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

协作研究：离散群的几何性质和解析性质--诺维科夫猜想和鲍姆-康纳斯猜想重点研究组

• 批准号: 0073812
• 负责人: Shmuel Weinberger
• 金额: $ 20.8万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-09-01 至 2003-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0073812&HistoricalAwards=false
• 关键词: Collaborative; Research; Geometric; Analytic; Properties

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.

Novikov猜想是流形理论中尚未解决的基本问题之一。它的历史是一段令人着迷的旅程，穿越了一幅非常多种多样的数学景观。这一猜想激起了不同学科之间的激烈思想交流，(就像其他著名的悬而未决的问题一样)，它本身也产生了生动的新数学。Baum Connes猜想将Novikov猜想的基本方面引入了算符代数理论，并与表象理论、自旋几何等领域进行了新的接触。最近，这两个猜想都取得了惊人的进展。维论、群的从属作用、Banach空间几何和组合学等方法和思想发挥了重要作用。一个不同寻常的令人兴奋的机会已经出现，可以在一些相距甚远的领域之间引发相互作用。一些核心问题是如此基本，以至于人们甚至可以期待在学生层面进行重要的交流。这些问题是如此广泛，以至于普通的数学方案，两个或三个人，协同努力将不会给出最快速和有效的进展。拟议方案的主要目标如下：马歇尔力量从拓扑、分析和从几个不太明显的领域对诺维科夫和鲍姆·康内斯猜想进行一般攻击。创造一种快速而有效的手段，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。