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Leonard Scott教授与德国斯图加特大学数学与计算机科学学院的Klaus Roggenkamp教授合作进行一项研究。这些数学家打算研究积分有限群环的一般面积和特征为p0的半单代数群的表示理论。更具体地说，他们的目标是研究格雷厄姆·希格曼（Graham Higman）在1940年首次提出的同构问题。简单地说，同构问题问的是，“两个积分有限群环的同构何时意味着群是同构的？”PI在解决这一重要问题方面的领导作用得到了应有的认可。1984年，他们部分肯定地解决了有限幂零群的问题，最近又为相关的Zassenhaus猜想提供了一个反例。研究者现在所处的阶段是，他们已经开发出了解决这个问题所必需的技术和工具，更重要的是，他们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.