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

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ABSTRACT The 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.