critical processes, indecomposable and unstoppable. The previous parts of this series introduced controlled spaces and their fundamental category. Here we study how to compute the latter. The homotopy structure of these spaces will be examined in Part IV.
{"title":"The topology of critical processes, III (Computing homotopy)","authors":"Marco Grandis","doi":"arxiv-2409.02972","DOIUrl":"https://doi.org/arxiv-2409.02972","url":null,"abstract":"Directed Algebraic Topology studies spaces equipped with a form of direction,\u0000to include models of non-reversible processes. In the present extension we also\u0000want to cover critical processes, indecomposable and unstoppable. The previous parts of this series introduced controlled spaces and their\u0000fundamental category. Here we study how to compute the latter. The homotopy\u0000structure of these spaces will be examined in Part IV.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195564","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}
Mahefa Ratsisetraina Ravelonanosy, Vlado Menkovski, Jacobus W. Portegies
We investigate the ability of Diffusion Variational Autoencoder ($Delta$VAE) with unit sphere $mathcal{S}^2$ as latent space to capture topological and geometrical structure and disentangle latent factors in datasets. For this, we introduce a new diagnostic of disentanglement: namely the topological degree of the encoder, which is a map from the data manifold to the latent space. By using tools from homology theory, we derive and implement an algorithm that computes this degree. We use the algorithm to compute the degree of the encoder of models that result from the training procedure. Our experimental results show that the $Delta$VAE achieves relatively small LSBD scores, and that regardless of the degree after initialization, the degree of the encoder after training becomes $-1$ or $+1$, which implies that the resulting encoder is at least homotopic to a homeomorphism.
{"title":"Topological degree as a discrete diagnostic for disentanglement, with applications to the $Δ$VAE","authors":"Mahefa Ratsisetraina Ravelonanosy, Vlado Menkovski, Jacobus W. Portegies","doi":"arxiv-2409.01303","DOIUrl":"https://doi.org/arxiv-2409.01303","url":null,"abstract":"We investigate the ability of Diffusion Variational Autoencoder ($Delta$VAE)\u0000with unit sphere $mathcal{S}^2$ as latent space to capture topological and\u0000geometrical structure and disentangle latent factors in datasets. For this, we\u0000introduce a new diagnostic of disentanglement: namely the topological degree of\u0000the encoder, which is a map from the data manifold to the latent space. By\u0000using tools from homology theory, we derive and implement an algorithm that\u0000computes this degree. We use the algorithm to compute the degree of the encoder\u0000of models that result from the training procedure. Our experimental results\u0000show that the $Delta$VAE achieves relatively small LSBD scores, and that\u0000regardless of the degree after initialization, the degree of the encoder after\u0000training becomes $-1$ or $+1$, which implies that the resulting encoder is at\u0000least homotopic to a homeomorphism.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195568","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：