Relative Rapid Decay and Applications

给定一个群，可以给它联想到一个群环，这个群环是由群元素的复线性组合而成的。幂等猜想是指如果一个群没有有限阶的元素，那么群环中唯一等于其自身平方的元素应该是0或1。这是一个完全的代数命题，它可以被重新表述为对某一有限方程集只有平凡解。然而，适当补全代数群环可以使其成为一个几何对象，幂等猜想可以用几何形式重新表述。如果群是可交换的，则幂等猜想是成立的，并等价于n维环面的连通性。我建议开发与群环的算子范数补全相关的分析工具，以更好地理解更一般的群类，例如，它包括特殊的线性群SL（n，Z）。群体出现在许多科学领域，因为它们编码了物理、生物或其他系统的对称性，因此研究它们很重要。我们不可能对所有的群体都说些有用的东西，所以我们把他们分成班级，每个班级分别学习。我感兴趣的是由几何物体对称而产生的群的种类。对这些物体几何形状的理解通常可以转化为群体的代数信息。我计划从代数和分析中开发工具来研究这些几何物体，特别强调理解只有在大规模上才能显现出来的几何性质。

Given a group, one can associate to it a group ring, which is formed from complex linear combinations of group elements. The idempotent conjecture is that if a group has no elements of finite order, then the only elements in the group ring which are equal to their own square should be zero or one. This is a completely algebraic statement, which can be re-phrased by saying that there are only trivial solutions to a certain finite set of equations. However, suitably completing the algebraic group ring can turn it into a geometric object, and the idempotent conjecture can be reformulated in geometric terms. If the group is commutative, the idempotent conjecture is known to be true, and to be equivalent to the connectedness of the n-dimensional torus. I propose to develop analytical tools related to the operator norm completion of the group ring to get a better understanding of a more general class of groups, which includes, for example, the special linear group SL(n,Z).Groups arise in many scientific fields because they encode symmetries ofphysical, biological or other systems, therefore it is important to studythem. It is impossible to say anything useful about all groups, so wedivide them into classes and study each class separately. I am interestedin classes of groups that arise as symmetries of geometric objects.Understanding something about the geometry of these objects often can betranslated into algebraic information about the group. I plan to developtools from algebra and analysis to study these geometric objects, with aparticular emphasis on understanding geometric properties which onlybecome apparent on a large scale.