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

摘要诺维科夫猜想是流形理论中尚未解决的基本问题之一。它的历史是一段引人入胜的旅程，穿越了一个非常不同的数学景观。这个猜想激起了不同学科之间激烈的思想交流，并且（像其他著名的未解决问题一样）产生了它自己的活泼的新数学。Baum cones猜想将Novikov猜想的基本方面引入算子代数理论，并与表示理论、自旋几何等领域建立了新的联系。最近，这两个猜想都取得了惊人的进展。涉及维数理论、群的可服从作用、巴拿赫空间几何和组合学的方法和思想发挥了重要作用。一个不同寻常的激动人心的机会出现了，它激发了一些相当分散的领域之间的相互作用。一些核心问题非常基本，甚至可以期待在学生层面进行重要的交流。问题是如此广泛，以至于普通的小数学方案，两三个人，协作的努力不会给最快速和有效的进展。该计划的主要目标如下：从拓扑学、分析和几个不太明显的领域对诺维科夫和鲍姆·康内斯猜想进行总攻击的马歇尔部队。创造一种快速有效的手段，为这一广泛领域的持续研究提供必要的工具。通过协调的访问计划，扩大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，让他们接触到不同寻常的数学思想和专业知识。

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.