Shreya Arya, Barbara Giunti, Abigail Hickok, Lida Kanari, Sarah McGuire, Katharine Turner
{"title":"Decomposing the Persistent Homology Transform of Star-Shaped Objects","authors":"Shreya Arya, Barbara Giunti, Abigail Hickok, Lida Kanari, Sarah McGuire, Katharine Turner","doi":"arxiv-2408.14995","DOIUrl":"https://doi.org/arxiv-2408.14995","url":null,"abstract":"In this paper, we study the geometric decomposition of the degree-$0$\u0000Persistent Homology Transform (PHT) as viewed as a persistence diagram bundle.\u0000We focus on star-shaped objects as they can be segmented into smaller, simpler\u0000regions known as ``sectors''. Algebraically, we demonstrate that the degree-$0$\u0000persistence diagram of a star-shaped object in $mathbb{R}^2$ can be derived\u0000from the degree-$0$ persistence diagrams of its sectors. Using this, we then\u0000establish sufficient conditions for star-shaped objects in $mathbb{R}^2$ so\u0000that they have ``trivial geometric monodromy''. Consequently, the PHT of such a\u0000shape can be decomposed as a union of curves parameterized by $S^1$, where the\u0000curves are given by the continuous movement of each point in the persistence\u0000diagrams that are parameterized by $S^{1}$. Finally, we discuss the current\u0000challenges of generalizing these results to higher dimensions.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195576","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}
{"title":"Gromov--Hausdorff Distance for Directed Spaces","authors":"Lisbeth Fajstrup, Brittany Terese Fasy, Wenwen Li, Lydia Mezrag, Tatum Rask, Francesca Tombari, Živa Urbančič","doi":"arxiv-2408.14394","DOIUrl":"https://doi.org/arxiv-2408.14394","url":null,"abstract":"The Gromov--Hausdorff distance measures the similarity between two metric\u0000spaces by isometrically embedding them into an ambient metric space. In this\u0000work, we introduce an analogue of this distance for metric spaces endowed with\u0000directed structures. The directed Gromov--Hausdorff distance measures the\u0000distance between two new (extended) metric spaces, where the new metric, on the\u0000same underlying space, is induced from the length of the zigzag paths. This\u0000distance is then computed by isometrically embedding the directed spaces,\u0000endowed with the zigzag metric, into an ambient directed space with respect to\u0000such zigzag distance. Analogously to the standard Gromov--Hausdorff distance,\u0000we propose alternative definitions based on the distortion of d-maps and\u0000d-correspondences. Unlike the classical case, these directed distances are not\u0000equivalent.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195605","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}
{"title":"Minimal projective resolution and magnitude homology of geodetic metric spaces","authors":"Yasuhiko Asao, Shun Wakatsuki","doi":"arxiv-2408.12147","DOIUrl":"https://doi.org/arxiv-2408.12147","url":null,"abstract":"Asao-Ivanov showed that magnitude homology is a Tor functor, hence we can\u0000compute it by giving a projective resolution of a certain module. In this\u0000article, we compute magnitude homology by constructing a minimal projective\u0000resolution. As a consequence, we determine magnitude homology of geodetic\u0000metric spaces. We show that it is a free $mathbb Z$-module, and give a\u0000recursive algorithm for constructing all cycles. As a corollary, we show that a\u0000finite geodetic metric space is diagonal if and only if it contains no 4-cuts.\u0000Moreover, we give explicit computations for cycle graphs, Petersen graph,\u0000Hoffman-Singleton graph, and a missing Moore graph. It includes another\u0000approach to the computation for cycle graphs, which has been studied by\u0000Hepworth--Willerton and Gu.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"87 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195577","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}
Mirjam Cvetič, Ron Donagi, Jonathan J. Heckman, Max Hübner, Ethan Torres
{"title":"Cornering Relative Symmetry Theories","authors":"Mirjam Cvetič, Ron Donagi, Jonathan J. Heckman, Max Hübner, Ethan Torres","doi":"arxiv-2408.12600","DOIUrl":"https://doi.org/arxiv-2408.12600","url":null,"abstract":"The symmetry data of a $d$-dimensional quantum field theory (QFT) can often\u0000be captured in terms of a higher-dimensional symmetry topological field theory\u0000(SymTFT). In top down (i.e., stringy) realizations of this structure, the QFT\u0000in question is localized in a higher-dimensional bulk. In many cases of\u0000interest, however, the associated $(d+1)$-dimensional bulk is not fully gapped\u0000and one must instead consider a filtration of theories to reach a gapped bulk\u0000in $D = d+m$ dimensions. Overall, this leads us to a nested structure of\u0000relative symmetry theories which descend to coupled edge modes, with the\u0000original QFT degrees of freedom localized at a corner of this $D$-dimensional\u0000bulk system. We present a bottom up characterization of this structure and also\u0000show how it naturally arises in a number of string-based constructions of QFTs\u0000with both finite and continuous symmetries.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195579","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}
{"title":"Persistent Homology via Ellipsoids","authors":"Sara Kališnik, Bastian Rieck, Ana Žegarac","doi":"arxiv-2408.11450","DOIUrl":"https://doi.org/arxiv-2408.11450","url":null,"abstract":"Persistent homology is one of the most popular methods in Topological Data\u0000Analysis. An initial step in any analysis with persistent homology involves\u0000constructing a nested sequence of simplicial complexes, called a filtration,\u0000from a point cloud. There is an abundance of different complexes to choose\u0000from, with Rips, Alpha, and witness complexes being popular choices. In this\u0000manuscript, we build a different type of a geometrically-informed simplicial\u0000complex, called an ellipsoid complex. This complex is based on the idea that\u0000ellipsoids aligned with tangent directions better approximate the data compared\u0000to conventional (Euclidean) balls centered at sample points that are used in\u0000the construction of Rips and Alpha complexes, for instance. We use Principal\u0000Component Analysis to estimate tangent spaces directly from samples and present\u0000algorithms as well as an implementation for computing ellipsoid barcodes, i.e.,\u0000topological descriptors based on ellipsoid complexes. Furthermore, we conduct\u0000extensive experiments and compare ellipsoid barcodes with standard Rips\u0000barcodes. Our findings indicate that ellipsoid complexes are particularly\u0000effective for estimating homology of manifolds and spaces with bottlenecks from\u0000samples. In particular, the persistence intervals corresponding to a\u0000ground-truth topological feature are longer compared to the intervals obtained\u0000when using the Rips complex of the data. Furthermore, ellipsoid barcodes lead\u0000to better classification results in sparsely-sampled point clouds. Finally, we\u0000demonstrate that ellipsoid barcodes outperform Rips barcodes in classification\u0000tasks.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195580","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}
Enrique G Alvarado, Robin Belton, Kang-Ju Lee, Sourabh Palande, Sarah Percival, Emilie Purvine, Sarah Tymochko
{"title":"Any Graph is a Mapper Graph","authors":"Enrique G Alvarado, Robin Belton, Kang-Ju Lee, Sourabh Palande, Sarah Percival, Emilie Purvine, Sarah Tymochko","doi":"arxiv-2408.11180","DOIUrl":"https://doi.org/arxiv-2408.11180","url":null,"abstract":"The Mapper algorithm is a popular tool for visualization and data exploration\u0000in topological data analysis. We investigate an inverse problem for the Mapper\u0000algorithm: Given a dataset $X$ and a graph $G$, does there exist a set of\u0000Mapper parameters such that the output Mapper graph of $X$ is isomorphic to\u0000$G$? We provide constructions that affirmatively answer this question. Our\u0000results demonstrate that it is possible to engineer Mapper parameters to\u0000generate a desired graph.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"169 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195603","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}
{"title":"Distributional Lusternik-Schnirelmann category of manifolds","authors":"Ekansh Jauhari","doi":"arxiv-2408.11036","DOIUrl":"https://doi.org/arxiv-2408.11036","url":null,"abstract":"We obtain several sufficient conditions for the distributional LS-category\u0000(dcat) of closed manifolds to be maximum, i.e., equal to their classical\u0000LS-category (cat). This gives us many new computations of dcat, especially for\u0000essential manifolds and (generalized) connected sums. In the process, we also\u0000determine the dcat of closed 3-manifolds having torsion-free fundamental groups\u0000and some closed geometrically decomposable 4-manifolds. Finally, we extend some\u0000of our results to closed Alexandrov spaces and discuss their cat and dcat in\u0000dimension 3.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195597","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}
{"title":"On $(n-2)$-connected $2n$-dimensional Poincaré complexes with torsion-free homology","authors":"Xueqi Wang","doi":"arxiv-2408.09996","DOIUrl":"https://doi.org/arxiv-2408.09996","url":null,"abstract":"Let $X$ be an $(n-2)$-connected $2n$-dimensional Poincar'e complex with\u0000torsion-free homology, where $ngeq 4$. We prove that $X$ can be decomposed\u0000into a connected sum of two Poincar'e complexes: one being $(n-1)$-connected,\u0000while the other having trivial $n$th homology group. Under the additional\u0000assumption that $H_n(X)=0$ and $Sq^2:H^{n-1}(X;mathbb{Z}_2)to\u0000H^{n+1}(X;mathbb{Z}_2)$ is trivial, we can prove that $X$ can be further\u0000decomposed into connected sums of Poincar'e complexes whose $(n-1)$th homology\u0000is isomorphic to $mathbb{Z}$. As an application of this result, we classify\u0000the homotopy types of such $2$-connected $8$-dimensional Poincar'e complexes.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225158","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}
{"title":"Identities for Whitehead products and infinite sums","authors":"Jeremy Brazas","doi":"arxiv-2408.10430","DOIUrl":"https://doi.org/arxiv-2408.10430","url":null,"abstract":"Whitehead products and natural infinite sums are prominent in the higher\u0000homotopy groups of the $n$-dimensional infinite earring space $mathbb{E}_n$\u0000and other locally complicated Peano continua. In this paper, we derive general\u0000identities for how these operations interact with each other. As an\u0000application, we consider a shrinking-wedge $X$ of $(n-1)$-connected finite\u0000CW-complexes $X_1,X_2,X_3,dots$ and compute the infinite-sum closure\u0000$mathcal{W}_{2n-1}(X)$ of the set of Whitehead products $[alpha,beta]$ in\u0000$pi_{2n-1}left(Xright)$ where $alpha,betainpi_n(X)$ are represented in\u0000respective sub-wedges that meet only at the basepoint. In particular, we show\u0000that $mathcal{W}_{2n-1}(X)$ is canonically isomorphic to\u0000$prod_{j=1}^{infty}left(pi_{n}(X_j)otimes prod_{k>j}pi_n(X_k)right)$.\u0000The insight provided by this computation motivates a conjecture about the\u0000isomorphism type of the elusive groups $pi_{2n-1}(mathbb{E}_n)$, $ngeq 2$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195598","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}
{"title":"An Exposition on the Algebra and Computation of Persistent Homology","authors":"Jason Ranoa","doi":"arxiv-2408.07899","DOIUrl":"https://doi.org/arxiv-2408.07899","url":null,"abstract":"We discuss the algebra behind the matrix reduction algorithm for persistent\u0000homology, as presented in the paper ''Computing Persistent Homology'' by Afra\u0000Zomorodian and Gunnar Carlsson, in the lens of the more modern characterization\u0000of persistence modules as functors from a poset category to a category of\u0000vector spaces over a field adopted by authors such as Peter Bubenik, Frederik\u0000Chazal, and Ulrich Bauer.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227907","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}
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