New directions in noncommutative geometry.

我们建议在现代纯数学的一个非常活跃和令人兴奋的领域举办一个研讨会，这个领域被称为非交换几何。非交换几何的一个基本前提是关于一个空间或一个群的信息可以从一个合适的算子代数的性质中得到。对于群G，一个有趣的空间的例子是它的不可约表示的空间。这些空间通常具有非常复杂的结构，并且已经开发了各种拓扑工具来帮助它们的研究。理解群的表示论的另一种方法是通过研究相关的群C*-代数，根据非交换几何的哲学，它扮演着群的不可约表示空间的连续函数代数的角色。例如，一个群的约化C*-代数包含了群G的不可约表示的信息，这些不可约表示构成了G的左正则表示. Baum-Connes猜想提出了一个利用约化C*-代数的K-理论来提取群表示空间的拓扑信息的方案.这一假设以一种非常巧妙的方式将群的分析性质与它以规定方式作用的空间几何联系起来。在过去的二十年里，这个猜想一直是人们深入研究的主题，并产生了许多奇妙的结果。最近出现的一些最令人兴奋的见解是猜想和朗兰兹纲领之间的联系，朗兰兹纲领是一个复杂的方案，描述了某一类群的表示空间的结构。我们的研讨会将汇集领先的数学家提供一个很好的机会，交流思想，可能导致解决一些有趣的和困难的问题，在这一领域。应邀参加我们会议的发言者同时在英国出席是很罕见的。

We propose to hold a workshop in a very active and exciting area of modern pure mathematics, which is known as noncommutative geometry. A fundamental premise of noncommutative geometry is the idea that thatinformation about a space or a group can be obtained from properties of a suitable operator algebra. For a group G, an example of an interesting space to consider is a space of its irreducible representations. These spaces typically have a very complicated structure and various topological tools have been developed to assist in their study.Another way to understand the representation theory of a group is through the study of associated group C*-algebras, which according to the philosophy of noncommutative geometry, play the role of the algebras of continuous functions of spaces of irreducible representationsof the group. For example, the reduced C*-algebra of a group contains information about the irreducible representations of the group G that make up theleft regular representation of G.The Baum-Connes conjecture proposes a scheme of extracting topological information about the space of representations of a group by means of K-theory of its reduced C*-algebra. This hypothesis links in a very ingenious way analytic properties of the group with the geometry of spaces on which it acts in a prescribed way. This conjecture has been the subject of intense study over the past two decades and has produced a number of wonderful results. Some of the most exciting insights that emerged recently is the link between the conjecture and the Langlands programme, which is a sophisticated scheme that describes the structure of spaces of representations of a certain class of groups. Our workshop will bring together leading mathematicians to provide an excellent opportunity for the exchange of ideas that are likely to lead to the resolution of a number of interesting and difficult problems in this area. It is rare that the speakers invited to our meeting are present in the UK at the same time.