{"title":"Extremal simplicial distributions on cycle scenarios with arbitrary outcomes","authors":"Aziz Kharoof, Cihan Okay, Selman Ipek","doi":"arxiv-2406.19961","DOIUrl":"https://doi.org/arxiv-2406.19961","url":null,"abstract":"Cycle scenarios are a significant class of contextuality scenarios, with the\u0000Clauser-Horne-Shimony-Holt (CHSH) scenario being a notable example. While\u0000binary outcome measurements in these scenarios are well understood, the\u0000generalization to arbitrary outcomes remains less explored, except in specific\u0000cases. In this work, we employ homotopical methods in the framework of\u0000simplicial distributions to characterize all contextual vertices of the\u0000non-signaling polytope corresponding to cycle scenarios with arbitrary\u0000outcomes. Additionally, our techniques utilize the bundle perspective on\u0000contextuality and the decomposition of measurement spaces. This enables us to\u0000extend beyond scenarios formed by gluing cycle scenarios and describe\u0000contextual extremal simplicial distributions in these generalized contexts.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515028","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":"Cohomology of character stacks via TQFTs","authors":"Jesse Vogel","doi":"arxiv-2406.19857","DOIUrl":"https://doi.org/arxiv-2406.19857","url":null,"abstract":"We study the cohomology of $G$-representation varieties and $G$-character\u0000stacks by means of a topological quantum field theory (TQFT). This TQFT is\u0000constructed as the composite of a so-called field theory and the 6-functor\u0000formalism of sheaves on topological stacks. We apply this framework to compute\u0000the cohomology of various $G$-representation varieties and $G$-character stacks\u0000of closed surfaces for $G = text{SU}(2), text{SO}(3)$ and $text{U}(2)$. This\u0000work can be seen as a categorification of earlier work, in which such a TQFT\u0000was constructed on the level of Grothendieck groups to compute the\u0000corresponding Euler characteristics.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515031","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":"Periodic phenomena in equivariant stable homotopy theory","authors":"Mark Behrens, Jack Carlisle","doi":"arxiv-2406.19352","DOIUrl":"https://doi.org/arxiv-2406.19352","url":null,"abstract":"Building off of many recent advances in the subject by many different\u0000researchers, we describe a picture of A-equivariant chromatic homotopy theory\u0000which mirrors the now classical non-equivariant picture of Morava,\u0000Miller-Ravenel-Wilson, and Devinatz-Hopkins-Smith, where A is a finite abelian\u0000p-group. Specifically, we review the structure of the Balmer spectrum of the\u0000category of A-spectra, and the work of Hausmann-Meier connecting this to MU_A\u0000and equivariant formal group laws. Generalizing work of\u0000Bhattacharya-Guillou-Li, we introduce equivariant analogs of v_n-self maps, and\u0000generalizing work of Carrick and Balderrama, we introduce equivariant analogs\u0000of the chromatic tower, and give equivariant analogs of the smash product and\u0000chromatic convergence theorems. The equivariant monochromatic theory is also\u0000discussed. We explore computational examples of this theory in the case of A =\u0000C_2, where we connect equivariant chromatic theory with redshift phenomena in\u0000Mahowald invariants.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508002","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":"Weight structures and formality","authors":"Coline Emprin, Geoffroy Horel","doi":"arxiv-2406.19142","DOIUrl":"https://doi.org/arxiv-2406.19142","url":null,"abstract":"This is a survey on formality results relying on weight structures. A weight\u0000structure is a naturally occurring grading on certain differential graded\u0000algebras. If this weight satisfies a purity property, one can deduce formality.\u0000Algebraic geometry provides us with such weight structures as the cohomology of\u0000algebraic varieties tends to present additional structures including a Hodge\u0000structure or a Galois action.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515032","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":"Symmetric powers of null motivic Euler characteristic","authors":"Dori Bejleri, Stephen McKean","doi":"arxiv-2406.19506","DOIUrl":"https://doi.org/arxiv-2406.19506","url":null,"abstract":"Let k be a field of characteristic not 2. We conjecture that if X is a\u0000quasi-projective k-variety with trivial motivic Euler characteristic, then\u0000Sym$^n$X has trivial motivic Euler characteristic for all n. Conditional on\u0000this conjecture, we show that the Grothendieck--Witt ring admits a power\u0000structure that is compatible with the motivic Euler characteristic and the\u0000power structure on the Grothendieck ring of varieties. We then discuss how\u0000these conditional results would imply an enrichment of G\"ottsche's formula for\u0000the Euler characteristics of Hilbert schemes.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515029","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":"Bridging between überhomology and double homology","authors":"Luigi Caputi, Daniele Celoria, Carlo Collari","doi":"arxiv-2406.18778","DOIUrl":"https://doi.org/arxiv-2406.18778","url":null,"abstract":"We establish an isomorphism between the 0-degree \"uberhomology and the\u0000double homology of finite simplicial complexes, using a Mayer-Vietoris spectral\u0000sequence argument. We clarify the correspondence between these theories by\u0000providing examples and some consequences; in particular, we show that\u0000\"uberhomology groups detect the standard simplex, and that the double\u0000homology's diagonal is related to the connected domination polynomial.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515033","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":"Efficient algorithms for optimal homology problems and their applications","authors":"Kostiantyn Lyman","doi":"arxiv-2406.19422","DOIUrl":"https://doi.org/arxiv-2406.19422","url":null,"abstract":"The multiscale simplicial flat norm (MSFN) of a d-cycle is a family of\u0000optimal homology problems indexed by a scale parameter {lambda} >= 0. Each\u0000instance (mSFN) optimizes the total weight of a homologous d-cycle and a\u0000bounded (d + 1)-chain, with one of the components being scaled by {lambda}.We\u0000propose a min-cost flow formulation for solving instances of mSFN at a given\u0000scale {lambda} in polynomial time in the case of (d + 1)-dimensional\u0000simplicial complexes embedded in {R^(d + 1)} and homology over Z. Furthermore,\u0000we establish the weak and strong dualities for mSFN, as well as the\u0000complementary slackness conditions. Additionally, we prove optimality\u0000conditions for directed flow formulations with cohomology over Z+. Next, we propose an approach based on the multiscale flat norm, a notion of\u0000distance between objects defined in the field of geometric measure theory, to\u0000compute the distance between a pair of planar geometric networks. Using a\u0000triangulation of the domain containing the input networks, the flat norm\u0000distance between two networks at a given scale can be computed by solving a\u0000linear program. In addition, this computation automatically identifies the 2D\u0000regions (patches) that capture where the two networks are different. We\u0000demonstrate through 2D examples that the flat norm distance can capture the\u0000variations of inputs more accurately than the commonly used Hausdorff distance.\u0000As a notion of stability, we also derive upper bounds on the flat norm distance\u0000between a simple 1D curve and its perturbed version as a function of the radius\u0000of perturbation for a restricted class of perturbations. We demonstrate our\u0000approach on a set of actual power networks from a county in the USA. Our\u0000approach can be extended to validate synthetic networks created for multiple\u0000infrastructures such as transportation, communication, water, and gas networks.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515030","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":"Parametrized topological complexity of spherical fibrations over spheres","authors":"Yuki Minowa","doi":"arxiv-2406.17227","DOIUrl":"https://doi.org/arxiv-2406.17227","url":null,"abstract":"Parametrized topological complexity is a homotopy invariant that represents\u0000the degree of instability of motion planning problem that involves external\u0000constraints. We consider the parametrized topological complexity in the case of\u0000spherical fibrations over spheres. We explicitly compute a lower bound in terms\u0000of weak category and determine the parametrized topological complexity of some\u0000spherical fibrations.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"88 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508003","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}
Parikshit Solunke, Vitoria Guardieiro, Joao Rulff, Peter Xenopoulos, Gromit Yeuk-Yin Chan, Brian Barr, Luis Gustavo Nonato, Claudio Silva
{"title":"MOUNTAINEER: Topology-Driven Visual Analytics for Comparing Local Explanations","authors":"Parikshit Solunke, Vitoria Guardieiro, Joao Rulff, Peter Xenopoulos, Gromit Yeuk-Yin Chan, Brian Barr, Luis Gustavo Nonato, Claudio Silva","doi":"arxiv-2406.15613","DOIUrl":"https://doi.org/arxiv-2406.15613","url":null,"abstract":"With the increasing use of black-box Machine Learning (ML) techniques in\u0000critical applications, there is a growing demand for methods that can provide\u0000transparency and accountability for model predictions. As a result, a large\u0000number of local explainability methods for black-box models have been developed\u0000and popularized. However, machine learning explanations are still hard to\u0000evaluate and compare due to the high dimensionality, heterogeneous\u0000representations, varying scales, and stochastic nature of some of these\u0000methods. Topological Data Analysis (TDA) can be an effective method in this\u0000domain since it can be used to transform attributions into uniform graph\u0000representations, providing a common ground for comparison across different\u0000explanation methods. We present a novel topology-driven visual analytics tool, Mountaineer, that\u0000allows ML practitioners to interactively analyze and compare these\u0000representations by linking the topological graphs back to the original data\u0000distribution, model predictions, and feature attributions. Mountaineer\u0000facilitates rapid and iterative exploration of ML explanations, enabling\u0000experts to gain deeper insights into the explanation techniques, understand the\u0000underlying data distributions, and thus reach well-founded conclusions about\u0000model behavior. Furthermore, we demonstrate the utility of Mountaineer through\u0000two case studies using real-world data. In the first, we show how Mountaineer\u0000enabled us to compare black-box ML explanations and discern regions of and\u0000causes of disagreements between different explanations. In the second, we\u0000demonstrate how the tool can be used to compare and understand ML models\u0000themselves. Finally, we conducted interviews with three industry experts to\u0000help us evaluate our work.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"164 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508004","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":"Formal groups over non-commutative rings","authors":"Christian Nassau","doi":"arxiv-2406.14247","DOIUrl":"https://doi.org/arxiv-2406.14247","url":null,"abstract":"We develop an extension of the usual theory of formal group laws where the\u0000base ring is not required to be commutative and where the formal variables need\u0000neither be central nor have to commute with each other. We show that this is the natural kind of formal group law for the needs of\u0000algebraic topology in the sense that a (possibly non-commutative) complex\u0000oriented ring spectrum is canonically equipped with just such a formal group\u0000law. The universal formal group law is carried by the Baker-Richter spectrum\u0000M{xi} which plays a role analogous to MU in this non-commutative context. As suggested by previous work of Morava the Hopf algebra B of \"formal\u0000diffeomorphisms of the non-commutative line\" of Brouder, Frabetti and\u0000Krattenthaler is central to the theory developed here. In particular, we verify\u0000Morava's conjecture that there is a representation of the Drinfeld\u0000quantum-double D(B) through cohomology operations in M{xi}.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"89 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508005","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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