Relative Rapid Decay and Applications

给定一个群，人们可以将它与群环相关联，群环由群元素的复杂线性组合形成。幂等猜想是，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零或一。这是一个完全代数的陈述，可以换句话说，对于某个有限的方程组，只有平凡的解。然而，适当地完成代数群环可以把它变成一个几何对象，幂等猜想可以在几何术语中重新表述。如果群是可交换的，则幂等猜想为真，并且等价于n维环面的连通性。我建议开发与群环的算子范数完备化相关的分析工具，以便更好地理解更一般的群类，例如，特殊的线性群SL（n，Z）。群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。不可能对所有的群体都说些有用的话，所以我们把他们分成几个班，分别研究每一个班。我对几何对象的对称性所产生的群的分类很感兴趣，对这些对象的几何学的理解通常可以转化为关于群的代数信息。我计划从代数和分析中开发工具来研究这些几何对象，特别强调理解只有在大规模上才变得明显的几何特性。

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.