U.S.- Federal Republic of Germany Cooperative Research in Mathematics: The Isomorphism Problem for Finite Group Rings

该奖项将支持弗吉尼亚大学的伦纳德·斯科特教授与德国斯图加特大学数学和计算机科学学院的克劳斯·罗根坎普教授进行研究合作。这些数学家致力于研究积分有限群环的一般面积和特征p0上的半单代数群的表示理论。更具体地说，他们致力于解决格雷厄姆·希格曼于1940年首次提出的同构问题。简而言之，同构问题是这样一个问题：“当两个整数有限群环的同构意味着这两个群是同构的？”PI在解决这一重要问题方面的领导作用理所当然地得到了认可。1984年，他们部分地解决了有限幂零群的正定问题，最近又给出了相关的Zassenhaus猜想的反例。调查人员现在已经开发了解决问题所需的技术和工具，更重要的是，提供了一个框架来证明同构问题“通常”有一个肯定的答案。

This award will support Professor Leonard Scott of the University of Virginia in a research collaboration with Professor Klaus Roggenkamp of the Faculty of Mathematics and Computer Science at the University of Stuttgart, Germany. These mathematicians intend to work on the general area of integral finite group rings and the representation theory of semisimple algebraic groups in characteristic p 0. More specifically, they aim to work on the isomorphism problem first posed by Graham Higman in 1940. Briefly stated, the isomorphism problem asks, "when does an isomorphism of two integral finite group rings imply the groups are isomorphic?" The PI's are justly recognized for their leadership in attacking this important problem. In 1984 they partially solved the problem in the affirmative for finite nilpotent groups and recently provided a counter-example to the related Zassenhaus conjecture. The investigators are now at the stage where they have developed the techniques and tools necessary to resolve the problem and, more important, to provide a framework for proving that the isomorphism problem "usually" has a positive answer.