The Baum-Connes Conjecture for Translation Algebras

分析和几何之间的前沿是数学研究中一个令人兴奋的领域，许多前沿工作正在进行。鲍姆和康尼斯以他们著名的猜想的形式提出的伟大见解是，群的某些微妙的解析不变量与更易牵引的几何不变量密切相关，实际上在许多情况下，这些不变量实际上是重合的。几何不变量和解析不变量分别称为猜想的LHS和RHS。Baum-Connes猜想是研究非交换几何的核心，在整个数学中有着重要的应用，它隐含着许多备受瞩目的猜想，包括Novikov高签名猜想、稳定的Gromov-Lawson-Rosenberg猜想和Kadison-Kaplansky猜想。Baum-Connes哲学也适用于更灵活的大规模度量几何世界，在那里粗略的Baum-Connes猜想再次将解析和几何不变量联系在一起，在这种情况下，在度量空间的世界中。在用于处理Baum-Connes猜想和粗略Baum-Connes猜想的方法之间存在着显著的分歧。前者通常使用分析方法(例如快速衰减法、Kasparov的KK理论)来攻击，而后者则使用更多的几何方法(有限渐近维度，Yu‘s性质A)来研究。最近提出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桥梁：人们可以将一个群拆分，并从几何上研究这些碎片，同时仍然保留了群结构提供的一些对称信息。用这种方法对群进行几何分解，使得Baum-Connes猜想(解析不变量)的RHS可以用类似于LHS的方法来计算，而发展这种方法就是本项目的目的。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方法。该项目的基础是在这个新的框架中开发和研究Baum、Connes等人的思想，提供一系列新的工具来解决国际上感兴趣的长期悬而未决的问题。

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.