The Baum-Connes Conjecture for Translation Algebras

翻译代数的鲍姆-康尼斯猜想

• 批准号: EP/J015806/1
• 负责人: Nicholas Wright
• 金额: $ 12.8万
• 依托单位: University of Southampton
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2013
• 资助国家: 英国
• 起止时间: 2013 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FJ015806%2F1
• 关键词: Baum; Connes; Conjecture; Translation; Algebras

The frontier between analysis and geometry is an exciting field of mathematical research where much leading-edge work is being carried out. The great insight of Baum and Connes, in the form of their celebrated conjecture, is that certain delicate analytical invariants of a group are intimately related to more tractible geometric invariants, and indeed in many cases these invariants actually coincide. The geometric and analytic invariants are referred to respectively as the LHS and the RHS of the conjecture. The Baum-Connes conjecture is central in the study of non-commutative geometry and has important applications throughout mathematics, implying numerous high profile conjectures including the Novikov Higher Signature Conjecture, the Stable Gromov-Lawson-Rosenberg Conjecture and the Kadison-Kaplansky Conjecture. The Baum-Connes philosophy also applies in the more flexible world of large-scale metric geometry, where the coarse Baum-Connes conjecture again relates analytical and geometric invariants, in this case in the world of metric spaces.There has been a significant divide between the approaches used to tackle the Baum-Connes and coarse Baum-Connes conjectures. The former is usually attacked using analytical methods (for instance Rapid Decay, Kasparov's KK-theory) while the latter is investigated using more geometric methods (finite asymptotic dimension, Yu's property A). The recent concept of partial translation algebras bridges the gap between the group and metric space worlds: one can take a group apart and study the pieces geometrically, while still retaining some of the symmetry information that the group structure provides. Decomposing a group geometrically in this way allows the RHS of the Baum-Connes conjecture (the analytical invariant) to be computed in an analogous way to the LHS, and developing this approach is the purpose of this project.The Baum-Connes conjecture for SL(3,Z) is a famous open problem, and tackling this is an ambitious aim of the project. Previous attempts to solve the problem have failed, as SL(3,Z) is not susceptible to the analytical methods mentioned above. The technology of translation structures provides a new way to examine the group C*-algebra, by computing invariants for subspaces of a group. The foundation of the project is to develop and study the ideas of Baum, Connes et al in this new framework, providing an armoury of new tools to tackle longstanding open problems which are of international interest.

分析和几何之间的边界是数学研究的一个令人兴奋的领域，许多前沿的工作正在进行。鲍姆和康纳斯的伟大见解，以他们著名的猜想的形式，是一个群体的某些微妙的分析不变量与更易追踪的几何不变量密切相关，事实上，在许多情况下，这些不变量实际上是一致的。几何和解析不变量分别称为猜想的LHS和RHS。鲍姆-康纳斯猜想是非对易几何研究的中心，在整个数学中有着重要的应用，包括诺维科夫高等签名猜想，稳定的格罗莫夫-劳森-罗森伯格猜想和凯迪森-卡普兰斯基猜想。Baum-Connes哲学也适用于更灵活的大尺度度量几何世界，其中粗糙的Baum-Connes猜想再次涉及分析和几何不变量，在这种情况下，在度量空间的世界中。在用于处理Baum-Connes和粗糙Baum-Connes猜想的方法之间存在显著的分歧。前者通常使用分析方法（例如快速衰减，卡斯帕罗夫的KK理论）进行攻击，而后者则使用更多的几何方法（有限渐近维数，Yu的性质A）进行研究。最近的部分平移代数的概念在群和度量空间世界之间架起了一座差距的桥梁：人们可以把一个群拆开，用几何学的方法研究各个部分，同时仍然保留了群结构所提供的一些对称信息。以这种方式几何地分解一个群允许以类似于LHS的方式计算Baum-Connes猜想的RHS（解析不变量），并且开发这种方法是本项目的目的。SL（3，Z）的Baum-Connes猜想是一个著名的开放问题，解决这个问题是本项目的一个雄心勃勃的目标。以前解决这个问题的尝试都失败了，因为SL（3，Z）不受上述分析方法的影响。平移结构技术通过计算群的子空间的不变量，为研究群C*-代数提供了一种新的途径。该项目的基础是在这个新的框架中发展和研究Baum，Connes等人的思想，提供一个新工具库来解决国际感兴趣的长期开放问题。

项目成果

Spaces of Graphs, Boundary Groupoids and the Coarse Baum-Connes Conjecture

• DOI: 10.1016/j.aim.2014.02.029
• 发表时间: 2012-08
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Martin Finn-Sell;N. Wright

The local spectrum of the Dirac operator for the universal cover of SL 2 ( R )

SL 2 ( R ) 通用覆盖的狄拉克算子的局部谱

• DOI: 10.1016/j.jfa.2015.10.010
• 发表时间: 2016
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Brodzki J

K-theory and exact sequences of partial translation algebras

K 理论和部分平移代数的精确序列

• DOI: 10.1016/j.aim.2014.12.023
• 发表时间: 2015
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Brodzki J

On groupoids with involutions and their cohomology

关于具有对合的群群及其上同调

• 期刊: New York Journal of Mathematics
• 影响因子: 0.6
• 作者: El-Kaïoum Moutuou (Author)