Collaborative Research: Geometric and Analytic Properties of Discrete Groups--A Focused Research Group on the Novikov Conjecture and the Baum-Connes Conjecture

诺维科夫猜想是流形理论中尚未解决的基本问题之一。它的历史是一个迷人的旅程，通过一个显着不同的数学景观。这个猜想已经引起了广泛分离的学科之间激烈的思想交流，并且（像其他著名的未解决的问题一样）它已经产生了自己的生动的新数学。BaumConnes猜想将Novikov猜想的基本方面转移到算子代数理论中，并与表示论，自旋几何和其他领域建立了新的联系。最近，这两项工作都取得了显著进展。方法和思想涉及维数理论，顺从行动的群体，Banach空间几何和组合学发挥了重要作用。一个异常激动人心的机会出现了，在一些相当分散的领域之间激发互动。一些核心问题是如此基本，人们甚至可以期待在学生层面的重要交流。这些问题是如此广泛，普通的数学方案的小，两个或三个人，合作的努力将不会给最迅速和有效的进展。拟议计划的主要目标如下：马歇尔部队从拓扑学，分析和从几个不太明显的地区对诺维科夫和鲍姆康纳斯的防御进行总攻。 创造一种快速有效的方法，为在这一广泛领域继续研究提供必要的工具。 通过一个协调的访问计划，促进美国和外国数学家之间的交流与合作。 为研究生提供有效的培训机会，让他们接触到数学思想和专业知识的不寻常的广度。

ABSTRACTThe Novikov conjecture is one of the fundamental unsolved problems of manifold theory. Its history is a fascinating journey through a remarkably varied mathematical landscape. The conjecture has provoked vigorous exchanges of ideas between widely separated subjects, and (like other famous unsolved problems) it has generated lively new mathematics of its own. The Baum Connes conjecture transports the fundamental aspects of the Novikov conjecture to operator algebra theory, and makes new contacts with representation theory,spin geometry and other areas. Very recently, striking progress has been made on the both conjectures. Methods and ideas involving dimension theory, amenable actions of groups, Banach space geometry and combinatorics have played essential roles. An unusually exciting opportunity has arisen tospark interaction among some quite widely separated fields. Some of the core questions are so basic that one can even expect important exchanges at the student level. The issues are so broad that the ordinary mathematical scheme of small, two or three person, collaborative efforts will not give the most rapid and efficient progress. The key objectives of the proposed program are as follows: Marshall forces from topology, analysis and from several less apparent areas for a general attack on the Novikov and Baum Connes conjectures. Create a rapid and effcient means of providing the essential tools for continuing research in this broad area. Broaden the communication and cooperation between US and foreign mathematicians through a coordinated program of visits. Offer effective training opportunities for graduate students, giving them exposure to an unusual breadth of mathematical ideas and expertise.