Cycle scenarios are a significant class of contextuality scenarios, with the Clauser-Horne-Shimony-Holt (CHSH) scenario being a notable example. While binary outcome measurements in these scenarios are well understood, the generalization to arbitrary outcomes remains less explored, except in specific cases. In this work, we employ homotopical methods in the framework of simplicial distributions to characterize all contextual vertices of the non-signaling polytope corresponding to cycle scenarios with arbitrary outcomes. Additionally, our techniques utilize the bundle perspective on contextuality and the decomposition of measurement spaces. This enables us to extend beyond scenarios formed by gluing cycle scenarios and describe contextual extremal simplicial distributions in these generalized contexts.
{"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}
We study the cohomology of $G$-representation varieties and $G$-character stacks by means of a topological quantum field theory (TQFT). This TQFT is constructed as the composite of a so-called field theory and the 6-functor formalism of sheaves on topological stacks. We apply this framework to compute the cohomology of various $G$-representation varieties and $G$-character stacks of closed surfaces for $G = text{SU}(2), text{SO}(3)$ and $text{U}(2)$. This work can be seen as a categorification of earlier work, in which such a TQFT was constructed on the level of Grothendieck groups to compute the corresponding Euler characteristics.
{"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}
Building off of many recent advances in the subject by many different researchers, we describe a picture of A-equivariant chromatic homotopy theory which mirrors the now classical non-equivariant picture of Morava, Miller-Ravenel-Wilson, and Devinatz-Hopkins-Smith, where A is a finite abelian p-group. Specifically, we review the structure of the Balmer spectrum of the category of A-spectra, and the work of Hausmann-Meier connecting this to MU_A and equivariant formal group laws. Generalizing work of Bhattacharya-Guillou-Li, we introduce equivariant analogs of v_n-self maps, and generalizing work of Carrick and Balderrama, we introduce equivariant analogs of the chromatic tower, and give equivariant analogs of the smash product and chromatic convergence theorems. The equivariant monochromatic theory is also discussed. We explore computational examples of this theory in the case of A = C_2, where we connect equivariant chromatic theory with redshift phenomena in Mahowald invariants.
基于许多不同研究者在这一主题上的最新进展，我们描述了 A-等变色同调理论的图景，它反映了莫拉瓦、米勒-拉文尔-威尔逊和德维纳茨-霍普金斯-史密斯现在经典的非等变图景，其中 A 是一个有限无性 p 群。具体地说，我们回顾了A谱范畴的巴尔默谱结构，以及豪斯曼-迈尔将其与MU_A和等变形式群律联系起来的工作。根据巴塔查里亚-吉卢-李的工作，我们引入了 v_n 自映射的等变类比；根据卡里克和巴尔德拉马的工作，我们引入了色度塔的等变类比，并给出了粉碎积和色度收敛定理的等变类比。我们还讨论了等变单色理论。我们探讨了该理论在 A =C_2 情况下的计算实例，并将等变色度理论与马霍瓦尔德不变式中的红移现象联系起来。
{"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}
This is a survey on formality results relying on weight structures. A weight structure is a naturally occurring grading on certain differential graded algebras. If this weight satisfies a purity property, one can deduce formality. Algebraic geometry provides us with such weight structures as the cohomology of algebraic varieties tends to present additional structures including a Hodge structure or a Galois action.
{"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}
Let k be a field of characteristic not 2. We conjecture that if X is a quasi-projective k-variety with trivial motivic Euler characteristic, then Sym$^n$X has trivial motivic Euler characteristic for all n. Conditional on this conjecture, we show that the Grothendieck--Witt ring admits a power structure that is compatible with the motivic Euler characteristic and the power structure on the Grothendieck ring of varieties. We then discuss how these conditional results would imply an enrichment of G"ottsche's formula for the Euler characteristics of Hilbert schemes.
让 k 是一个特性不为 2 的域。我们猜想，如果 X 是具有微不足道的动机欧拉特征的类投影 k 素数，那么对于所有 n，Sym$^n$X 都具有微不足道的动机欧拉特征。在这一猜想的条件下，我们证明了格罗登第克--维特环具有与动机欧拉特征和格罗登第克素数环上的动力结构相容的动力结构。然后，我们讨论了这些条件结果将如何意味着对希尔伯特方案欧拉特征的 G"ottsche 公式的丰富。
{"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}
We establish an isomorphism between the 0-degree "uberhomology and the double homology of finite simplicial complexes, using a Mayer-Vietoris spectral sequence argument. We clarify the correspondence between these theories by providing examples and some consequences; in particular, we show that "uberhomology groups detect the standard simplex, and that the double homology's diagonal is related to the connected domination polynomial.
{"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}
The multiscale simplicial flat norm (MSFN) of a d-cycle is a family of optimal homology problems indexed by a scale parameter {lambda} >= 0. Each instance (mSFN) optimizes the total weight of a homologous d-cycle and a bounded (d + 1)-chain, with one of the components being scaled by {lambda}.We propose a min-cost flow formulation for solving instances of mSFN at a given scale {lambda} in polynomial time in the case of (d + 1)-dimensional simplicial complexes embedded in {R^(d + 1)} and homology over Z. Furthermore, we establish the weak and strong dualities for mSFN, as well as the complementary slackness conditions. Additionally, we prove optimality conditions for directed flow formulations with cohomology over Z+. Next, we propose an approach based on the multiscale flat norm, a notion of distance between objects defined in the field of geometric measure theory, to compute the distance between a pair of planar geometric networks. Using a triangulation of the domain containing the input networks, the flat norm distance between two networks at a given scale can be computed by solving a linear program. In addition, this computation automatically identifies the 2D regions (patches) that capture where the two networks are different. We demonstrate through 2D examples that the flat norm distance can capture the variations of inputs more accurately than the commonly used Hausdorff distance. As a notion of stability, we also derive upper bounds on the flat norm distance between a simple 1D curve and its perturbed version as a function of the radius of perturbation for a restricted class of perturbations. We demonstrate our approach on a set of actual power networks from a county in the USA. Our approach can be extended to validate synthetic networks created for multiple infrastructures such as transportation, communication, water, and gas networks.
d 循环的多尺度简单平面规范（MSFN）是由尺度参数 {lambda} >= 0 索引的最优同构问题族。每个实例（mSFN）优化同构 d 循环和有边（d + 1）链的总权重，其中一个分量的尺度为 {lambda} 。在嵌入{R^(d + 1)}的 (d + 1)维简单复数和 Z 上同调的情况下，我们提出了一种最小成本流公式，用于在给定规模 {lambda} 下以多项式时间求解 mSFN 的实例。此外，我们还证明了具有 Z+ 上同调的有向流公式的最优性条件。接下来，我们提出了一种基于多尺度平面规范的方法，即几何度量理论领域定义的对象间距离概念，来计算一对平面几何网络之间的距离。通过对包含输入网络的域进行三角剖分，可以通过求解线性方程来计算两个网络在给定尺度下的平面法线距离。此外，这种计算方法还能自动识别捕捉两个网络不同之处的二维区域（斑块）。作为稳定性的一个概念，我们还推导出了简单一维曲线与其扰动版本之间的平规范距离的上限，它是扰动半径对受限扰动类别的函数。我们在美国一个县的一组实际电力网络上演示了我们的方法。我们的方法可以扩展到验证为多种基础设施（如交通、通信、水和天然气网络）创建的合成网络。
{"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}
Parametrized topological complexity is a homotopy invariant that represents the degree of instability of motion planning problem that involves external constraints. We consider the parametrized topological complexity in the case of spherical fibrations over spheres. We explicitly compute a lower bound in terms of weak category and determine the parametrized topological complexity of some spherical fibrations.
{"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
With the increasing use of black-box Machine Learning (ML) techniques in critical applications, there is a growing demand for methods that can provide transparency and accountability for model predictions. As a result, a large number of local explainability methods for black-box models have been developed and popularized. However, machine learning explanations are still hard to evaluate and compare due to the high dimensionality, heterogeneous representations, varying scales, and stochastic nature of some of these methods. Topological Data Analysis (TDA) can be an effective method in this domain since it can be used to transform attributions into uniform graph representations, providing a common ground for comparison across different explanation methods. We present a novel topology-driven visual analytics tool, Mountaineer, that allows ML practitioners to interactively analyze and compare these representations by linking the topological graphs back to the original data distribution, model predictions, and feature attributions. Mountaineer facilitates rapid and iterative exploration of ML explanations, enabling experts to gain deeper insights into the explanation techniques, understand the underlying data distributions, and thus reach well-founded conclusions about model behavior. Furthermore, we demonstrate the utility of Mountaineer through two case studies using real-world data. In the first, we show how Mountaineer enabled us to compare black-box ML explanations and discern regions of and causes of disagreements between different explanations. In the second, we demonstrate how the tool can be used to compare and understand ML models themselves. Finally, we conducted interviews with three industry experts to help us evaluate our work.
随着黑盒机器学习（ML）技术在关键应用中的使用越来越多，人们对能够为模型预测提供透明度和责任感的方法的需求也越来越大。因此，大量针对黑盒模型的局部可解释性方法得到了开发和推广。然而，由于一些方法的高维性、异质性、不同尺度和随机性，机器学习解释仍然难以评估和比较。拓扑数据分析（Topological Data Analysis，TDA）是这一领域的有效方法，因为它可以用来将归因转化为统一的图表示，为不同解释方法之间的比较提供共同基础。我们介绍了一种新颖的拓扑驱动可视化分析工具 Mountaineer，它允许人工智能从业人员通过将拓扑图与原始数据分布、模型预测和特征归因联系起来，以交互方式分析和比较这些表示。登山者有助于对 ML 解释进行快速、反复的探索，使专家能够深入了解解释技术，理解基本数据分布，从而对模型行为得出有理有据的结论。此外，我们还通过两个使用真实世界数据的案例研究展示了 Mountaineer 的实用性。在第一个案例中，我们展示了登山者如何帮助我们比较黑盒子 ML 解释，并找出不同解释之间存在分歧的区域和原因。其次，我们展示了如何使用该工具来比较和理解 ML 模型本身。最后，我们对三位行业专家进行了访谈，以帮助我们评估自己的工作。
{"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}
We develop an extension of the usual theory of formal group laws where the base ring is not required to be commutative and where the formal variables need neither 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 algebraic topology in the sense that a (possibly non-commutative) complex oriented ring spectrum is canonically equipped with just such a formal group law. The universal formal group law is carried by the Baker-Richter spectrum M{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 diffeomorphisms of the non-commutative line" of Brouder, Frabetti and Krattenthaler is central to the theory developed here. In particular, we verify Morava's conjecture that there is a representation of the Drinfeld quantum-double D(B) through cohomology operations in M{xi}.
我们发展了形式群法的通常理论的一个扩展，在这个扩展中，基环不要求是交换的，形式变量既不需要是中心变量，也不需要彼此交换。我们证明，对于代数拓扑学的需要来说，这是一种自然的形式群法，因为面向复环谱（可能是非交换的）就是典型地配备了这样一种形式群法。通用形式群法由贝克-里克特谱M{/xi}承载，它在这种非交换背景下扮演着类似于MU的角色。正如莫拉瓦之前的工作所建议的，布劳德、弗拉贝蒂和克拉滕塔勒的 "非交换线的形式衍变 "的霍普夫代数 B 是本文所发展的理论的核心。特别是，我们验证了莫拉瓦的猜想，即通过 M{xi} 中的同调运算，存在德林费尔德量子偶 D(B) 的表示。
{"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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