Geometry and Groups: Enumeration and Finite Representations

自然界中最丰富的对称群是由二维球体的所有旋转的集合或欧几里得平面的所有刚性运动的集合来例证的。按顺序应用两个对称性，以及将一个对称性倒置以获得另一个对称性的操作，赋予这些对称性集合以在数学上称为一个群的结构。许多其他组作为这些连续运动组的子组出现，以平面瓷砖的刚性运动组及其与欧几里得平面的整个等轴测组的关系为模型。数论产生了许多这样的例子，其结构类似于整数位于实数行内的离散的、完全分开的方式。其中一个要追求的项目试图找到一种计算上可行的方法来列出最重要的连续运动组内所有算术定义的离散运动子群。上面提到的连续运动组在数学上被称为半单李群；索菲斯·李是19世纪的数学家，他发现了这种连续变换组的基本结构事实，而半单性是大多数重要例子所共有的代数性质。计划的第一行工作将递归地枚举半单李群中的算术格，为这类群的同构问题提供解决方案，并解决Belolipetsky和Lubotzky关于算术流形的不同有限覆盖之间的等距数目的猜想。第二条线涉及流形与其有限薄片覆盖空间之间的关系，这一关系编码在流形的基本群的无限完备性中。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The richest symmetry groups in nature are exemplified by the collection of all rotations of a two-dimensional sphere or the collection of all rigid motions of the Euclidean plan. The operations of applying two symmetries in order, and of inverting a symmetry to get another one, endow these sets of symmetries with the structure known in mathematics as a group. Many other groups arise as subgroups of these groups of continuous motions, modeled on the group of rigid motions of a planar tiling and its relationship to the full group of isometries of the Euclidean plan. Number theory give rise to many of these examples, with constructions that are analogous to the discrete, widely separated way that the integers sit within the real number line. One of the projects to be pursued seeks to find a computationally feasible way to list all of the arithmetically defined discrete subgroups of motion within the most important groups of continuous motions.The continuous groups of motion referred to above are known in mathematics as semisimple Lie groups; Sophus Lie was the nineteenth-century mathematician who discovered the basic structural facts of such groups of continuous transformations, and semisimplicity is an algebraic property shared by most of the important examples. The first line of work planned will recursively enumerate arithmetic lattices in semisimple Lie groups, providing a solution to the isomorphism problem for this class of groups and addressing a conjecture of Belolipetsky and Lubotzky on the number of isometries between distinct finite covers of an arithmetic manifold. A second line of investigation concerns the relationship between a manifold and its finite-sheeted covering spaces, which is encoded in the profinite completion of the manifold's fundamental group.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.