arXiv - MATH - Algebraic Topology最新文献_第10页

Relative nonhomogeneous Koszul duality for PROPs associated to nonaugmented operads 与非碎片操作数相关的 PROPs 的相对非均质科斯祖尔对偶性

arXiv - MATH - Algebraic Topology Pub Date : 2024-06-12 DOI: arxiv-2406.08132

Geoffrey Powell

{"title":"Relative nonhomogeneous Koszul duality for PROPs associated to nonaugmented operads","authors":"Geoffrey Powell","doi":"arxiv-2406.08132","DOIUrl":"https://doi.org/arxiv-2406.08132","url":null,"abstract":"The purpose of this paper is to show how Positselski's relative\u0000nonhomogeneous Koszul duality theory applies when studying the linear category\u0000underlying the PROP associated to a (non-augmented) operad of a certain form,\u0000in particular assuming that the reduced part of the operad is binary quadratic.\u0000In this case, the linear category has both a left augmentation and a right\u0000augmentation (corresponding to different units), using Positselski's\u0000terminology. The general theory provides two associated linear differential graded (DG)\u0000categories; indeed, in this framework, one can work entirely within the DG\u0000realm, as opposed to the curved setting required for Positselski's general\u0000theory. Moreover, DG modules over DG categories are related by adjunctions. When the reduced part of the operad is Koszul (working over a field of\u0000characteristic zero), the relative Koszul duality theory shows that there is a\u0000Koszul-type equivalence between the appropriate homotopy categories of DG\u0000modules. This gives a form of Koszul duality relationship between the above DG\u0000categories. This is illustrated by the case of the operad encoding unital, commutative\u0000associative algebras, extending the classical Koszul duality between\u0000commutative associative algebras and Lie algebras. In this case, the associated\u0000linear category is the linearization of the category of finite sets and all\u0000maps. The relative nonhomogeneous Koszul duality theory relates its derived\u0000category to the respective homotopy categories of modules over two explicit\u0000linear DG categories.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515043","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

D-GRIL: End-to-End Topological Learning with 2-parameter Persistence D-GRIL：具有双参数持久性的端到端拓扑学习

arXiv - MATH - Algebraic Topology Pub Date : 2024-06-11 DOI: arxiv-2406.07100

Soham Mukherjee, Shreyas N. Samaga, Cheng Xin, Steve Oudot, Tamal K. Dey

引用次数: 0

Nerve Models of Subdivision Bifiltrations 细分双层神经模型

arXiv - MATH - Algebraic Topology Pub Date : 2024-06-11 DOI: arxiv-2406.07679

Michael Lesnick, Ken McCabe

{"title":"Nerve Models of Subdivision Bifiltrations","authors":"Michael Lesnick, Ken McCabe","doi":"arxiv-2406.07679","DOIUrl":"https://doi.org/arxiv-2406.07679","url":null,"abstract":"We study the size of Sheehy's subdivision bifiltrations, up to homotopy. We\u0000focus in particular on the subdivision-Rips bifiltration $mathcal{SR}(X)$ of a\u0000metric space $X$, the only density-sensitive bifiltration on metric spaces\u0000known to satisfy a strong robustness property. Given a simplicial filtration\u0000$mathcal{F}$ with a total of $m$ maximal simplices across all indices, we\u0000introduce a nerve-based simplicial model for its subdivision bifiltration\u0000$mathcal{SF}$ whose $k$-skeleton has size $O(m^{k+1})$. We also show that the\u0000$0$-skeleton of any simplicial model of $mathcal{SF}$ has size at least $m$.\u0000We give several applications: For an arbitrary metric space $X$, we introduce a\u0000$sqrt{2}$-approximation to $mathcal{SR}(X)$, denoted $mathcal{J}(X)$, whose\u0000$k$-skeleton has size $O(|X|^{k+2})$. This improves on the previous best\u0000approximation bound of $sqrt{3}$, achieved by the degree-Rips bifiltration,\u0000which implies that $mathcal{J}(X)$ is more robust than degree-Rips. Moreover,\u0000we show that the approximation factor of $sqrt{2}$ is tight; in particular,\u0000there exists no exact model of $mathcal{SR}(X)$ with poly-size skeleta. On the\u0000other hand, we show that for $X$ in a fixed-dimensional Euclidean space with\u0000the $ell_p$-metric, there exists an exact model of $mathcal{SR}(X)$ with\u0000poly-size skeleta for $pin {1, infty}$, as well as a\u0000$(1+epsilon)$-approximation to $mathcal{SR}(X)$ with poly-size skeleta for\u0000any $p in (1, infty)$ and fixed ${epsilon > 0}$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515044","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

Splitting of abelian varieties in motivic stable homotopy category 动机稳定同调范畴中的无常变分

arXiv - MATH - Algebraic Topology Pub Date : 2024-06-09 DOI: arxiv-2406.05674

Haoyang Liu

引用次数: 0

The special unitary groups $SU(2n)$ as framed manifolds 作为框架流形的特殊单元群 $SU(2n)$

arXiv - MATH - Algebraic Topology Pub Date : 2024-06-08 DOI: arxiv-2406.11878

Haruo Minami

引用次数: 0

Tangent spaces of diffeological spaces and their variants 差分空间的切线空间及其变体

arXiv - MATH - Algebraic Topology Pub Date : 2024-06-07 DOI: arxiv-2406.04703

Masaki Taho

引用次数: 0

Combinatorial Complex Score-based Diffusion Modelling through Stochastic Differential Equations 通过随机微分方程建立基于复杂分数的组合扩散模型

arXiv - MATH - Algebraic Topology Pub Date : 2024-06-07 DOI: arxiv-2406.04916

Adrien Carrel

{"title":"Combinatorial Complex Score-based Diffusion Modelling through Stochastic Differential Equations","authors":"Adrien Carrel","doi":"arxiv-2406.04916","DOIUrl":"https://doi.org/arxiv-2406.04916","url":null,"abstract":"Graph structures offer a versatile framework for representing diverse\u0000patterns in nature and complex systems, applicable across domains like\u0000molecular chemistry, social networks, and transportation systems. While\u0000diffusion models have excelled in generating various objects, generating graphs\u0000remains challenging. This thesis explores the potential of score-based\u0000generative models in generating such objects through a modelization as\u0000combinatorial complexes, which are powerful topological structures that\u0000encompass higher-order relationships. In this thesis, we propose a unified framework by employing stochastic\u0000differential equations. We not only generalize the generation of complex\u0000objects such as graphs and hypergraphs, but we also unify existing generative\u0000modelling approaches such as Score Matching with Langevin dynamics and\u0000Denoising Diffusion Probabilistic Models. This innovation overcomes limitations\u0000in existing frameworks that focus solely on graph generation, opening up new\u0000possibilities in generative AI. The experiment results showed that our framework could generate these complex\u0000objects, and could also compete against state-of-the-art approaches for mere\u0000graph and molecule generation tasks.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515046","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

Ultrasolid Homotopical Algebra 超固同位代数

arXiv - MATH - Algebraic Topology Pub Date : 2024-06-06 DOI: arxiv-2406.04063

Sofía Marlasca Aparicio

引用次数: 0

On the automorphism groups of smooth Fano threefolds 论光滑法诺三围的自形群

arXiv - MATH - Algebraic Topology Pub Date : 2024-06-05 DOI: arxiv-2406.03584

Nikolay Konovalov

引用次数: 0

Homotopy similarity of maps. Maps of the circle 地图的同调相似性圆的映射

arXiv - MATH - Algebraic Topology Pub Date : 2024-06-04 DOI: arxiv-2406.02526

S. S. Podkorytov

引用次数: 0